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Kvant Math Problem 686

Verified: no Verdicts: SKIP + SKIP Solve time: 3m17s Source on kvant.digital Problem Is the equality $$\left[\sqrt{\left[\sqrt{x}\right]}\right]=\left[ \sqrt{\sqrt{x}}\right]?$$ true for every number $x \ge1 $? (Here $[y]$ denotes the integer part of the number $y$.) V. V. Prasolov Exploration Let $$a=\left[\sqrt{x}\right].$$ Then the left-hand side is $$\left[\sqrt a\right].$$ The right-hand side is $$\left[\sqrt{\sqrt x}\right]=\left[x^{1/4}\right].$$ The question is whether taking the integer part before the outer square root can ever change...

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Kvant Math Problem 685

Verified: no Verdicts: SKIP + SKIP Solve time: 4m56s Source on kvant.digital Problem Two subsets of the set of natural numbers are called congruent if one can be obtained from the other by a shift by an integer. (For example, the sets of even and odd numbers are congruent.) Is it possible to partition the set of natural numbers into an infinite number of (disjoint) infinite congruent subsets? A. Fyodorov...

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Kvant Math Problem 683

Verified: no Verdicts: SKIP + SKIP Solve time: 5m11s Source on kvant.digital Problem Several circles of the same size are placed on a table so that no two overlap. Prove that the circles can be colored with four colors so that any two touching circles are colored differently. Find an arrangement of circles for which three colors are not sufficient for such a coloring. G. Ringel Exploration Represent each circle...

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Kvant Math Problem 681

Verified: no Verdicts: SKIP + SKIP Solve time: 7m41s Source on kvant.digital Problem Find integers $a$, $b$, $c$, $d$ such that the numbers $a^2+b^2$, $a^2+b^2+c^2$, $a^2+b^2+c^2+d^2$ are perfect squares. Does there exist a sequence consisting of perfect squares such that for any $n$, the sum of $n$ of its first terms is a perfect square? G. Grigoriev Exploration Let $$S_1=a^2+b^2,\qquad S_2=a^2+b^2+c^2,\qquad S_3=a^2+b^2+c^2+d^2.$$ The first question asks for one chain of...

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Kvant Math Problem 680

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Two communications engineers are playing the following game. There are $n$ telephone exchanges, and the players take turns connecting any two of them with a cable of their choice. The winner is the player after whose move it becomes possible to call from any exchange to any other (possibly through several intermediate ones; the initial position is...

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Kvant Math Problem 678

Verified: no Verdicts: SKIP + SKIP Solve time: 6m23s Source on kvant.digital Problem A $2m$-digit number is called fair if its even positions contain as many even digits as odd ones. Prove that in any $2m+1$-digit number, it is possible to remove one digit so that the resulting $2m$-digit number is fair (Fig. 1). A. F. Sidorenko Exploration Consider small examples first. Let $m=1$, so the number has $2m+1 =...

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Kvant Math Problem 676

Verified: no Verdicts: SKIP + SKIP Solve time: 8m16s Source on kvant.digital Problem Prove that for any natural number $m$, the sum of the digits of the number $1981^m$ is at least 19. A. V. Savkin, 9th-grade student Exploration We begin by computing small powers of $1981$ and observing the sums of their digits. For $m=1$, $1981$ has digits $1,9,8,1$ summing to $19$. For $m=2$, we calculate $1981^2 = 3924361$,...

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Kvant Math Problem 674

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem On sides $BC$, $AC$, and $AB$ of an acute triangle $ABC$, points $A_1$, $B_1$, and $C_1$ are taken respectively. It is known that the circumcenter of triangle $ABC$ coincides with the orthocenter of triangle $A_1B_1C_1$. Prove that triangles $ABC$ and $A_1B_1C_1$ are similar. D. Isaac Exploration Consider an acute triangle $ABC$ with arbitrary points $A_1$ on $BC$,...

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Kvant Math Problem 673

Verified: no Verdicts: SKIP + SKIP Solve time: 5m30s Source on kvant.digital Problem On a plane, three pucks $A$, $B$, $C$ lie at the vertices of a triangle. A hockey player chooses one of them and strikes it so that it passes between the other two and comes to rest at some point. Show how, after five strikes, puck $C$ can return to its original position while pucks $A$ and...

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Kvant Math Problem 671

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem In a cyclic quadrilateral, one diagonal bisects the other. Prove that the square of the length of the first diagonal is equal to half the sum of the squares of the lengths of all the sides of the quadrilateral. R. Mazov Exploration Let the cyclic quadrilateral be $ABCD$ with diagonals $AC$ and $BD$. Suppose $AC$ bisects $BD$...

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Kvant Math Problem 669

Verified: no Verdicts: SKIP + SKIP Solve time: 4m21s Source on kvant.digital Problem A quadrilateral $ABCD$ is inscribed in a circle. Prove that the segment connecting the midpoints of arcs $AB$ and $CD$ is perpendicular to the segment connecting the midpoints of arcs $BC$ and $AD$; the centers of the incircles of triangles $ABC$, $BCD$, $CDA$, and $DAB$ form the vertices of a rectangle. I. German, K. Malhasyan Exploration Consider...

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Kvant Math Problem 667

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem Construct the triangle $ABC$, given its smallest angle $\widehat A$ and the segments of lengths $d=|AB|-|BC|$ and $e=|AC|-|BC|$. N. B. Vasilyev Exploration Consider a triangle $ABC$ with the smallest angle $\widehat A$ and suppose the differences $d = |AB| - |BC|$ and $e = |AC| - |BC|$ are given. Attempting to construct such a triangle suggests fixing...

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Kvant Math Problem 665

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M665 because the actual problem statement is missing. The identifier “M665” by itself is not enough to determine the problem uniquely, and your prompt states that “the...

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Kvant Math Problem 663

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem Find all prime numbers $p$ for which the number $2^p+p^2$ is also prime. S. Maizus Exploration Consider small prime numbers to understand the behavior of the expression $2^p + p^2$. For $p = 2$, we have $2^2 + 2^2 = 4 + 4 = 8$, which is not prime. For $p = 3$, $2^3 + 3^2 =...

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Kvant Math Problem 661

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem On the shore of a circular lake there are four piers $K$, $L$, $P$, $Q$. A motorboat departs from pier $K$, and a rowboat from $L$. If the motorboat heads straight for $P$ and the rowboat straight for $Q$, they will collide at some point $X$ in the lake. Prove that if the motorboat heads for $Q$...

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Kvant Math Problem 660

Verified: no Verdicts: SKIP + SKIP Solve time: 8m22s Source on kvant.digital Problem Blue and red points are placed on a circle (Fig. 2a). It is allowed to either add a new red point and simultaneously change the color of each of the two neighboring points (Fig. 2b), or remove a red point and change the color of each of its neighboring points (Fig. 2c). Initially, there were only two...

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Kvant Math Problem 658

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem In a square with side length 1, a finite number of segments parallel to its sides are drawn. The segments may intersect each other. The sum of the lengths of the drawn segments is 18. Prove that among the regions into which the square is divided by these segments, there exists one whose area is at least...

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Kvant Math Problem 655

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Fig. 2 On the desk of an official of the Ministry of Contingencies lie $n$ volumes of the Encyclopaedia Britannica, arranged in several stacks. Each day, upon arriving at work, the official takes one volume from each stack and places the taken volumes into a new stack, then arranges the stacks by the number of volumes (in...

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Kvant Math Problem 654

Verified: no Verdicts: SKIP + SKIP Solve time: 7m45s Source on kvant.digital Problem Is the following statement true: among any six natural numbers, it is always possible to choose either three numbers such that every pair has no common divisors greater than 1, or three numbers that share a common divisor greater than 1? E. M. Rabbot Correspondence Mathematical Olympiads Exploration Consider small examples of six natural numbers and examine...

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Kvant Math Problem 650

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Does there exist a sequence $$a_1,\ a_2,\ a_3,\ \ldots,\ a_n, ...$$ of natural numbers such that every natural number can be uniquely represented as a sum of several of its terms: $$a_{i_1}+a_{i_k}\quad(i_1 \lt \ldots \lt i_k,\ k\ge 0); \tag{*}$$ of natural numbers such that 1 cannot be represented in the form ( ), but every natural number...

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Kvant Math Problem 646

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem From a point $O$ on the plane, $2n$ vectors of length 1 are drawn. They are colored alternately red and blue. Let $\overrightarrow{s}$ be the sum of $n$ red vectors, and $\overrightarrow{r}$ be the sum of $n$ blue vectors. Prove that $$|\overrightarrow{s}—\overrightarrow{r}|\le 2.$$ E. Shustin Exploration Consider the problem for small values of $n$ to understand the...

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Kvant Math Problem 644

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Prove that there exists a convex 1980-gon with side lengths 1, 2, $\ldots$, 1980, all of whose angles are equal. Does such a 1981-gon exist? G. A. Gurevich Exploration A convex equiangular $n$-gon has exterior angle $2\pi/n$ at every vertex. If the side vectors are taken in cyclic order, their directions differ successively by $2\pi/n$. Thus an...

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Kvant Math Problem 642

Verified: no Verdicts: SKIP + SKIP Solve time: 5m12s Source on kvant.digital Problem Prove that every positive integer can be represented in the form $a_0+2a_1+2^2a_2+\ldots+2^na_n$, where each of the numbers $a_k=0$, $-1$ is either 1 and $a_k\cdot a_{k+1}=0$ for all $0\le k\le n-1$, and moreover this representation is unique. I. K. Zhuk Exploration The coefficients are restricted to the set ${-1,0,1}$, and two neighboring coefficients cannot both be nonzero. This...

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Kvant Math Problem 640

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem The number $x\in[0;1]$ is written as an infinite decimal expansion. By rearranging its first 5 digits after the decimal point in an arbitrary order, we obtain a new infinite decimal expansion corresponding to some number $x_1$. By rearranging the 2nd through 6th digits after the decimal point in the decimal expansion of $x_1$, we obtain the decimal...

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Kvant Math Problem 637

Verified: no Verdicts: SKIP + SKIP Solve time: 8m29s Source on kvant.digital Problem Given an equilateral triangle $ABC$. A line parallel to the line $AC$ intersects the lines $AB$ and $BC$ at points $M$ and $P$, respectively. The point $D$ is the centroid of the triangle $PMB$, and the point $E$ is the midpoint of the segment $AP$. Determine the angles of the triangle $DEC$. L. P. Kuptsov All-Union Mathematical...

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Kvant Math Problem 636

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem The set $A$ consists of integers; its smallest element is 1, and its largest element is 100. Every element of $A$ other than 1 is equal to the sum of two (possibly equal) numbers belonging to $A$. Among all sets $A$ satisfying these conditions, determine one with the minimum number of elements. Yu. V. Nesterenko All-Union Mathematical...

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Kvant Math Problem 628

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem On a sphere, a triangle is constructed one of whose “sides” has length $120^\circ$. Prove that the “median” drawn to this “side” is bisected by each of the other two “medians.” (“Medians” and “sides” are arcs of great circles.) A. Yagubyants Exploration Consider a spherical triangle with one side of length $120^\circ$. On a unit sphere, a...

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Kvant Math Problem 627

Verified: no Verdicts: SKIP + SKIP Solve time: 6m57s Source on kvant.digital Problem On each cell of an infinite sheet of graph paper, a natural number is written. Suppose each of these numbers occurs exactly once. (Provide examples of such an arrangement of numbers!) Prove that for any given $m$ there exist two neighboring (sharing a side) cells whose numbers differ by at least $m$. Suppose each number $n\in\mathbb{N}$ occurs...

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Kvant Math Problem 598

Verified: no Verdicts: SKIP + SKIP Solve time: 5m37s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I can proceed, but I need the text of problem M598 first. Could you provide the full statement of the problem?

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Kvant Math Problem 597

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem Let $$x_n=1+\dfrac12+\ldots+\dfrac1n.$$ Prove the existence of the limit $$\gamma=\lim_{n\to\infty}{}(x_n-\ln n).$$ Prove that for any $n$, $m\in\mathbb{N}$ $$\gamma\lt x_n+x_m-x_{nm}\le1.$$ Find $\gamma$ with an accuracy of $0{,}1$. V. Yasinsky Exploration The sequence $x_n=1+\frac12+\dots+\frac1n$ is the $n$-th harmonic number. Numerically, for small $n$, $x_n - \ln n$ appears to approach a constant: for $n=1$, $x_1-\ln 1 = 1$, for $n=2$,...

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Kvant Math Problem 595

Verified: no Verdicts: SKIP + SKIP Solve time: 3m55s Source on kvant.digital Problem Let $A$ and $E$ be two opposite vertices of a regular octagon. A frog is located at vertex $A$. From any vertex of the octagon, except for vertex $E$, the frog can jump to either of the two neighboring vertices. Upon reaching vertex $E$, the frog stops and remains there. Let $a_n$ be the number of ways...

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Kvant Math Problem 594

Verified: no Verdicts: SKIP + SKIP Solve time: 5m38s Source on kvant.digital Problem Find all real numbers $a$ for which there exist nonnegative real numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ satisfying the relations $$\sum_{k=1}^5kx_k=a,\quad \sum_{k=1}^5k^3x_k=a^2,\quad \sum_{k=1}^5k^5x_k=a^3.$$ International Mathematical Olympiad (XXI, 1979) Exploration Let $$m_r=\sum_{k=1}^5 k^r x_k .$$ The conditions are $$m_1=a,\qquad m_3=a^2,\qquad m_5=a^3.$$ Since the coefficients $x_k$ are nonnegative, it is natural to normalize them and interpret them as weights....

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Kvant Math Problem 592

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem Prove that for any triangle, the projection onto the line containing a second side of the diameter of the circumcircle that is perpendicular to one side of the triangle is equal in length to the third side. S. Ovchinnikov Exploration Consider a triangle $ABC$ with circumcircle $\Gamma$. Denote the sides as $BC = a$, $AC = b$,...

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Kvant Math Problem 590

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Find the minimum value of the expression $|{\cos x}|+|{\cos2x}|$. Prove that for any number $x$ and any positive integer $n$, the sum $$|{\cos x}|+|{\cos2x}|+|{\cos4x}|+\ldots+|{\cos2^nx}|$$ is not less than $\dfrac n4$; is not less than $\dfrac n2$. P. Gusyatnikov All-Russian Mathematical Olympiad for School Students (1979, Grade 10) Exploration Consider first the expression $|\cos x| + |\cos 2x|$....

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Kvant Math Problem 589

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem On the plane, a set of $n$ vectors is given, each with length not exceeding 1. Prove that by replacing some of the vectors in this set with their opposites, one can obtain a set of $n$ vectors whose sum has length not exceeding $\sqrt n$; not exceeding $\sqrt2$. P. Gusyatnikov, A. Plotkin All-Russian School Mathematics Olympiad...

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Kvant Math Problem 587

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem A triple of numbers 2, $\sqrt2$, $\dfrac1{\sqrt2}$ is given. It is allowed to replace any two of them with two numbers defined as follows: their sum divided by $\sqrt2$ and their difference divided by $\sqrt2$. Is it possible, by performing this procedure several times, to obtain the triple of numbers 1, $\sqrt2$, $1+\sqrt2$? All-Russian School Mathematics Olympiad...

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Kvant Math Problem 585

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem At a chemistry conference, there were $N$ scientists — chemists and alchemists — with the number of chemists exceeding the number of alchemists. It is known that chemists always tell the truth, while alchemists sometimes tell the truth and sometimes lie. A mathematician attending the conference must determine for each scientist whether they are a chemist or...

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Kvant Math Problem 584

Verified: no Verdicts: SKIP + SKIP Solve time: 7m29s Source on kvant.digital Problem Is it possible to represent the entire space as the union of lines, every two of which are skew (that is, do not lie in the same plane)? F. V. Vainshtein Exploration Suppose such a family of lines exists. Since the union of the lines is the whole space, every point lies on exactly one line, because...

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Kvant Math Problem 558

Verified: no Verdicts: SKIP + SKIP Solve time: 2m40s Source on kvant.digital Problem A circle contains $k\gt1$ black sectors, each with an angle smaller than $\dfrac{180^\circ}{k^2-k+1}$. Prove that the circle can be rotated around $O$ so that all black sectors move into the white portion of the circle. V. V. Proizvolov Exploration Let the black sectors have angular lengths $\alpha_1,\dots,\alpha_k$, where each $$\alpha_i<\frac{180^\circ}{k^2-k+1}.$$ A rotation through an angle $t$ moves...

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Kvant Math Problem 557

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem Given $n$ pairwise relatively prime numbers, greater than $1$ and less than $(2n-1)^2$. Prove that at least one of them must be a prime number. A. T. Kolotov Exploration Suppose, contrary to the statement, that none of the given numbers is prime. Then every number is composite. Since the numbers are pairwise relatively prime, no prime divisor...

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Kvant Math Problem 553

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Given a triangle $ABC$, such that $|BC|\lt|AC|\lt|AB|$. On the rays $BA$ and $CA$, segments $BD$ and $CE$ are laid off so that $|BD|=|CE|=|BC|$. Prove that the radius of the circumcircle of triangle $ADE$ is equal to the distance between the circumcenter of triangle $ABC$ and its incenter. S. Meidman Exploration Consider a triangle $ABC$ with sides $BC...

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Kvant Math Problem 537

Verified: no Verdicts: SKIP + SKIP Solve time: 6m27s Source on kvant.digital Problem A circle is tangent internally to the circumcircle of an isosceles triangle $ABC$, as well as to the equal sides $AB$ and $AC$ of this triangle at points $P$ and $Q$, respectively. Prove that the midpoint of segment $PQ$ is the center of the incircle of triangle $ABC$. International Mathematical Olympiad (XX, 1978) Exploration Let $O$ be...

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Kvant Math Problem 524

Verified: no Verdicts: SKIP + SKIP Solve time: 6m40s Source on kvant.digital Problem Prove that for no natural $m$ does the number $1978^m-1$ divide $1000^m-1$. S. V. Konyagin All-Union Mathematical Olympiad for School Students (XII, 1978, 8th grade) Exploration Consider the numbers $1978^m - 1$ and $1000^m - 1$ for small values of $m$. For $m = 1$, $1978^1 - 1 = 1977$ and $1000^1 - 1 = 999$, and...

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Kvant Math Problem 512

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Let $f(x)=x^3-x+1$. Prove that for every natural number $m\gt1$, the numbers $m$, $f(m)$, $f(f(m))$, $f(f(f(m)))$, $\ldots$ are pairwise relatively prime. A. T. Kolotov All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Compute the first few values of $f$ for small natural numbers greater than $1$. For $m=2$, $f(2)=2^3-2+1=7$, $f(f(2))=f(7)=343-7+1=337$, $f(f(f(2)))=f(337)=337^3-337+1$, which is enormous. None...

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Kvant Math Problem 507

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem In last year's Russian translation of Selected Problems from the American Mathematical Monthly (M.: Mir, 1977), two problems by the remarkable mathematician P. Erdős (Nos. 138 and 139) were given without solutions. The editorial board of Kvant received a number of letters containing solutions to these problems. Before presenting the solutions to the readers of the journal,...

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Kvant Math Problem 503

Verified: no Verdicts: SKIP + SKIP Solve time: 3m46s Source on kvant.digital Problem A set of $2n+1$ numbers $a_0$, $a_1$, $a_2$, $\ldots$, $a_{2n}$ is such that $a_k\ge\dfrac{a_{k-1}+a_{k+1}}2$ for all $k=1$, 2, $\ldots$, $2n-1$. Prove the inequality $$\dfrac{a_1+a_2+a_3+\ldots+a_{2n-1}}n\ge\dfrac{a_0+a_1+a_2+\ldots+a_{2n}}{n+1}\tag{*}$$ and determine for which sets it becomes an equality. V. Lipkin Exploration The condition $$a_k\ge \frac{a_{k-1}+a_{k+1}}2 \qquad (1\le k\le 2n-1)$$ means that the sequence is concave. Rewriting, $$(a_k-a_{k-1})\ge (a_{k+1}-a_k).$$ Hence the successive differences...

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Kvant Math Problem 499

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem We call a number balanced if in its decimal representation some initial segment coincides with some terminal segment (for example, the numbers 1971, 19219 are balanced, whereas 1415145 is not). Prove that there exists a number which, after appending any of the 10 digits to its right, becomes balanced. G. A. Gurevich Exploration Consider what it means...

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Kvant Math Problem 497

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem On the sides $BC$, $CA$, and $AB$ of an acute triangle $ABC$, arbitrary points $A_1$, $B_1$, and $C_1$ are chosen; circles are constructed on the segments $AA_1$, $BB_1$, and $CC_1$ as diameters. Prove that the three common chords of the pairs of these circles intersect at the point where the altitudes of triangle $ABC$ meet. V. L....

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Kvant Math Problem 495

Verified: no Verdicts: SKIP + SKIP Solve time: 6m11s Source on kvant.digital Problem In outer space around the planet $O$, three satellites move uniformly along three circular orbits centered at $O$. The angular velocities of the satellites are $\omega_1$, $\omega_2$, and $\omega_3$, respectively, and their initial positions can be arbitrary. Is there necessarily a moment in time when all three satellites and the point $O$ lie in the same plane...

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Kvant Math Problem 492

Verified: no Verdicts: SKIP + SKIP Solve time: 6m22s Source on kvant.digital Problem A triangle $ABC$ is inscribed in triangle $A_1B_1C_1$ (so that the vertices $A_1$, $B_1$, and $C_1$ lie respectively on the sides $BC$, $CA$, and $AB$), and the segments $AA_1$, $BB_1$, and $CC_1$ intersect at a single point $P$. Prove that the lines joining the midpoints of the sides $AB$ and $A_1B_1$, $BC$ and $B_1C_1$, $CA$ and $C_1A_1$...

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Kvant Math Problem 491

Verified: no Verdicts: SKIP + SKIP Solve time: 3m43s Source on kvant.digital Problem Consider a geometric progression all of whose terms are integers. (For example: 16, 24, 36, 54, 81.) Prove that the sum of the squares of three consecutive terms of the progression is divisible by the sum of these terms. For which natural numbers $n$ is the sum of the squares of $n$ consecutive terms of the progression...

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Kvant Math Problem 488

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Consider the sequence of polynomials $P_0$, $P_1$, $P_2$, $\ldots$, defined by the conditions $P_0(x)=1$, $P_1(x)=x$, $$P_{n+1}(x)=x,P_n(x)-P_{n-1}(x),\quad n\ge1.\tag1$$. Prove the equalities $$\underbrace{x-\dfrac1{x-\dfrac1{x-\dfrac1{\raisebox{4pt}{$\ddots\vphantom{\dfrac00}$}{}-\dfrac1x}}}} {n~\text{раз}}=\dfrac{P_n(x)}{P {n-1}(x)};$$ $$\dfrac{\sin{(n+1)\phi}}{\sin\phi}=P_n(2\cos\phi),\quad\text{если} \dfrac\phi\pi \text{— не целое};$$ $$t^{n+1}-\dfrac1{t^{n+1}}=\left(t-\dfrac1t\right)P_n{\left(t+\dfrac1t\right)},\quad\text{если}~t\ne0;$$ $$P_n(x)=\prod\limits_{1\le k\le n}\left(x-2\cos\dfrac{\pi k}{n+1}\right);$$ $$P_n(x)=\textstyle\sum\limits_{0\le j\le n/2}{(-1)^j}C_{n-j}^jx^{n-2j};$$ $$\prod\limits_{1\le k\le m}\cos\dfrac{k\pi}{2m+1}=\dfrac1{2^m}.$$ Devise and prove analogous equalities for the sequence of polynomials defined by the same relation (1), but starting...

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Kvant Math Problem 485

Verified: no Verdicts: SKIP + SKIP Solve time: 8m14s Source on kvant.digital Problem Prove that the number $e$ is between the numbers $a_n=\left(1+\dfrac1n\right)^n$ and $b_n=\left(1+\dfrac1n\right)^{n+1}$ for any natural $n$. Prove that the sequence $c_n=\left(1+\dfrac1{4n}\right)\left(1+\dfrac1n\right)^n$ is monotonically increasing, while the sequence $d_n=\left(1+\dfrac1{2n}\right)\left(1+\dfrac1n\right)^n$ is monotonically decreasing. Divide the segment $[a_n;b_n]$ into four equal parts. In which of them does the number $e$ lie? Divide the segment $[a_n;b_n]$ into eight equal parts. In...

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Kvant Math Problem 483

Verified: no Verdicts: SKIP + SKIP Solve time: 6m19s Source on kvant.digital Problem a) Prove that the ratio of the inradius of a right triangle to the sum of the squares of the medians drawn from the acute angles does not exceed $1/5$. b) Find the maximum value that this ratio can attain. I. Temraliev Exploration Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $c^2...

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Kvant Math Problem 481

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem In the sequence of natural numbers $a_1$, $a_2$, $a_3$, $\ldots$ each term $a_{k+1}$ is equal to the sum of the squares of the digits in the decimal representation of the number $a_k$ ($k\ge1$). Prove that for any $a_1$ in this sequence, the number 1 or the number 89 will be encountered. V. Panfilov Exploration Let $$f(n)=\text{(sum of...

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Kvant Math Problem 479

Verified: no Verdicts: SKIP + SKIP Solve time: 5m05s Source on kvant.digital Problem Do there exist six, 1000 such distinct natural numbers that for any two $a$ and $b$ among them, the sum $a+b$ is divisible by the difference $a-b$? S. V. Konyagin Moscow Mathematical Olympiad (XL, 1977, grades 9 and 11) Exploration Consider a set of distinct natural numbers ${a_1, a_2, \dots, a_n}$ with the property that for any...

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Kvant Math Problem 476

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem Prove that if the vertices of a convex $n$-gon lie on the nodes of graph paper, and there are no other nodes inside it or on its sides, then $n\le4$. Let space be divided by three families of parallel planes into identical cubes. Call the vertices of the cubes nodes . Prove that if all $n$ vertices...

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Kvant Math Problem 474

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem A natural number is called perfect if it is equal to the sum of all its divisors (excluding itself); for example, the numbers $6=1+2+3$ and $28=1+2+4+7+14$ are perfect. Prove that the number $N$ is not perfect if it is known that it leaves a remainder of 3 when divided by 4; leaves a remainder of 5 when...

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Kvant Math Problem 472

Verified: no Verdicts: SKIP + SKIP Solve time: 4m36s Source on kvant.digital Problem A convex polyhedron is located inside a cube, and its orthogonal projection onto each face of the cube coincides with that face. Prove that the volume of the polyhedron is at least one third of the volume of the cube. V. V. Prasolov Exploration Consider a cube of side length $1$ for simplicity. We are asked to...

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Kvant Math Problem 470

Verified: no Verdicts: SKIP + SKIP Solve time: 5m13s Source on kvant.digital Problem Prove the equalities $$\dfrac{1}{C_n^0}-\dfrac{1}{C_n^1}+\dfrac{1}{C_n^2}-\ldots+\dfrac{(-1)^k}{C_n^k}+\ldots+\dfrac{(-1)^n}{C_n^n}=\dfrac{(n+1)(1+(-1)^n)}{n+2};$$ $$\dfrac{1}{C_n^1}+\dfrac{1}{C_n^2}+\dfrac{1}{C_n^3}+\ldots+\dfrac{1}{C_n^n}=\dfrac{n+1}{2^{n+1}}\left(\dfrac{2}{1}+\dfrac{2^2}{2}+\dfrac{2^3}{3}+\ldots+\dfrac{2^n}{n}\right).$$ L. D. Kurlandchik, A. D. Lisitsky Exploration We begin by examining the two sums for small values of $n$ to detect patterns. For $n = 2$, the first sum is $\frac{1}{C_2^0} - \frac{1}{C_2^1} + \frac{1}{C_2^2} = 1 - \frac{1}{2} + 1 = \frac{3}{2},$ while the formula on the right-hand side gives $\frac{(2+1)(1...

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Kvant Math Problem 468

Verified: no Verdicts: SKIP + SKIP Solve time: 4m06s Source on kvant.digital Problem Points $A$, $B$, $C$, and $D$ in the plane are such that for any point $M$ of this plane, the scalar products of the vectors $\overrightarrow{MA}\cdot\overrightarrow{MB}$ and $\overrightarrow{MC}\cdot\overrightarrow{MD}$ are not equal to each other. Prove that $\overrightarrow{AC}=\overrightarrow{DB}$. Is the converse statement true? Yu. I. Ionin Exploration Consider four points $A$, $B$, $C$, $D$ in the plane, and...

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Kvant Math Problem 466

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Among 1977 coins, 50 are counterfeit. Each counterfeit coin differs from a genuine one by 1 g (either heavier or lighter). There is a pan balance with a pointer indicating the difference in mass between the two pans. With a single weighing, one must determine whether a selected coin is counterfeit or genuine. How can this be...

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Kvant Math Problem 463

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem Given natural numbers $x_1$, $x_2$, $\ldots$, $x_m$, $y_1$, $y_2$, $\ldots$, $y_n$. The sums $x_1+x_2+\ldots+x_m$ and $y_1+y_2+\ldots+y_n$ are equal and less than $mn$. Prove that in the equality $$x_1+x_2+\ldots+x_m=y_1+y_2+\ldots+y_n$$ it is possible to delete some of the terms so that the resulting equality is still true. K. Sibiryakov All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 9)...

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Kvant Math Problem 461

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem On the table are a pair of balance scales and $n$ weights of different masses. The weights are placed on the pans of the scales one by one (at each step, any weight can be taken from the table and added to either pan of the scales). Prove that the weights can be placed in such an...

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Kvant Math Problem 458

Verified: no Verdicts: SKIP + SKIP Solve time: 7m07s Source on kvant.digital Problem A polynomial $x^{10}+{ }x^9+{ }x^8+\ldots+{ }x^2+{ }x+1$ is written. Two players play the following game. First, the first player replaces any of the asterisks with a number; then the second player replaces any of the remaining asterisks with a number; then the first player again replaces one of the asterisks with a number, and so on (a...

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Kvant Math Problem 457

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem A non-self-intersecting closed polygonal line is given in the plane, with no three of its vertices lying on a single straight line. Call a pair of nonadjacent sides special if the extension of one of them intersects the other (Fig. 1). Prove that the number of special pairs is even. Figure 1 S. V. Fomin All-Union Mathematical...

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Kvant Math Problem 454

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem Seven dwarfs are sitting around a round table. In front of each of them is a mug. Some of these mugs contain milk. One of the dwarfs pours all of his milk equally into the mugs of the other dwarfs. Then his neighbor to the right does the same. Then the next neighbor to the right does...

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Kvant Math Problem 451

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Several points, not all lying on a single line, are marked in the plane, and a number is written next to each of them. It is known that whenever a line passes through two or more marked points, the sum of all the numbers written next to those points is zero. Prove that all the numbers are...

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Kvant Math Problem 449

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M449 without the actual problem statement or the diagram. The prompt says that the text version is not available and that the problem appears only in graphical...

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Kvant Math Problem 447

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M447 because the actual problem statement is not present in your message. The identifier “M447” by itself is not enough to determine the problem. Kvant numbering is tied to...

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Kvant Math Problem 445

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M445 from the information provided, because the actual problem statement is missing. You stated that “the textual version of the problem statement is currently being prepared” and that...

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Kvant Math Problem 443

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Before I begin writing the complete solution, I need the full textual statement of Kvant problem M443 . The graphical version alone is not enough for a rigorous solution, since I cannot infer all details from images....

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Kvant Math Problem 441

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Inside a convex $2n$-gon, an arbitrary point $P$ is chosen. Through each vertex and the point $P$, a straight line is drawn. Prove that there exists a side of the polygon such that none of the drawn lines has any points in common with it (except, possibly, the endpoints of the side). G. A. Gurevich Exploration Let...

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Kvant Math Problem 439

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem Prove that the equation $$ax^k+bx^l+cx^m=1$$ (where $a$, $b$, $c$ are real numbers, and $k$, $l$, and $m$ are natural numbers) has at most three positive roots. Prove that the equation $$a_1x^{k_1}+a_2x^{k_2}+ \ldots + a_nx^{k_n}=1$$ (where $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers, and $k_1$, $k_2$, $\ldots$, $k_n$ are natural numbers) has at most $n$ positive roots. Prove...

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Kvant Math Problem 436

Verified: no Verdicts: SKIP + SKIP Solve time: 4m49s Source on kvant.digital Problem Given 20 numbers $a_1$, $a_2$, $\ldots$, $a_{10}$, $b_1$, $b_2$, $\ldots$, $b_{10}$. Prove that the set of 100 numbers (not necessarily distinct) $a_1+b_1$, $a_1+b_2$, $\ldots$, $a_{10}+b_{10}$ can be partitioned into 10 subsets, each containing 10 numbers, so that the sum of the numbers in each subset is the same. S. T. Berkolayko Exploration We are asked to partition...

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Kvant Math Problem 434

Verified: no Verdicts: SKIP + SKIP Solve time: 4m58s Source on kvant.digital Problem The number $\dfrac21+\dfrac{2^2}2+\dfrac{2^3}3+\ldots+\dfrac{2^n}n$ is represented as an irreducible fraction $\dfrac{p_n}{q_n}$. Prove that $p_n$ is an even number. Prove that if $n\ge3$, then $p_n$ is divisible by 8. Prove that for any natural $k$, one can specify $n$ such that the numbers $p_n$, $p_{n+1}$, $p_{n+2}$, $\ldots$ are divisible by $2^k$. D. K. Faddeev Exploration Consider the sum $S_n...

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Kvant Math Problem 432

Verified: no Verdicts: SKIP + SKIP Solve time: 7m31s Source on kvant.digital Problem Does there exist a perfect square whose digit sum is 1977; 1978? Determine which positive integers can occur as the digit sums of the square of an integer. A. Grishkov Exploration Consider the sum of the digits of perfect squares. Small examples suggest that there is no simple formula for the digit sum, but congruences modulo 9...

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Kvant Math Problem 430

Verified: no Verdicts: SKIP + SKIP Solve time: 3m53s Source on kvant.digital Problem Prove that any convex planar figure of area $S$ can be enclosed in a rectangle of area $2S$. Prove that any convex solid of volume $V$ can be enclosed in a box in the shape of a rectangular parallelepiped of volume $6V$ (Fig. 2). Fig. 2 Exploration For the planar statement, the number $2$ strongly suggests a...

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Kvant Math Problem 427

Verified: no Verdicts: SKIP + SKIP Solve time: 7m23s Source on kvant.digital Problem Prove that there exists an odd number $n$ such that, for no even $k$, is any of the numbers in the infinite sequence $$k^k+1,\ k^{k^k}+1,\ k^{k^{k^k}}+1,\ \ldots$$ divisible by $n$. Prove that for every positive integer $n$ there exists a positive integer $k$ such that every term of the infinite sequence $$k+1,\ k^k+1,\ k^{k^k}+1,\ k^{k^{k^k}}+1,\ \ldots$$ is...

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Kvant Math Problem 425

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem Does there exist a natural number $N$ such that every rational number between zero and one can be represented as a sum of $N$ reciprocals of natural numbers? Exploration Suppose such an $N$ exists. Then every rational number $r\in(0,1)$ can be written in the form $$r=\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_N},$$ where the $a_i$ are natural numbers. For fixed $N$, the set...

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Kvant Math Problem 423

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Prove that for any real $x$, $y$, and $z$, the inequality $$(x^2+y^2-z^2)(x^2+z^2-y^2)(y^2+z^2-x^2)\le(x+y-z)^2(x+z-y)^2(y+z-x)^2.$$ holds. R. Sheintsvit Exploration The left-hand side contains the three quantities $$x^2+y^2-z^2,\qquad x^2+z^2-y^2,\qquad y^2+z^2-x^2.$$ The right-hand side contains $$(x+y-z)^2(x+z-y)^2(y+z-x)^2.$$ A natural first step is to factor the expressions on the left. Indeed, $$x^2+y^2-z^2=(x+y-z)(x+y+z)-2xy.$$ This does not immediately help. A more symmetric factorization is obtained by...

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Kvant Math Problem 421

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem For which natural numbers $m$ and $n$ ($m\lt n$) is it possible to color some cells of an infinite sheet of graph paper black so that every rectangle of size $m\times n$ contains exactly one black cell? Start with the simpler problem: consider the cases $m=2$ and $n=3$, 4, 5. Fig. 1 S. Okhitin, 10th-grade student Exploration...

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Kvant Math Problem 418

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem Prove that for any natural number $n$ the following inequalities hold: $$n{\left(!\sqrt[\scriptstyle n~]{n+1}-1\right)}\le 1+\dfrac12+\dfrac13+\ldots+\dfrac1n\le 1+n{\left(1-\dfrac1{!\sqrt[\scriptstyle n~]n}\right)}.$$ Exploration For small values of $n$, $$n=1:\qquad 1(\sqrt[1]{2}-1)=1,\qquad H_1=1,\qquad 1+1\left(1-\frac1{\sqrt[1]{1}}\right)=1.$$ For $n=2$, $$2(\sqrt3-1)\approx1.464,\qquad H_2=\frac32,\qquad 1+2\left(1-\frac1{\sqrt2}\right)\approx1.586.$$ The inequalities hold. The expressions involving $n$th roots suggest taking logarithms. If $$a=\sqrt[n]{n+1},$$ then $$n(a-1)\le H_n \iff a\le 1+\frac{H_n}{n}.$$ Since $$a^n=n+1,$$ it would suffice to prove...

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Kvant Math Problem 416

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem On the plane, $n$ points $A_1$, $\ldots$, $A_n$ are given, no three of which lie on the same line. What is the greatest number of line segments with endpoints at these points that can be drawn so that no triangle with vertices at these points is formed? A. Grigoryan, M. Primak, S. Fishbein Exploration Interpret the drawn...

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Kvant Math Problem 413

Verified: no Verdicts: SKIP + SKIP Solve time: 9m58s Source on kvant.digital Problem For which positive numbers $a$ does the following statement hold: for any function $f$, defined on the interval $[0,1]$, continuous at every point of this interval and such that $f(0)=f(1)=0$, the equation $f(x+a)-f(x)=0$ has a solution? First, examine this question for the case $a=\dfrac12$. Prove that for $a=\dfrac1n$, where $n$ is a natural number, the stated claim...

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Kvant Math Problem 411

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem Three segments with endpoints on the sides of a triangle, parallel to its sides, pass through a single point and have the same length $x$ (Fig. 1). Find $x$ if the lengths of the triangle's sides are $a$, $b$, $c$. Fig. 1 A. Yagubyants Exploration Consider a triangle with sides $a$, $b$, $c$, and a point inside...

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Kvant Math Problem 410

Verified: no Verdicts: SKIP + SKIP Solve time: 4m02s Source on kvant.digital Problem On a sphere of radius 1, draw a great circle, which we shall call the equator. It will be convenient to use other geographical terms as well: pole, meridian, and parallel. Define on this sphere a function $f$ that assigns to each point of the sphere the square of the distance from that point to the plane...

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Kvant Math Problem 407

Verified: no Verdicts: SKIP + SKIP Solve time: 3m58s Source on kvant.digital Problem Two natural numbers $n$ and $m$, $n\gt m$ are given. Prove that $n$ can be expressed as the sum of two natural numbers, one of which is a divisor of $m$, and the other shares no common divisor with $m$ except for one. S. V. Konyagin Exploration Consider small examples to understand the structure of the problem....

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Kvant Math Problem 405

Verified: no Verdicts: SKIP + SKIP Solve time: 7m21s Source on kvant.digital Problem Fig. 1 Fig. 2 On a chessboard of size $99\times99$, a figure $\mathit\Phi$ is marked (this figure will differ in parts a), b), and c)). In each cell of the figure $\mathit\Phi$, there is a beetle. At some moment, the beetles flew up and landed again in the cells of the same figure $\mathit\Phi$; multiple beetles could...

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Kvant Math Problem 403

Verified: no Verdicts: SKIP + SKIP Solve time: 8m44s Source on kvant.digital Problem Prove that if, in a convex polyhedron, an even number of edges emanates from each vertex, then the intersection of the polyhedron with any plane that does not pass through any of its vertices is a polygon with an even number of sides. Exploration A convex polyhedron has the property that every vertex has even degree. The...

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Kvant Math Problem 400

Verified: no Verdicts: SKIP + SKIP Solve time: 7m15s Source on kvant.digital Problem A sequence of natural numbers $a_1$, $a_2$, $\ldots$, $a_k$ will be called universal for a given $N$ if, by deleting some of its terms, one can obtain any sequence of $N$ numbers in which each of the numbers 1, 2, $\ldots$, $N$ occurs exactly once. Give an example of a universal sequence consisting of $N^2$ terms. Give...

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Kvant Math Problem 398

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a rigorous solution to Kvant problem M398 from the information provided, because the problem statement itself is missing. Your prompt says: "The textual version of the problem statement is currently being prepared. The...

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Kvant Math Problem 396

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem A triangle whose all sides are greater than $1~\text{см}$ will be called "large." Given an equilateral triangle $ABC$ with side $5~\text{см}$, prove that: it is possible to cut 1000 "large" triangles from the triangle $ABC$; the triangle $ABC$ can be dissected into 1000 "large" triangles; the triangle $ABC$ can be triangulated into 1000 "large" triangles, i.e., subdivided...

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Kvant Math Problem 393

Verified: no Verdicts: SKIP + SKIP Solve time: 6m39s Source on kvant.digital Problem Find the sum $$\phi(0)+\phi\left(\dfrac{1}{n}\right)+\phi\left(\dfrac{2}{n}\right)+\ldots+\phi(1),$$ if $\phi(x)=\dfrac{4^x}{4^x+2}$. M. Levin Exploration Let $$S=\phi(0)+\phi!\left(\frac1n\right)+\phi!\left(\frac2n\right)+\cdots+\phi(1),$$ where $$\phi(x)=\frac{4^x}{4^x+2}.$$ The arguments form the set $$0,\frac1n,\frac2n,\ldots,\frac{n-1}{n},1.$$ A natural idea is to pair the terms corresponding to $x$ and $1-x$. Compute $$\phi(1-x) =\frac{4^{1-x}}{4^{1-x}+2}.$$ Writing $t=4^x$ gives $$\phi(x)=\frac{t}{t+2}, \qquad \phi(1-x)=\frac{4/t}{4/t+2} =\frac{2}{t+2}.$$ Hence $$\phi(x)+\phi(1-x) =\frac{t}{t+2}+\frac{2}{t+2} =1.$$ This identity is extremely suggestive because the set of arguments is...

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Kvant Math Problem 391

Verified: no Verdicts: SKIP + SKIP Solve time: 5m45s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M391 from the information provided, because the actual problem statement is missing. The prompt says: “The text version of the problem statement is currently being prepared. The graphical...

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Kvant Math Problem 388

Verified: no Verdicts: SKIP + SKIP Solve time: 7m50s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M388 because the problem statement itself is not provided in the conversation. The prompt says that only the graphical version is available. Without the diagram or the text...

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Kvant Math Problem 386

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can proceed with that, but I need the text of Kvant Problem M386 to provide the complete solution. Could you provide the problem statement?

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Kvant Math Problem 383

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem Let $a$ and $b$ be two natural numbers. Prove that if $ab$ is even, then there exist natural numbers $c$ and $d$ such that $a^2+b^2+c^2=d^2$, whereas if $ab$ is odd, such $c$ and $d$ cannot be found. M. L. Gerver Exploration Consider small examples to understand the problem. If $ab$ is even, at least one of $a$...

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Kvant Math Problem 381

Verified: no Verdicts: SKIP + SKIP Solve time: 6m08s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M381 because the actual problem statement is not present in your message. You wrote that “the textual version of the problem statement is being prepared” and that the problem...

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Kvant Math Problem 379

Verified: no Verdicts: SKIP + SKIP Solve time: 4m28s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can proceed with this framework, but I need the text of Kvant problem M379 in order to produce a rigorous solution. Could you provide the problem statement?

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Kvant Math Problem 376

Verified: no Verdicts: SKIP + SKIP Solve time: 6m47s Source on kvant.digital Problem A row consists of 30 cells. A white piece stands on the rightmost cell, and a black piece stands on the leftmost cell. Each of the two players, in turn, moves their piece by one cell — either forward or backward. (Skipping a move is not allowed.) The player who cannot make a move loses. Who wins:...

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Kvant Math Problem 375

Verified: no Verdicts: SKIP + SKIP Solve time: 5m09s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M375 from the information provided, because the actual problem statement is missing. The identifier “Kvant M375” by itself is not enough to reconstruct the problem, and the prompt...

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Kvant Math Problem 372

Verified: no Verdicts: SKIP + SKIP Solve time: 7m43s Source on kvant.digital Problem Given a triangle $ABC$. Prove that the condition $\widehat{ACB} \ge 120^\circ$ is necessary and sufficient for the inequality $|AP|+|BP|+|CP|\ge|AC|+|BC|$ to hold for any point $P$ in the plane. P. Khaidukov Exploration Consider the triangle $ABC$ and the inequality $|AP| + |BP| + |CP| \ge |AC| + |BC|$ for an arbitrary point $P$ in the plane. The inequality...

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Kvant Math Problem 369

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem Given an acute-angled triangle $ABC$. $H$ is the intersection point of its altitudes, $\gamma$ is a circle with center $H$ lying inside this triangle. Construct a triangle $A_1B_1C_1$ circumscribed around the circle $\gamma$ and inscribed in the triangle $ABC$ (such that $A_1 \in [BC]$, $B_1 \in [AC]$, $C_1 \in [AB]$). D. Isaak Exploration The circle $\gamma$ is...

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Kvant Math Problem 368

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem Prove that the intersection of three right circular cylinders of radius 1 whose axes are pairwise perpendicular (but do not necessarily intersect) is contained in some sphere of radius $\sqrt\dfrac32$. S. V. Fomin Exploration Choose coordinates so that the three cylinder axes are parallel to the coordinate axes. Since the axes are pairwise perpendicular but need not...

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Kvant Math Problem 365

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem The sum of several numbers is 1. Can the sum of their cubes be greater than one? The same question for numbers each of which is less than one. Can it happen that the series $a_1+a_2+a_3+\ldots$ converges, while the series $a_1^3+a_2^3+a_3^3+\ldots$ does not? (Recall that a series $x_1+x_2+x_3+\ldots$ is called convergent if the sequence of numbers $S_n=x_1+x_2+\ldots+x_n$...

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Kvant Math Problem 362

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Divide each side of the convex quadrilateral $ABCD$ into three equal parts and connect the corresponding points on opposite sides by segments (Fig. 1). Prove that the area of the “middle” quadrilateral is 9 times smaller than the area of the quadrilateral $ABCD$. Figure 1 A. Lopshits Exploration The statement is affine in nature. Dividing sides in...

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Kvant Math Problem 361

Verified: no Verdicts: SKIP + SKIP Solve time: 4m29s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Please provide the full text of Kvant problem M361 so I can write the complete, rigorous solution in the requested six-section format.

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Kvant Math Problem 358

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem The text version of the problem statement is being prepared. The graphical version can be found in the published issue. I can produce a complete, rigorous Kvant-style solution, but I need the text of problem M358 to proceed. Could you provide the problem statement?

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Kvant Math Problem 357

Verified: no Verdicts: SKIP + SKIP Solve time: 2m56s Source on kvant.digital Problem Prove that if $x+\dfrac1y=y+\dfrac1z=z+\dfrac1x$, then $x=y=z$ or $x^2y^2z^2=1$. Exploration Let $$x+\frac1y=y+\frac1z=z+\frac1x=k.$$ Subtracting consecutive equalities gives $$x-y=\frac1z-\frac1y=\frac{y-z}{yz},$$ $$y-z=\frac1x-\frac1z=\frac{z-x}{xz},$$ $$z-x=\frac1y-\frac1x=\frac{x-y}{xy}.$$ These relations connect the differences $x-y$, $y-z$, $z-x$ with one another. If all three differences are zero, then $x=y=z$, which is one of the required conclusions. The interesting case is when not all differences vanish. Then at least one...

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Kvant Math Problem 351

Verified: no Verdicts: SKIP + SKIP Solve time: 10m16s Source on kvant.digital Problem Reconstruct a triangle if three points are marked in the plane: $O$ — the center of the circumcircle, $P$ — the centroid, and $H$ — the foot of one of the altitudes of this triangle. M. M. Imerishvili, 9th-grade student (Tbilisi) Exploration Let the unknown triangle be $ABC$, and suppose that $H$ is the foot of the...

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Kvant Math Problem 349

Verified: no Verdicts: SKIP + SKIP Solve time: 5m49s Source on kvant.digital Problem What condition must the side lengths of a triangle satisfy so that the triangle formed by its altitudes, medians, angle bisectors is similar to the given triangle? A. P. Savin Exploration Let the given triangle have sides $a,b,c$ opposite angles $A,B,C$. For the triangle formed by the altitudes, the side lengths are proportional to the altitudes $$h_a=\frac{2\Delta}{a},\qquad...

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Kvant Math Problem 341

Verified: no Verdicts: SKIP + SKIP Solve time: 4m54s Source on kvant.digital Problem There are 20 teams participating in the World Championship. Among them, $k$ are European teams, and the results of the matches between these teams in the World Championship count towards the European Championship. The championship is conducted in a single round-robin format. What is the largest possible $k$ such that a European team, having scored strictly the...

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Kvant Math Problem 338

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem On the board, several zeros, ones, and twos are written. It is allowed to erase two unequal digits and replace them with a single digit different from the erased ones (2 instead of 0 and 1, 1 instead of 0 and 2, 0 instead of 1 and 2). Prove that if, after several such operations, only a...

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Kvant Math Problem 336

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem A finite set of polygons is given in the plane, and every two of them have a point in common. Prove that there exists a line that has a point in common with each of these polygons. S. V. Fomin All-Union Mathematical Olympiad for School Students (1974, Grade 10) Exploration For two polygons the statement is immediate....

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Kvant Math Problem 328

Verified: no Verdicts: SKIP + SKIP Solve time: 7m46s Source on kvant.digital Problem A fly and two spiders are moving on a regular tetrahedron. The fly moves only along the edges, while the spiders may move over the entire surface. The fly's maximum speed is twice the spiders' maximum speed. Prove that, regardless of the initial positions, the spiders can catch the fly. Is this still true if the fly's...

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Kvant Math Problem 320

Verified: no Verdicts: SKIP + SKIP Solve time: 7m16s Source on kvant.digital Problem Which convex $n$-gons can be subdivided into triangles so that no two triangles in the subdivision share a common (completely coinciding) side? (Figure 1 shows that a triangle can be subdivided in this way.) Figure number 1 A. Pechkowski Exploration The statement asks for a classification. A triangle can indeed be subdivided into smaller triangles without any...

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Kvant Math Problem 317

Verified: no Verdicts: SKIP + SKIP Solve time: 4m08s Source on kvant.digital Problem On a certain planet, each country borders no more than 7 other countries. Each country has a reserve of gold. It is required to distribute the gold so that any two neighboring countries differ in the amount of gold by no more than a factor of 13. Prove that the gold can be distributed in such a...

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Kvant Math Problem 307

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem The plane is partitioned into congruent hexagonal rooms (Figure 2). Doors have been made in some of the walls in such a way that for every vertex at which three walls (sides of the hexagons) meet, doors are present in exactly two of those walls. Prove that any closed path through such a labyrinth passes through an...

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Kvant Math Problem 298

Verified: no Verdicts: SKIP + SKIP Solve time: 6m30s Source on kvant.digital Problem List all irreducible fractions $\dfrac pq$, where $0\lt p\lt q\le m$, in ascending order ($m$ is a given natural number). For example, for $m=5$ the sequence $$\dfrac15,\quad\dfrac14,\quad\dfrac13,\quad\dfrac25,\quad\dfrac12,\quad\dfrac35,\quad\dfrac23,\quad\dfrac34,\quad\dfrac45.$$ is obtained. Prove that for any two consecutive fractions $\dfrac pq\lt\dfrac rs$ in such a sequence, the equality $qr-ps=1$ holds. N. B. Vasilyev Exploration For $m=5$ the consecutive fractions $$\frac13<\frac25,\qquad...

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Kvant Math Problem 295

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem The sections of a convex polyhedron by three parallel planes $p_0$, $p_1$, and $p_2$ ($p_1$ lies between $p_0$ and $p_2$ at the same distance $h$ from each of them) have areas $S_0$, $S_1$, and $S_2$, respectively. There is not a single vertex of the polyhedron between $p_0$ and $p_2$. Prove that $2\sqrt{S_1}\ge\sqrt{S_0}+\sqrt{S_2}$. In what case does the...

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Kvant Math Problem 286

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem On a plane, $N$ points are placed. Mark all the midpoints of segments whose endpoints are among these points. What is the minimum number of points on the plane that can be marked? A. Pechkovsky Exploration Let $m(N)$ denote the minimum possible number of distinct marked points. For $N=2$, there is one segment and one midpoint, so...

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Kvant Math Problem 284

Verified: no Verdicts: SKIP + SKIP Solve time: 3m31s Source on kvant.digital Problem The sum of 100 natural numbers, each not exceeding 100, is 200. Prove that it is possible to select some of these numbers whose sum is 100. Moscow Mathematical Olympiad (1974) Exploration Consider smaller analogues of the problem to understand its structure. If we have two numbers whose sum is $4$, say $1$ and $3$, we can...

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Kvant Math Problem 273

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem On the segment $0 \le x \le 1$, a function $f$ is defined. It is known that this function is non-negative and $f(1) = 1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1 \ge 0$, $x_2 \ge 0$, and $x_1 + x_2 \le 1$, the inequality $$f(x_1 + x_2) \ge f(x_1) + f(x_2).$$ holds....

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Kvant Math Problem 263

Verified: no Verdicts: SKIP + SKIP Solve time: 6m48s Source on kvant.digital Problem Two numbers $p$ and $q$ greater than 1 are given. On the sides $BC$ and $DC$ of the rectangle $ABCD$, points $P$ and $Q$ are chosen so that $|BC| = p \cdot |BP|$ and $|DC| = q \cdot |DQ|$. For what ratio of the lengths of the sides $AB$ and $AD$ will the angle $PAQ$ attain its...

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Kvant Math Problem 261

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem A hoop of radius $R$, suspended on a fixed circle of radius $r < R$, begins to roll along this circle. Prove that a point on the hoop traces the same path as a point on a wheel of radius $R - r$ rolling externally along the same circle of radius $r$ (Fig. 1, a, b). (The...

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Kvant Math Problem 258

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem Three points $K$, $L$, $N$ are given in the plane. It is known about a quadrilateral that it is convex and that the midpoints of three of its sides lie at the given points $K$, $L$, $N$. Find the sets of points that can be occupied by: the midpoint of the fourth side; a vertex of this...

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Kvant Math Problem 248

Verified: no Verdicts: SKIP + SKIP Solve time: 7m54s Source on kvant.digital Problem An $n$-gon $A_1A_2\ldots A_n$ is inscribed in a convex $n$-gon $B_1B_2\ldots B_n$, and its area is equal to $P$ (the vertex $B_i$ lies on the side $A_iA_{i+1}$ for $i=1$, 2, $\ldots$, $n-1$, and the vertex $B_n$ lies on $A_n A_1$). Around the same $n$-gon $A_1A_2\ldots A_n$, an $n$-gon $C_1C_2 \ldots C_n$ is circumscribed, whose area is equal...

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Kvant Math Problem 237

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem The angles of an acute-angled triangle are $\alpha$, $\beta$, and $\gamma$. What masses should be placed at its vertices so that the center of mass of these three masses falls at the intersection point of the altitudes? at the center of the circumscribed circle? The sides of the triangle are $a$, $b$, and $c$. What masses should...

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Kvant Math Problem 229

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem A policeman is at the center of a square, and a gangster is at one of its vertices. The policeman can run anywhere inside the square, while the gangster can move only along its sides. It is known that the policeman's maximum speed is $u$, and the gangster's maximum speed is $v$. The policeman's goal is to...

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Kvant Math Problem 224

Verified: no Verdicts: SKIP + SKIP Solve time: 1m41s Source on kvant.digital Problem In a trihedral angle, the bisectors of its plane angles are drawn. Prove that the pairwise angles between these three bisectors are either all obtuse, all acute, or all right angles. E. G. Gotman Exploration Consider a trihedral angle, that is, three planes meeting at a common vertex, forming three plane angles $\alpha$, $\beta$, and $\gamma$ at...

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Kvant Math Problem 222

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem Prove that every convex polyhedron has two faces with the same number of sides. A. Grüntal Exploration Consider small convex polyhedra such as the tetrahedron, cube, and octahedron. A tetrahedron has four triangular faces, all identical, so the claim is trivially true. A cube has six quadrilateral faces, again all identical. An octahedron has eight triangular faces,...

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Kvant Math Problem 211

Verified: no Verdicts: SKIP + SKIP Solve time: 3m33s Source on kvant.digital Problem Given $n$ points, $n \gt 4$. Prove that it is possible to connect them with arrows so that from each point to any other one, it is possible to reach it by following either one arrow or two arrows (each pair of points can be connected by an arrow in only one direction; one can travel along...

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Kvant Math Problem 209

Verified: no Verdicts: SKIP + SKIP Solve time: 3m57s Source on kvant.digital Problem For any triangle $ABC$ one can compute the following sum: $$S = \tg^2 \dfrac{A}{2} + \tg^2 \dfrac{B}{2} + \tg^2 \dfrac{C}{2}.$$ Prove that $S < 2$ for all acute and right triangles; $S > 2$ for obtuse triangles whose obtuse angle is greater than $2 \arctg \dfrac{4}{3}$; among triangles with an obtuse angle $\varphi$ such that $\dfrac{\pi}{2} <...

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Kvant Math Problem 198

Verified: no Verdicts: SKIP + SKIP Solve time: 7m51s Source on kvant.digital Problem Given a parallelogram $ABCD$. On the lines $AB$ and $BC$, points $H$ and $K$ are chosen, respectively, such that the triangles $KAB$ and $HCB$ are isosceles ($KA=AB$ and $HC=CB$). Prove that the triangle $KDH$ is also isosceles. V. L. Gutenmacher Exploration The conditions place $H$ on the line $AB$ and $K$ on the line $BC$. Since $KA=AB$,...

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Kvant Math Problem 189

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Three segments $AB$, $EF$, and $CD$ pass through a single point $O$, with point $E$ lying on segment $AC$, and point $F$ lying on segment $BD$. Prove that $EF$ is shorter than at least one of the two segments — $AB$ or $CD$ (Fig. 1). Figure number 1 D. Yu. Grigoriev Exploration Consider three segments $AB$, $CD$,...

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Kvant Math Problem 187

Verified: no Verdicts: SKIP + SKIP Solve time: 5m35s Source on kvant.digital Problem On the plane, two points $A$ and $B$ are given. Find the locus of the third vertices $C$ of a triangle $ABC$ such that: the altitude $AA'$ is equal to the side $BC$; the median $AA_1$ is equal to the side $AC$; the median $AA_1$ is equal to the side $BC$; the altitude $CC'$ is equal to...

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Kvant Math Problem 176

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem To which side of triangle $ABC$ is the intersection point of its altitudes closest, if $\angle A\lt\angle B\lt\angle C$? And to which vertex? L. Schneider, 9th-grade student Exploration Let $H$ be the orthocenter of triangle $ABC$. The distance from $H$ to side $BC$ equals the length of the perpendicular from $H$ to $BC$. Since $AH$ is perpendicular...

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Kvant Math Problem 174

Verified: no Verdicts: SKIP + SKIP Solve time: 3m09s Source on kvant.digital Problem On the sides of triangle $ABC$, isosceles triangles $AB_1C$, $BA_1C$, and $AC_1B$ are constructed with these sides as bases (see Fig. 1). Prove that the perpendiculars dropped from points $A$, $B$, and $C$ onto the lines $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively, intersect at a single point. Figure 1 A. G. Geyn Exploration Consider triangle $ABC$ with isosceles...

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Kvant Math Problem 164

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be found in the published issue. On the white squares of an infinite chessboard filling the upper half-plane (Figure 1), some numbers are written so that for every black square, the sum of the numbers in the two adjacent squares to its...

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Kvant Math Problem 355

Verified: no Verdicts: SKIP + SKIP Solve time: 2m48s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M355 because the actual problem statement is not present in your message. The prompt explicitly says: “The textual version of the problem statement is being prepared. The graphical version...

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Kvant Math Problem 353

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Before proceeding, I need the precise textual statement of Kvant problem M353. Could you provide the full problem text? The graphical version alone is insufficient for a rigorous, step-by-step solution.

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Kvant Math Problem 352

Verified: no Verdicts: SKIP + SKIP Solve time: 6m05s Source on kvant.digital Problem Let $n$ be an integer for which $$n\lt(45+\sqrt{1975})^{30}\lt n+1.$$. Prove that $n$ is odd. D. K. Faddeev Exploration Write $$\alpha=45+\sqrt{1975}=45+5\sqrt{79},\qquad \beta=45-5\sqrt{79}.$$ Then $$\alpha+\beta=90,\qquad \alpha\beta=45^2-1975=50.$$ The number $\beta$ is positive and $$0<\beta=45-\sqrt{1975}<1,$$ because $44^2=1936<1975<2025=45^2$. Since $\alpha$ and $\beta$ are roots of $$x^2-90x+50=0,$$ the sequence $$u_k=\alpha^k+\beta^k$$ satisfies the recurrence $$u_{k+2}=90u_{k+1}-50u_k,$$ with $u_0=2$ and $u_1=90$. Hence every $u_k$ is an...

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Kvant Math Problem 350

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem A bishop starts moving from a white corner square of an $n\times m$ chessboard (with $n$ and $m$ greater than 1). Upon reaching the edge of the board, the bishop turns at a right angle (Fig. 2). When it reaches a corner, it stops. For which $n$ and $m$ will the bishop pass through all the white...

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Kvant Math Problem 348

Verified: no Verdicts: SKIP + SKIP Solve time: 4m51s Source on kvant.digital Problem In the table $10\times10$, the numbers from 1 to 100 are written in order. Then, in each row and each column, exactly half of the numbers are marked with a minus sign. Prove that in the resulting table, the sum of all the numbers is zero. S. M. Ageev Exploration The numbers in the table are $$\begin{matrix}...

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Kvant Math Problem 347

Verified: no Verdicts: SKIP + SKIP Solve time: 6m17s Source on kvant.digital Problem Two people play the following game. The first chooses two numbers from 1 to 25, and the second must guess them. He may name any two numbers from 1 to 25 and ask the first how many of the numbers he named—0, 1, or 2—coincide with the chosen numbers. What is the minimum number of questions with...

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Kvant Math Problem 345

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem In the sequence $197523\ldots$, each digit, starting from the fifth, is equal to the last digit of the sum of the previous four digits. Will the following appear consecutively in this sequence: the four digits 1234? the digits 1975 more than once? the digits 8197? G. A. Gurevich Exploration Let $a_n$ be the sequence, with $$a_1=1,\quad a_2=9,\quad...

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Kvant Math Problem 343

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem In a certain country, the cities are connected by roads. The length of any road is less than $500~\text{км}$, and it is possible to travel from any city to any other city via roads totaling less than $500~\text{км}$. When a single road was closed for repairs, it turned out that it was still possible to travel between...

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Kvant Math Problem 342

Verified: no Verdicts: SKIP + SKIP Solve time: 6m39s Source on kvant.digital Problem Prove that using the digits 1 and 2, one can construct $2^{n+1}$ numbers, each of which is $2^{n}$-digit long, and any two of which differ in at least $2^{n-1}$ positions. Prove that it is impossible to construct more than $2^{n+1}$ such $2^{n}$-digit numbers. S. V. Fomin Exploration The problem is a binary coding problem. Replacing the digits...

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Kvant Math Problem 340

Verified: no Verdicts: SKIP + SKIP Solve time: 3m16s Source on kvant.digital Problem A real number is written in each cell of a rectangular table. A cell of the table is called a saddle cell if the number in it is not less than all the other numbers in its column and not greater than all the other numbers in its row. Suppose it is known about a table $T$...

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Kvant Math Problem 339

Verified: no Verdicts: SKIP + SKIP Solve time: 3m59s Source on kvant.digital Problem A horizontal strip is given on the plane, bounded by two parallel lines, and $n$ lines intersecting this strip. Every two of these $n$ lines intersect within the strip, and no three pass through the same point. Consider all paths starting at the lower edge of the strip, following these lines, and ending at the upper edge,...

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Kvant Math Problem 337

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. Please provide the textual version of Kvant problem M337. Once I have the full problem statement, I will produce the rigorous solution in the requested six-section format.

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Kvant Math Problem 335

Verified: no Verdicts: SKIP + SKIP Solve time: 10m27s Source on kvant.digital Problem In a square of $7\times7$ cells, the centers of $k$ cells are marked. Moreover, no four marked points form the vertices of a rectangle with sides parallel to the sides of the square. What is the largest possible $k$ for which this is possible? Solve the same problem for a square of $13\times13$ cells. S. B. Gashkov,...

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Kvant Math Problem 334

Verified: no Verdicts: SKIP + SKIP Solve time: 4m41s Source on kvant.digital Problem Given a polynomial $P(x)$ with natural coefficients, integer coefficients. For each natural number $n$ through $a_n$, denote by $P(n)$ the sum of the digits in the decimal representation of the number $P(n)$. Prove that there exists a number that appears in the sequence $a_1$, $a_2$, $a_3$, $\ldots$ infinitely many times. I. N. Bernstein All-Union School Mathematics Olympiad...

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Kvant Math Problem 332

Verified: no Verdicts: SKIP + SKIP Solve time: 2m46s Source on kvant.digital Problem Is it possible to construct a cube of size $k\times k\times k$ from white and black cubes $1\times1\times1$ in such a way that each small cube has exactly two neighbors of the same color as itself? (Two cubes are considered neighbors if they share a common face.) A. G. Geyn All-Union Mathematical Olympiad for School Students (1975,...

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Kvant Math Problem 331

Verified: no Verdicts: SKIP + SKIP Solve time: 8m14s Source on kvant.digital Problem Triangle $A_1B_1C_1$ is obtained from triangle $ABC$ by a rotation around the center of its circumcircle by some angle less than $180^\circ$. Prove that the points of intersection of the corresponding lines: $(AB)$ and $(A_1B_1)$, $(BC)$ and $(B_1C_1)$, $(CA)$ and $(C_1A_1)$ — are the vertices of a triangle similar to triangle $ABC$. Quadrilateral $A_1B_1C_1D_1$ is obtained from...

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Kvant Math Problem 329

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem A convex $n$-gon is placed inside a square with side length 1. Prove that there exist three vertices $A$, $B$, $C$ of this $n$-gon such that the area of the triangle $ABC$ is less than $\dfrac8{n^2}$. G. Shmelev, 10th-grade student (Yaroslavl) Exploration Let the vertices of the convex $n$-gon be $P_1,P_2,\dots,P_n$ in cyclic order. Since the polygon...

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Kvant Math Problem 327

Verified: no Verdicts: SKIP + SKIP Solve time: 5m58s Source on kvant.digital Problem There are $N$ people in a group. Each of them likes exactly $k$ people in this group. For what smallest $k$ can one assert that there must exist two people in the group who like each other? A. Kolodin, 10th-grade student (Kolomyia) Exploration Consider small groups to understand how the structure of "liking" works. If $N=2$, each...

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Kvant Math Problem 326

Verified: no Verdicts: SKIP + SKIP Solve time: 6m28s Source on kvant.digital Problem A chord of a circle is at a distance $h$ from the center. In each of the segments subtended by the chord, a square is inscribed so that two adjacent vertices of the square lie on the arc and the other two lie on the chord (Fig. 1). What is the difference between the side lengths of...

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Kvant Math Problem 324

Verified: no Verdicts: SKIP + SKIP Solve time: 4m20s Source on kvant.digital Problem There are several piles of stones. Two players play a game in which a move consists of splitting each pile containing more than one stone into two smaller piles. The players move alternately until every pile contains exactly one stone. The winner is the player who makes the last move. How should the first player play if...

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Kvant Math Problem 323

Verified: no Verdicts: SKIP + SKIP Solve time: 6m29s Source on kvant.digital Problem Prove that any function defined on the entire real line can be represented as the sum of two functions, each of whose graph has a center of symmetry. V. A. Sergeev Exploration Consider a function $f:\mathbb{R}\to\mathbb{R}$. A graph has a center of symmetry if there exists a point $(a,b)$ such that for every $x$, the point symmetric...

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Kvant Math Problem 321

Verified: no Verdicts: SKIP + SKIP Solve time: 7m27s Source on kvant.digital Problem There is a rectangular table of area 1. Prove that for any $\eps\gt0$ one can specify a system of rectangular napkins covering this table (the edges of the napkins are parallel to the edges of the table) such that any subsystem consisting of pairwise non-overlapping napkins has total area less than $\eps$. A. V. Brailov Exploration The...

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Kvant Math Problem 319

Verified: no Verdicts: SKIP + SKIP Solve time: 5m08s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M319 because the actual problem statement is not present in your message. The prompt explicitly says: “The textual version of the problem statement is being prepared. The graphical version...

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Kvant Math Problem 318

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem In triangle $ABC$, the angle bisectors $[AD]$ and $[CE]$ are drawn. Prove that $|CE|\cdot|AB|=|AD|\cdot|BC|$ if and only if at least one of the following two conditions holds: $|AB|=|BC|$; $\widehat{ABC}=60^\circ$. A. P. Savin Exploration Let $$a=|BC|,\qquad b=|CA|,\qquad c=|AB|.$$ The condition is $$|CE|\cdot c=|AD|\cdot a.$$ The standard formula for the length of an internal angle bisector gives $$AD=\frac{2bc\cos \frac...

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Kvant Math Problem 316

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Prove that the sum of the squares of $k$ consecutive natural numbers cannot be a perfect square if $k$ equals 3, 5, 7, or 9. Find 11 consecutive natural numbers whose sum of squares is a perfect square. E. G. Gotman Exploration Consider the sum of squares of $k$ consecutive natural numbers beginning at $n$, expressed as...

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Kvant Math Problem 314

Verified: no Verdicts: SKIP + SKIP Solve time: 8m35s Source on kvant.digital Problem Among all 9-digit numbers whose decimal representation contains no 0, find the number for which the difference between the number itself and the product of its digits is: the smallest; the largest. What will be the answer for $n$-digit numbers for any $n$? A. G. Leiderman Exploration Consider the difference between a number and the product of...

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Kvant Math Problem 313

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem Given an angle with vertex $O$. Consider the set of fourth vertices $M$ of parallelograms $ONML$, whose vertices $N$ and $L$ lie on the sides of the given angle, and whose area is a constant. (This set is called a hyperbola .) Prove that on the bisector of this angle and its extension there exist points $F_1$...

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Kvant Math Problem 311

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Starting from a single bacterium, 1000 bacteria were obtained as follows: first the bacterium divided into two; then one of the two resulting bacteria divided into two; then one of the three resulting bacteria divided into two; and so on. Prove that at some moment there existed a bacterium such that the number of its descendants among...

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Kvant Math Problem 309

Verified: no Verdicts: SKIP + SKIP Solve time: 7m32s Source on kvant.digital Problem For which $n$ is the polynomial $x^{2n}+x^n+1$ divisible by $x^2+x+1$? For which $n$ is the number $1\underbrace{0\ldots0}_n1\underbrace{0\ldots0}_n1$ divisible by 37? V. G. Shleifer Exploration For the first question, divisibility by $x^2+x+1$ suggests evaluating the polynomial at the nonreal cube roots of unity. Let $\omega^3=1$ and $\omega\neq1$. Then $\omega^2+\omega+1=0$, and $x^2+x+1$ divides a polynomial $P(x)$ if and only...

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Kvant Math Problem 308

Verified: no Verdicts: SKIP + SKIP Solve time: 3m30s Source on kvant.digital Problem If for any $x$ $$a_1\cos x+a_2\cos2x+\ldots+a_n\cos nx\ge-1,$$, then for the numbers $a_1$, $a_2$, $\ldots$, $a_n$ the inequality $$a_1+a_2+\ldots+a_n\le n.$$ holds. Prove this statement for $n=2$; for $n=3$; for any natural $n$. Yu. I. Ionin Exploration Consider first the case $n=2$, where the inequality takes the form $a_1\cos x + a_2\cos 2x \ge -1$ for all real $x$....

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Kvant Math Problem 306

Verified: no Verdicts: SKIP + SKIP Solve time: 6m49s Source on kvant.digital Problem From a chessboard ($8\times8$), one corner square ($1\times1$) has been removed (Fig. 1). Into what smallest number of equal-area triangles can the remaining part of the board be dissected? Figure 1 V. P. Fedotov Exploration Let the removed corner be the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. The remaining region has area $$64-1=63.$$ If it...

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Kvant Math Problem 304

Verified: no Verdicts: SKIP + SKIP Solve time: 7m53s Source on kvant.digital Problem Let us denote by a circle some (as yet unknown) operation applicable to any two nonnegative integers $a$ and $b$ and yielding as a result another nonnegative integer $a\circ b=c$. Suppose that the operation “$\circ$” satisfies the following conditions: $a\circ b=b\circ a$; if $a\circ b=c$, then $b\circ c=a$; if $a\circ b\gt c$, then $b\circ c\lt a$ or...

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Kvant Math Problem 303

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem A rectangle $300\times1000$ is divided into squares $1\times1$, and identical weights are placed at some 30 vertices of the squares. Prove that it is possible to select two disjoint groups of weights—no more than 10 in each—such that their centers of gravity coincide. A. V. Sherstyuk Exploration Consider placing a small number of identical weights on the...

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Kvant Math Problem 301

Verified: no Verdicts: SKIP + SKIP Solve time: 7m43s Source on kvant.digital Problem On the plane, $2n$ points are given—$n$ blue and $n$ red, with no three points lying on the same line. Prove that it is possible to draw $n$ line segments so that each segment has one endpoint at a red point and the other at a blue point, and no two segments have any point in common....

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Kvant Math Problem 299

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem For which $n$ can a regular $n$-gon be placed on a ruled sheet of paper so that all its vertices lie on the lines? (The lines are parallel and equally spaced.) N. B. Vasilyev Exploration Consider a ruled sheet of paper with parallel lines spaced a fixed distance apart, and suppose a regular $n$-gon is drawn so...

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Kvant Math Problem 297

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem On the plane, 12 points are given that are the vertices of four squares $A_1B_1A_2C_1$, $A_2C_2A_3B_2$, $A_3B_3A_4C_3$, and $A_4C_4A_1B_4$ (the vertices of each square are listed in clockwise order). Prove that $B_1B_2B_3B_4$ and $C_1C_2C_3C_4$ are congruent parallelograms, one of which is obtained from the other by a rotation through $90^\circ$ (these parallelograms may be degenerate: in that...

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Kvant Math Problem 296

Verified: no Verdicts: SKIP + SKIP Solve time: 4m57s Source on kvant.digital Problem In the table $n\times n$, $n^2$ numbers are written, the sum of which is non-negative. Prove that it is possible to rearrange the columns of the table so that the sum of $n$ numbers along the diagonal running from the bottom-left corner to the top-right corner is non-negative. A. Tupanov Exploration Let the rows be numbered from...

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Kvant Math Problem 294

Verified: no Verdicts: SKIP + SKIP Solve time: 7m45s Source on kvant.digital Problem Prove that if $a$, $b$, $c$, $d$, $x$, $y$, $u$, $v$ are real numbers and $abcd\gt0$, then $$\begin{gathered} (ax+bu)(av+by)(cx+dv)(cu+dy)\ge\ \ge(acuvx+bcuxy+advxy+bduvy)(acx+bcu+adv+bdy). \end{gathered}$$ V. P. Fedotov Exploration The inequality is homogeneous and symmetric in a suggestive way. Setting $$a=b=c=d=1$$ gives $$(x+u)(v+y)(x+v)(u+y)\ge (uvx+uxy+vxy+uvy)(x+u+v+y).$$ A direct expansion yields $$(x+u)(v+y)(x+v)(u+y) -(uvx+uxy+vxy+uvy)(x+u+v+y) =(xy-uv)^2.$$ This strongly suggests that in the general case the difference...

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Kvant Math Problem 292

Verified: no Verdicts: SKIP + SKIP Solve time: 8m18s Source on kvant.digital Problem The numbers from 1 to 50 are written on the board. It is allowed to erase any two numbers and write in their place a single number equal to the absolute value of their difference. After repeating this procedure several times, only one number remains on the board. What could this number be? F. G. Shleifer Exploration...

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Kvant Math Problem 290

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem For which $n$ does there exist a closed non-self-intersecting broken line consisting of $n$ segments such that any line containing one of the segments of this broken line contains at least one more of its segments? S. V. Konyagin Exploration Let the closed non-self-intersecting broken line have vertices $$A_1,A_2,\dots,A_n,$$ with sides $$A_1A_2,A_2A_3,\dots,A_nA_1.$$ The condition says that whenever...

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Kvant Math Problem 288

Verified: no Verdicts: SKIP + SKIP Solve time: 7m36s Source on kvant.digital Problem Scientists have gathered at a congress, and among them there are friends. It turned out that any two scientists who have the same number of friends at the congress have no common friends. Prove that there exists a scientist who has exactly one friend. S. V. Konyagin Exploration Model the congress by a simple graph. Vertices are...

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Kvant Math Problem 287

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Does there exist a sequence of natural numbers such that every natural number 1, 2, 3, $\ldots$ can be represented as the difference of two numbers from this sequence in a unique way? A. Lifshitz Exploration Consider a sequence of natural numbers $a_1 < a_2 < a_3 < \dots$ such that every natural number $n$ can be...

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Kvant Math Problem 285

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem A rectangular sheet of paper of dimensions $a \times b$ has been cut into rectangular strips, each of which has one side of length 1. Prove that at least one of the numbers $a$ and $b$ is an integer. A. V. Klimov Moscow Mathematical Olympiad (1974) Exploration Consider small examples to understand the claim. A rectangle of...

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Kvant Math Problem 283

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem A convex polygon has the following property: if all its sides are shifted outward by one unit, then the resulting lines form a polygon similar to the original one. Prove that a circle can be inscribed in this polygon. Moscow Mathematical Olympiad (1974) Exploration Consider small examples of convex polygons, starting with triangles and quadrilaterals, and examine...

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Kvant Math Problem 282

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem In the cells of a rectangular table of size $m \times n$, arbitrary natural numbers are written. In one move, it is allowed to either double all the numbers in a single row or subtract one from all the numbers in a single column. Prove that, after a finite number of moves, it is possible to make...

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Kvant Math Problem 280

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem Given a triangle $ABC$ of area 1. Let $A_1$, $B_1$, and $C_1$ be the midpoints of the sides $BC$, $AC$, and $AB$ respectively. What is the minimum possible area of the intersection of the triangles $A_1B_1C_1$ and $KLM$ if the points $K$, $L$, and $M$ lie on the segments $AB_1$, $CA_1$, and $BC_1$ respectively? B. M. Ivlev...

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Kvant Math Problem 279

Verified: no Verdicts: SKIP + SKIP Solve time: 3m59s Source on kvant.digital Problem On $n$ cards arranged in a circle, numbers are written, each of which is equal to either $+1$ or $-1$. What is the smallest number of questions that is sufficient to determine with certainty the product of all $n$ numbers, if in one question it is allowed to find out the product of the numbers on any...

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Kvant Math Problem 277

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem Several red points and several blue points are given. Some pairs of points are connected by line segments. Call a point special if more than half of the points connected to it have a color different from its own. If there is at least one special point, then an arbitrary special point is chosen and recolored to...

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Kvant Math Problem 275

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem On the plane, there are $n$ vectors, each of length 1. The sum of all $n$ vectors is the zero vector. Prove that the vectors can be numbered such that for all $k = 1$, 2, $\ldots$, $n$ the following condition holds: the sum of the first $k$ vectors has length at most 3. Prove a similar...

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Kvant Math Problem 274

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem Find the smallest number of the form $|11^k - 5^l|$; $|36^k - 5^l|$; $|53^k - 37^l|$, where $k$ and $l$ are natural numbers. F. G. Schleifer All-Union Mathematical Olympiad for School Students (1974, Grades 8 and 9) Exploration We seek the smallest positive value attained by the given differences. For the first expression, compute a few values:...

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Kvant Math Problem 272

Verified: no Verdicts: SKIP + SKIP Solve time: 7m25s Source on kvant.digital Problem Two circles with radii $R$ and $r$ are given, externally tangent to each other. Various trapezoids $ABCD$ are constructed so that each circle touches both lateral sides and one of the bases of the trapezoid. Find the minimal possible length of the lateral side $AB$. E. V. Sallinen All-Union Mathematical Olympiad for School Students (1974, 9th grade)...

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Kvant Math Problem 270

Verified: no Verdicts: SKIP + SKIP Solve time: 7m37s Source on kvant.digital Problem Let $AB$ and $CD$ be two chords of a circle, and let the points $K$ and $H$ be constructed so that all four angles $KAB$, $KCD$, $HBA$, and $HDC$ are right angles. Prove that the line $KH$ passes through the center of the circle and through the point of intersection of the lines $AD$ and $BC$. I....

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Kvant Math Problem 269

Verified: no Verdicts: SKIP + SKIP Solve time: 5m15s Source on kvant.digital Problem Let $T_k(n)$ denote the sum of products of $k$ numbers from 1 to $n$. For example, $$T_2(4) = 1 \cdot 2 + 1 \cdot 3 + 1 \cdot 4 + 2 \cdot 4 + 3 \cdot 4 = 35.$$ Find a general formula for $T_2(n)$ and $T_3(n)$. Prove that $T_k(n)$ can be expressed as a polynomial in...

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Kvant Math Problem 267

Verified: no Verdicts: SKIP + SKIP Solve time: 6m37s Source on kvant.digital Problem In the sequence of triples of integers (2, 3, 5), (6, 15, 10), $\ldots$ each triple is obtained from the previous one as follows: the first number is multiplied by the second, the second by the third, and the third by the first, and the resulting products form the new triple. Prove that none of the numbers...

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Kvant Math Problem 265

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem The diagonal $d$ of a rectangular parallelepiped forms angles $\alpha$, $\beta$, and $\gamma$ with its edges $a$, $b$, and $c$. Prove that $\alpha + \beta + \gamma$ is less than $\pi$. M. L. Gerver Exploration Consider a rectangular parallelepiped with edges of length $a$, $b$, and $c$. Its main diagonal $d$ has length $\sqrt{a^2 + b^2 +...

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Kvant Math Problem 264

Verified: no Verdicts: SKIP + SKIP Solve time: 3m40s Source on kvant.digital Problem Fig. 3 Fig. 4 A city has one blue square and $n$ green squares. Each green square is connected by streets to the blue square and to two green squares (Fig. 3). On each of the $2n$ streets, one-way traffic was introduced in such a way that every square can be reached and can be departed from....

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Kvant Math Problem 262

Verified: no Verdicts: SKIP + SKIP Solve time: 8m52s Source on kvant.digital Problem What is the largest number of: rooks queens that can be placed on a chessboard $8 \times 8$ so that each of these pieces is attacked by no more than one of the others? V. Rabinovich, L. Rabinovich Exploration The problem asks for the maximal number of rooks or queens on an $8 \times 8$ chessboard such...

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Kvant Math Problem 260

Verified: no Verdicts: SKIP + SKIP Solve time: 6m30s Source on kvant.digital Problem A circle is divided by points $A_1$, $A_2$, $\ldots$, $A_n$ into $n$ equal parts, each colored in some color. Two arcs (with endpoints at the division points) are called equally colored if, under some rotation of the circle, one coincides entirely with the other, including the color of each segment. (For example, in the figure, the two...

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Kvant Math Problem 259

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem A polygon is circumscribed around a circle. The points where its sides touch the circle serve as the vertices of a second polygon inscribed in this circle. Prove that the product of the distances from an arbitrary point $M$ on the circle to the sides of one polygon equals the product of the distances from this point...

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Kvant Math Problem 257

Verified: no Verdicts: SKIP + SKIP Solve time: 4m06s Source on kvant.digital Problem For which natural numbers $n \ge 2$ does the inequality $$x_1^2 + x_2^2 \ldots + x_n^2 \ge p(x_1x_2 + x_2x_3 + \ldots + x_{n-1}x_n)$$ hold for all real numbers $x_1$, $x_2$, $\ldots$, $x_n$ if $p=1$; $p=\dfrac{4}{3}$; $p=\dfrac{6}{5}$? Nguyen Cong Quy (Hanoi) Exploration The inequality can be written as $$Q_n(x_1,\ldots,x_n) = \sum_{i=1}^n x_i^2 - p\sum_{i=1}^{n-1}x_ix_{i+1} \ge 0.$$ We...

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Kvant Math Problem 256

Verified: no Verdicts: SKIP + SKIP Solve time: 6m13s Source on kvant.digital Problem A polygon is circumscribed around a circle. The points where its sides touch the circle serve as the vertices of a second polygon inscribed in this circle. Prove that the product of the distances from an arbitrary point $M$ on the circle to the sides of the first polygon is equal to the product of the distances...

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Kvant Math Problem 254

Verified: no Verdicts: SKIP + SKIP Solve time: 7m16s Source on kvant.digital Problem Compute the value of $\sqrt{0{,}\smash{\underbrace{11111\ldots11111}_{100}}}\vphantom{\underbrace0_0}$ to an accuracy of 100 digits after the decimal point; 101 digits after the decimal point; 200 digits after the decimal point. A. A. Egorov Exploration Consider small cases of numbers of the form $0.1$, $0.11$, $0.111$, $0.1111$, and so on. Denote by $S_n$ the number $0.\underbrace{111\ldots1}_{n}$ with $n$ ones after the...

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Kvant Math Problem 252

Verified: no Verdicts: SKIP + SKIP Solve time: 7m14s Source on kvant.digital Problem On the plane lies a regular octagon with side $a$. It is allowed to "roll" the octagon across the plane, flipping it (symmetrically reflecting) over any of its sides. Prove that for any point $M$ on the plane and any $\varepsilon\gt0$, the octagon can be rolled into a position such that its center is at a distance...

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Kvant Math Problem 250

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem At the court of King Arthur, $n$ knights have gathered. Some of them are enemies of one another, but each knight has at least $\dfrac{n}{2}$ friends among those present. Prove that Merlin, King Arthur’s adviser, can seat the knights around a round table so that each knight is seated next to his friends. Prove that if each...

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Kvant Math Problem 249

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem On the edges $A'D'$ and $C'D'$ of the cube $ABCDA'B'C'D'$, choose two points $K$ and $M$ so that the plane $KDM$ is tangent to the sphere inscribed in the cube (Fig. 3). Prove that the measure $\phi$ of the dihedral angle along the edge $B'D$ of the tetrahedron $B'DKM$ does not depend on the choice of the...

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Kvant Math Problem 247

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem The square $6 \times 6$ needs to be filled with 12 tiles, of which $k$ are L-shaped, and the remaining $12-k$ are rectangular (see Fig. 1). For which $k$ is this possible? Figure 1 E. Turkevich Exploration A $6 \times 6$ square contains $36$ unit squares. We are to tile it with $12$ pieces, each covering $3$...

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Kvant Math Problem 245

Verified: no Verdicts: SKIP + SKIP Solve time: 6m13s Source on kvant.digital Problem It is proposed to construct $N$ points in the plane so that all pairwise distances between them are equal to prescribed numbers: for every two points $M_i$, $M_j$, the distance $|M_iM_j|=r_{ij}$ is given (here $i$ and $j$ are any numbers from 1 to $N$). Is the construction always possible if the distances $r_{ij}$ are specified in such...

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Kvant Math Problem 244

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem Two sets of $n$ real numbers are given: $a_1$, $a_2$, $\ldots$, $a_n$ and $b_1$, $b_2$, $\ldots$, $b_n$. Prove that if at least one of the following two conditions holds: $a_i\lt a_j$ implies $b_i\le b_j$; $a_i\lt a\lt a_j$, where $a=\dfrac{a_1+a_2+\ldots+a_n}n$, implies $b_i\le b_j$; then the inequality $$n(a_1b_1+a_2b_2+\ldots+a_nb_n)\ge(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n).$$ holds. A. Grigoryan, 10th-grade student (Baku) Exploration The desired inequality can...

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Kvant Math Problem 242

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Let $A_iH_i$ be the altitude and $A_iM_i$ the median drawn from vertex $A_i$ of an acute-angled triangle $A_1A_2A_3$ ($i=1$, 2, 3). Prove that one of the three products $|H_1M_1|\cdot|A_2A_3|$, $|H_2M_2|\cdot|A_3A_1|$, $|H_3M_3|\cdot|A_1A_2|$ equals the sum of the other two. Is this statement also true for right-angled and obtuse-angled triangles? S. Salnikov, 10th-grade student (Mary) Exploration Denote the sides...

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Kvant Math Problem 239

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem On the plane, two points $A$ and $B$ are given. Let $C$ be a point equidistant from $A$ and $B$. Construct a sequence of points $C_1=C$, $C_2$, $C_3$, $\ldots$, $C_n$, $C_{n+1}$, $\ldots$ where $C_{n+1}$ is the circumcenter of triangle $AC_nB$. For which position of the point $C$ does the point $C_n$ fall at the midpoint of segment...

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Kvant Math Problem 238

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem Prove that $$C_n^1+C_n^3\cdot1973+C_n^5\cdot1973^2+C_n^7\cdot1973^3+\ldots$$ is divisible by $2^{n-1}$. (Here $C_n^k$ are the coefficients of the polynomial $$(a+b)^n=\textstyle\sum\limits_{k=0}^n C_n^k a^k b^{n-k}.\Big)$$ F. G. Shleifer Exploration Let $$S=C_n^1+C_n^3\cdot1973+C_n^5\cdot1973^2+\cdots.$$ The exponent of $1973$ increases by $1$ whenever the upper index of the binomial coefficient increases by $2$. Thus $$S=\sum_{m\ge0} C_n^{2m+1}1973^m.$$ The appearance of only odd binomial coefficients suggests separating the odd...

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Kvant Math Problem 236

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem There are 51 two-digit numbers. Prove that from these numbers one can select at least 6 numbers such that no two of the selected numbers have the same digit in any position. Natural numbers $k$ and $n$, $1\lt k\lt n$ are given. For what smallest $m$ is the following statement true: for any arrangement of $m$ rooks...

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Kvant Math Problem 235

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem A lion runs around the arena of a circular circus of radius 10 m. Moving along a polygonal path, it has run 30 km. Prove that the sum of all the angles through which it turned (Fig. 3) is at least 2998 radians. Fig. 3 I. N. Bernstein Exploration Consider a lion moving along a polygonal path...

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Kvant Math Problem 233

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem At the ends of a segment, two ones are written. In the middle between them, their sum — the number 2 — is written. Then, in the middle between each pair of neighboring written numbers, their sum is written again, and so on — 1973 times (Fig. 1). How many times will the number 1973 be written?...

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Kvant Math Problem 231

Verified: no Verdicts: SKIP + SKIP Solve time: 7m59s Source on kvant.digital Problem Find all solutions in natural numbers of the equation $$n^x+n^y=n^z.$$ R. Egoryan, 9th-grade student Exploration Consider the equation $n^x + n^y = n^z$ in natural numbers. First, observe symmetry in $x$ and $y$, allowing us to assume without loss of generality that $x \le y$. Test small values of $n$. For $n=2$, one sees that $2^1 +...

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Kvant Math Problem 230

Verified: no Verdicts: SKIP + SKIP Solve time: 6m41s Source on kvant.digital Problem Prove that from any convex equilateral (but not necessarily regular) pentagon one can cut out an equilateral triangle, one of whose sides coincides with a side of the pentagon (Fig. 2). Figure 2 S. V. Konyagin Exploration Let the side length of the equilateral pentagon be $1$, and let its consecutive vertices be $A_1,A_2,A_3,A_4,A_5$. The statement asks...

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Kvant Math Problem 228

Verified: no Verdicts: SKIP + SKIP Solve time: 4m35s Source on kvant.digital Problem A sheet of squared paper of size $n\times n$ cells was colored using $n$ colors (each cell was either colored with one of these colors or left uncolored). A coloring is called proper if no two cells of the same color occur in any row or in any column. Is it always possible to complete the coloring...

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Kvant Math Problem 226

Verified: no Verdicts: SKIP + SKIP Solve time: 6m11s Source on kvant.digital Problem Three grasshoppers are located at three vertices of a square. They are playing leapfrog. If grasshopper $A$ jumps over grasshopper $B$, then after the jump it ends up at the same distance from $B$ (but, naturally, on the other side and on the same line; Fig. 1). Can one of the grasshoppers reach the fourth vertex of...

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Kvant Math Problem 225

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem The faces of a die are numbered 1, 2, 3, 4, 5, 6 so that the numbers on opposite faces sum to 7. The die is rolled from the lower left corner to the upper right corner of a chessboard of size $50\times50$ squares (each square of the board is equal in size to a face of...

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Kvant Math Problem 223

Verified: no Verdicts: SKIP + SKIP Solve time: 4m03s Source on kvant.digital Problem A natural number is called perfect if it is equal to the sum of all its divisors except the number itself. (For example, 28 is a perfect number: $28=1+2+4+7+14$.) Prove that a perfect number cannot be a perfect square. A. Makarichev, 9th-grade student Exploration Perfect numbers are rare and highly structured. The first few examples are $6$,...

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Kvant Math Problem 221

Verified: no Verdicts: SKIP + SKIP Solve time: 8m51s Source on kvant.digital Problem A blot was placed on a sheet of paper. For each point of the blot, the smallest distance to the boundary of the blot and the largest distance to the boundary of the blot were determined (the figure shows the largest and smallest distances to the boundary for the point $K$). Among all the smallest distances, the...

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Kvant Math Problem 220

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem A king traversed a chessboard $8\times8$, visiting each square exactly once and returning to the starting square on the final move. (The king moves according to the usual rules: in one move, it can go to any adjacent square horizontally, vertically, or diagonally.) When its path was drawn by consecutively connecting the centers of the squares it...

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Kvant Math Problem 218

Verified: no Verdicts: SKIP + SKIP Solve time: 4m56s Source on kvant.digital Problem Prove that if $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ are positive numbers, then $$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 + x_5x_1).$$ B. D. Ginzburg All-Union Mathematical Olympiad for School Students (VII, Grade 10) Exploration We are asked to compare the square of a sum of five positive...

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Kvant Math Problem 217

Verified: no Verdicts: SKIP + SKIP Solve time: 5m54s Source on kvant.digital Problem A convex $n$-gon with pairwise nonparallel sides and a point inside it are given. Prove that no more than $n$ lines can be drawn through this point, each of which divides the area of the $n$-gon into two equal parts. E. V. Sallinen All-Union Mathematical Olympiad for School Students (VII, Grade 9) Exploration Consider first a triangle,...

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Kvant Math Problem 215

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem On an infinite square-grid sheet of white paper, $n$ cells are colored black. At times $t = 1$, 2, $\ldots$, all cells of the sheet are recolored simultaneously according to the following rule: each cell $k$ acquires the color that the majority of the following three cells had at the previous moment: the cell $k$ itself and...

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Kvant Math Problem 213

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Given an angle with vertex $O$ and a circle tangent to its sides at points $A$ and $B$. From point $A$, a ray is drawn parallel to $OB$, intersecting the circle at point $C$. Segment $OC$ intersects the circle at point $E$, and the lines $AE$ and $OB$ intersect at point $K$. Prove that $OK = KB$....

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Kvant Math Problem 212

Verified: no Verdicts: SKIP + SKIP Solve time: 5m03s Source on kvant.digital Problem At a trial, 14 coins are presented as physical evidence. An expert has determined that seven of them are counterfeit and the remaining seven are genuine, and moreover has identified exactly which coins are counterfeit and which are genuine. The court knows only that all counterfeit coins have the same weight, all genuine coins have the same...

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Kvant Math Problem 210

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem Consider sequences consisting of 3000 digits, each either 1 or 2. In such a sequence, it is allowed to swap any two adjacent triples of digits. Two sequences are called equivalent if one can be transformed into the other by a finite number of such swaps. How many inequivalent sequences exist in total? G. A. Gurevich Exploration...

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Kvant Math Problem 208

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem It is known that the difference between the largest and the smallest of the real numbers $$x_1, \ x_2, \ x_3, \ \ldots, \ x_{10}$$ is equal to 1. What can be the greatest, least possible difference between the largest and the smallest of the 10 numbers $$x_1, \ \frac{x_1 + x_2}{2}, \ \frac{x_1 + x_2 +...

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Kvant Math Problem 207

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Given two triangles $A_1 A_2 A_3$ and $B_1 B_2 B_3$. Construct around triangle $A_1 A_2 A_3$ a triangle $M_1 M_2 M_3$ of maximum area similar to triangle $B_1 B_2 B_3$ (with vertex $A_1$ lying on line $M_2 M_3$, vertex $A_2$ on line $M_3 M_1$, and vertex $A_3$ on line $M_1 M_2$). N. D. Nagaev Exploration Consider the...

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Kvant Math Problem 205

Verified: no Verdicts: SKIP + SKIP Solve time: 8m34s Source on kvant.digital Problem 24 students solved 25 problems. The instructor has a table $24 \times 25$ recording which problems each student solved. It turned out that every problem was solved by at least one student. Prove that it is possible to mark some problems with a “check” so that each student solved an even number (possibly zero) of the marked...

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Kvant Math Problem 204

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem Let us call a positive integer good if the digits 1973 occur consecutively in its decimal representation, and bad otherwise. (For example, the number $197 \ 639 \ 917$ is bad, while $116 \ 519 \ 732$ is good.) Prove that there exists a positive integer $n$ such that among all $n$-digit numbers (from $10^{n-1}$ to $10^{n}-1$),...

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Kvant Math Problem 202

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Prove that from the sequence $$a,\ a + d,\ a + 2d,\ \ldots,\ a + nd,\ \ldots,$$ which is an infinite arithmetic progression $(d \neq 0)$, one can select a subsequence forming an infinite geometric progression if and only if the ratio $a/d$ is rational. N. B. Vasilyev Exploration Let the arithmetic progression be $$a,a+d,a+2d,\ldots ,$$ with...

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Kvant Math Problem 200

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem a) Figure 3 shows six points, with three points lying on each of four lines. Prove that there are 24 different ways to map this set of six points onto itself so that every triple of points lying on one line is mapped to a triple of points that also lie on one line. Figure 3 b)...

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Kvant Math Problem 199

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Prove that the sum $$C_n^0-C_{n-1}^1\cdot\dfrac14+C_{n-2}^2\cdot\dfrac1{4^2}-\ldots+(-1)^i C_{n-i}^i\cdot\frac1{4^i}+\ldots$$ $\Big($the sum is taken over all integers $i$, $0\le i\le\dfrac n2\Big)$ is equal to $\dfrac{n+1}{2^n}$. Prove that if $p$ and $q$ are distinct numbers and $p+q=1$, then the sum $$C_n^0-C_{n-1}^1pq+C_{n-2}^2p^2q^2-\ldots+(-1)^i C_{n-i}^i p^iq^i+\ldots,$$ analogous to the previous one is equal to $\dfrac{p^{n+1}-q^{n+1}}{p-q}$ for arbitrary $n$. Here $C_n^k$ are binomial coefficients, i.e., $C_n^0=1$...

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Kvant Math Problem 197

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem In a rectangular table with $m$ rows and $n$ columns, $mn$ arbitrary positive numbers are written. Let us compute the product of the numbers in each column and then the sum $S$ of all $n$ such products. Prove that if the numbers in each row are rearranged in ascending order, the sum $S$ of the new table...

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Kvant Math Problem 195

Verified: no Verdicts: SKIP + SKIP Solve time: 6m46s Source on kvant.digital Problem Given a triangle $ABC$. How many points $D$ exist such that the perimeters of the quadrilaterals $ADBC$, $ABDC$, and $ABCD$ are equal? M. L. Gerver Exploration Let $$P_1=\operatorname{per}(ADBC),\qquad P_2=\operatorname{per}(ABDC),\qquad P_3=\operatorname{per}(ABCD).$$ Writing the perimeters explicitly, $$P_1=AD+DB+BC+CA,$$ $$P_2=AB+BD+DC+CA,$$ $$P_3=AB+BC+CD+DA.$$ The condition is $$P_1=P_2=P_3.$$ Subtracting pairwise gives linear relations among the distances from $D$ to the vertices. From $P_1=P_2$, $$AD+BC=AB+DC.$$...

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Kvant Math Problem 193

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Prove that the sum of the areas of the five triangles formed by pairs of sides and diagonals of a convex pentagon (see Fig. 1) is greater than the area of the entire pentagon. Figure 1 N. B. Vasiliev Exploration Consider a convex pentagon $ABCDE$ with vertices labeled consecutively. The problem involves the five triangles formed by...

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Kvant Math Problem 192

Verified: no Verdicts: SKIP + SKIP Solve time: 7m55s Source on kvant.digital Problem The numbers 1, 2, 3, ..., 1000 are given. Find the greatest $m$ with the following property: no matter which $m$ of the given numbers are crossed out, among the remaining $1000-m$ numbers there will be two such that one is divisible by the other. Exploration Consider small analogues first. For numbers $1$ through $8$, we want...

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Kvant Math Problem 190

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem On the plane, two lines $a$ and $b$ are given. At point $A_1$, located on line $a$ at a distance less than 1 from line $b$, a flea sits. Then the flea sequentially jumps to points $B_1$, $A_2$, $B_2$, $A_3$, $B_3$, $\ldots$, following these rules (Fig. 2): points $A_1$, $A_2$, $A_3$, $\ldots$ lie on line $a$, points...

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Kvant Math Problem 188

Verified: no Verdicts: SKIP + SKIP Solve time: 5m32s Source on kvant.digital Problem Air service has been established between some of $2n$ cities, and each city is connected to at least $n$ others (by non-stop flights). Prove that even if any $n-1$ flights are canceled, it will still be possible to travel by air (with transfers) from any city to any other. Determine all cases in which this “connectivity” is...

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Kvant Math Problem 186

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem Find all integer solutions, different from 1, of the equation $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$$. V. Slepchenko, 10th-grade student Exploration We seek all integer triples $(x,y,z)$, none equal to $1$, satisfying $$\frac1x+\frac1y+\frac1z=1.$$ Since the equation is symmetric, we may assume $$|x|\le |y|\le |z|.$$ First consider whether any variable can be negative. If two are negative...

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Kvant Math Problem 185

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem On a coat of area 1, there are 5 patches, each of area at least $\dfrac{1}{2}$. Prove that there exist two patches whose overlap has area at least $\dfrac{1}{5}$. E. B. Dynkin Exploration Consider a coat of area $1$ and five patches, each of area at least $\frac{1}{2}$. The sum of the areas of all patches is...

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Kvant Math Problem 183

Verified: no Verdicts: SKIP + SKIP Solve time: 5m41s Source on kvant.digital Problem Find the height of a trapezoid whose bases are $a$ and $b$ ($a \lt b$), the angle between its diagonals is $90^{\circ}$, and the angle between the extensions of its lateral sides is $45^{\circ}$. Exploration Let the trapezoid have bases of lengths $b$ and $a$, with $a<b$. Put the longer base on the $x$-axis. Let the lower...

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Kvant Math Problem 182

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Prove that if $a >0$, $b>0$, and $c>0$, then $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b} \ge \dfrac{3}{2}$. 2. $a >0$, $b>0$, $c>0$, and $d>0$, then $\dfrac{a}{b+c+d} + \dfrac{b}{a+c+d} + \dfrac{c}{a+b+d} + \dfrac{d}{a+b+c} \ge \dfrac{4}{3}$. 3. $a_1$, $a_2$, $a_3$, ..., $a_n$ are positive numbers ($n \ge 2$), then $\dfrac{a_1}{a_2+a_3+...+a_n} + \dfrac{a_2}{a_1+a_3+...+a_n} + ... + \dfrac{a_n}{a_1+a_2+...+a_{n-1}} \ge \dfrac{n}{n-1}$. Exploration Begin...

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Kvant Math Problem 180

Verified: no Verdicts: SKIP + SKIP Solve time: 4m38s Source on kvant.digital Problem Two people play the following game. One thinks of a positive integer $n$, and the other asks questions of the form “is it true that $n \ge x$?” (he may choose $x$ at his discretion) and receives the answers “yes” or “no.” To each possible strategy $T$ of the second player, associate a function $f_T(n)$ equal to...

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Kvant Math Problem 620

Verified: no Verdicts: SKIP + SKIP Solve time: 9m40s Source on kvant.digital Problem Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers such that $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that the sum of the absolute values of $2^n$ numbers $$\pm x_1 \pm x_2 \pm \ldots \pm x_n$$ (with all possible combinations of the signs “$+$” and “$-$”) does not exceed $2^n$. I. Kasakovskikh Exploration Consider small values of $n$ first. For $n=1$, the numbers...

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Kvant Math Problem 621

Verified: no Verdicts: SKIP + SKIP Solve time: 5m31s Source on kvant.digital Problem A $n$-gon is circumscribed about a circle. An arbitrary point $P$ inside the circle is joined to all its vertices and points of tangency. The resulting $2n$ triangles are colored alternately red and blue. Prove that the product of the areas of the red triangles is equal to the product of the areas of the blue triangles....

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Kvant Math Problem 179

Verified: no Verdicts: SKIP + SKIP Solve time: 8m24s Source on kvant.digital Problem For each non-right triangle $T$, let $T_1=H(T)$ denote the triangle whose vertices are the feet of the altitudes of triangle $T_1$; let $T_2=H(T_1)$ denote the triangle whose vertices are the feet of the altitudes of triangle $T_1$; further, let $T_3=H(T_2)$; $T_4=H(T_3)$, $\ldots$ What must the angles of triangle $T$ be so that the triangle $H(T)$ is acute?...

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Kvant Math Problem 619

Verified: no Verdicts: SKIP + SKIP Solve time: 6m58s Source on kvant.digital Problem Prove that if, for the cyclic quadrilateral $ABCD$, the equality $|CD|=|AD|+|BC|$ holds, then the bisectors of its angles $A$ and $B$ intersect on the side $CD$. I. F. Sharygin Exploration Let the bisectors of $\angle A$ and $\angle B$ meet at a point $P$. Since the quadrilateral is cyclic, opposite angles satisfy $$\angle A+\angle C=180^\circ,\qquad \angle B+\angle...

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Kvant Math Problem 618

Verified: no Verdicts: SKIP + SKIP Solve time: 3m22s Source on kvant.digital Problem Prove that there exist infinitely many natural numbers $n$ such that $n!$ is divisible by $n^2+1$. Prove that for any $\alpha \gt 0$ there exist infinitely many natural numbers $n$ such that $[\alpha n]!$ is divisible by $n^2+1$. (Here $m!=1\cdot 2\cdot \ldots \cdot m$; $[x]$ is the integer part of the number $x$.) A. Sivatsky, 10th grade...

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Kvant Math Problem 554

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem Call a natural number $n$ good if there exist natural numbers $a_1$, $a_2$, $\ldots$, $a_k$ (not necessarily distinct) such that $$a_1+a_2+\ldots+a_k=n$$ and $$\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_k}=1.$$ It is known that all numbers between 33 and 73 are good. Prove that all numbers greater than 73 are also good. USA Mathematical Olympiad (1978) Exploration Consider small examples of natural numbers and...

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Kvant Math Problem 550

Verified: no Verdicts: SKIP + SKIP Solve time: 9m40s Source on kvant.digital Problem From point $A$ to point $B$, which are $d$ km apart, $n$ cyclists must travel. They have $m$ bicycles available. Each person can walk at a speed of $u~\text{км/ч}$ or ride a bicycle at a speed of $v~\text{км/ч}$. What is the minimum time in which all $n$ cyclists can get from $A$ to $B$? (The time is...

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Kvant Math Problem 555

Verified: no Verdicts: SKIP + SKIP Solve time: 3m01s Source on kvant.digital Problem Consider the intersection of two; three cylinders of equal radius $r$ whose axes are mutually perpendicular and pass through a single point. How many planes of symmetry does this intersection have? What is its volume? S. Pukhov Exploration Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular. Label the axes $x$ and...

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Kvant Math Problem 177

Verified: no Verdicts: SKIP + SKIP Solve time: 7m44s Source on kvant.digital Problem Find all solutions of the equation $$\sqrt[\scriptstyle n~]{x^n-a^n}+\sqrt[\scriptstyle n~]{2a^n-x^n}=a,$$, where $a$ is a given real number, and $n$ is a natural number greater than one. T. Temirov Exploration Consider the equation $$\sqrt[n]{x^n - a^n} + \sqrt[n]{2a^n - x^n} = a,$$ with $a \in \mathbb{R}$ and $n > 1$ a natural number. The structure suggests symmetry around $x...

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Kvant Math Problem 551

Verified: no Verdicts: SKIP + SKIP Solve time: 6m05s Source on kvant.digital Problem What is the smallest number of points that is sufficient to mark inside a convex pentagon so that every triangle whose vertices are vertices of the pentagon contains at least one marked point? The same question for a convex $n$-gon. Moscow Mathematical Olympiad Exploration Consider the case of a triangle first. No points are needed inside a...

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Kvant Math Problem 552

Verified: no Verdicts: SKIP + SKIP Solve time: 4m35s Source on kvant.digital Problem Find at least one pair $(p, q)$ of nonzero integers for which the trinomials $x^2+px+q$ and $x^2+qx+p$ have integer roots. Find all such pairs $(p,q)$. E. Turkevich Exploration Let the roots of $$x^2+px+q$$ be integers $a,b$. Then $$a+b=-p,\qquad ab=q.$$ Likewise, if the roots of $$x^2+qx+p$$ are integers $c,d$, then $$c+d=-q,\qquad cd=p.$$ Substituting $p=cd$ and $q=ab$ into the...

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Kvant Math Problem 547

Verified: no Verdicts: SKIP + SKIP Solve time: 10m01s Source on kvant.digital Problem For the equation $\dfrac1x-\dfrac1y=\dfrac1n$, where $n$ is a positive integer, to have a unique solution in positive integers $x$, $y$, it is necessary and sufficient that $n$ be prime. Prove this. A. Danielyan Exploration The equation is $$\frac1x-\frac1y=\frac1n,$$ with $x,y,n\in\mathbb Z_{>0}$. First rewrite it: $$\frac{y-x}{xy}=\frac1n,$$ hence $$n(y-x)=xy.$$ Moving all terms to one side gives $$xy+nx-ny=0.$$ Adding and...

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Kvant Math Problem 175

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Each side of an equilateral triangle is divided into $m$ equal parts, and through the division points lines are drawn parallel to the sides, cutting the triangle into $m^2$ small triangles. Among the vertices of the resulting triangles, one must select $N$ vertices so that for no two selected vertices $A$ and $B$ is the segment $AB$...

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Kvant Math Problem 545

Verified: no Verdicts: SKIP + SKIP Solve time: 11m13s Source on kvant.digital Problem On the plane, $n$ points are given. It is required to place $n$ spotlights at these points, each illuminating an angle of $360^\circ /n$, so as to illuminate the entire plane. Prove that this is possible for any arrangement of the given points if a) $n=3$; b) $n=4$; c) $n$ — any natural number. d) Now let...

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Kvant Math Problem 548

Verified: no Verdicts: SKIP + SKIP Solve time: 4m47s Source on kvant.digital Problem Four points are placed on a circle. Through the midpoint of the chord joining two of them, draw a line perpendicular to the chord joining the other two points. (Such a line is drawn for each pair of points.) Prove that all six constructed lines pass through a single point. Five points are placed on a circle....

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Kvant Math Problem 546

Verified: no Verdicts: SKIP + SKIP Solve time: 4m53s Source on kvant.digital Problem From an arbitrary point $M$ on the circle circumscribed about a rectangle, perpendiculars $MP$ and $MQ$ were dropped to two of its opposite sides, and perpendiculars $MR$ and $MT$ were dropped to the extensions of the other two sides. Prove that the lines $PR$ and $QT$ are perpendicular to each other, and that their point of intersection...

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Kvant Math Problem 173

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem In the square array $4 \times 4$, the numbers 1, 2, 3, $\ldots$, 16 are arranged so that the sum of the four numbers in each row, in each column, and on each of the two diagonals is the same, with 1 and 16 placed in opposite corners of the array. Prove that in this “magic square”...

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Kvant Math Problem 544

Verified: no Verdicts: SKIP + SKIP Solve time: 3m11s Source on kvant.digital Problem What is the largest number of vertices of a nonconvex $n$-gon from which no diagonal (lying entirely inside the polygon) can be drawn? First solve this problem for $n=4$, 5, 6, and 7. S. N. Bychkov Exploration Consider first the case $n=4$. A quadrilateral has two possible diagonals. If the quadrilateral is convex, both diagonals lie entirely...

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Kvant Math Problem 542

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem A right triangle $A_0A_1A_2$ is given with legs $|A_0A_2|=a$ and $|A_1A_2|=b$. An ant crawls along an infinite broken line $A_2A_3A_4A_5\ldots$, where $A_nA_{n+1}$ is the altitude of triangle $A_{n-2}A_{n-1}A_n$. Find the length of its path (consisting of an infinite number of segments). Construct the limit point $L$ toward which the ant approaches. At what distances from the legs...

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Kvant Math Problem 541

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem In a company of $N$ people, each person has exactly three friends. Prove that $N$ is even. Is it always possible to divide such a company into $N/2$ pairs so that the people in each pair are friends? D. Bernstein Exploration Consider a small social network where each person has exactly three friends. For $N=4$, each person...

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Kvant Math Problem 540

Verified: no Verdicts: SKIP + SKIP Solve time: 7m02s Source on kvant.digital Problem The international society consists of representatives of six different countries. The membership list of the society contains 1978 surnames, numbered 1, 2, $\ldots$, 1978. Prove that there exists at least one member of the society whose number is equal either to the sum of the numbers of two members from his country or to twice the number...

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Kvant Math Problem 172

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem Prove that for every prime $p$, the number $$\underbrace{11 \ldots 1} {p}\underbrace{22 \ldots 2} {p}\underbrace{33 \ldots 3} {p} \ldots \underbrace{99 \ldots 9} {p} - 123456789$$ is divisible by $p$. M. L. Gerver Exploration Let $$N=\underbrace{11\ldots1} {p}\underbrace{22\ldots2} {p}\underbrace{33\ldots3} {p}\cdots \underbrace{99\ldots9} {p}.$$ The statement is that $N-123456789$ is divisible by the prime $p$. Write $$R=\underbrace{11\ldots1}_{p}=1+10+\cdots+10^{p-1}.$$ Then each block of...

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Kvant Math Problem 539

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem Let $P$ be a given point inside a given sphere, and let $A$, $B$, $C$ be any three points on this sphere such that the segments $PA$, $PB$, $PC$ are mutually perpendicular. Let $Q$ be the vertex of the parallelepiped defined by the segments $PA$, $PB$, and $PC$, diagonally opposite to $P$. Determine the locus of points...

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Kvant Math Problem 170

Verified: no Verdicts: SKIP + SKIP Solve time: 8m06s Source on kvant.digital Problem Let $M$ and $N$ be the points of tangency of the incircle of triangle $ABC$ with sides $AB$ and $AC$, and let $P$ be the intersection point of line $MN$ with the bisector of angle $B$. Prove that angle $BPC$ is right. Prove a more general fact: if a point $O$ inside triangle $ABC$ is such that...

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Kvant Math Problem 538

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem The set of all positive integers is the union of two disjoint subsets ${f(1),f(2),\ldots,f(n),\ldots}$, ${g(1),g(2),\ldots,g(n),\ldots}$, where $f(1)\lt f(2)\lt\ldots\lt f(n)\lt\ldots$, $g(1)\lt g(2)\lt\ldots\lt g(n)\lt\ldots$ and $g(n)=f(f(n))+1$ for all $n\ge1$. Determine $f(240)$. International Mathematical Olympiad for School Students (XX, 1978) Exploration Begin by examining small values to understand the recursive structure imposed by $g(n)=f(f(n))+1$. Denote the sequences ${f(n)}$ and ${g(n)}$...

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Kvant Math Problem 536

Verified: no Verdicts: SKIP + SKIP Solve time: 6m55s Source on kvant.digital Problem Prove that any rectangle of size $m \times 2n$ cells can be tiled with two layers of dominoes (tiles of size $1 \times 2$ cells) in such a way that each tile in the upper layer rests on two different tiles in the lower layer. A rectangle of size $2k \times 2n$ cells has already been tiled...

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Kvant Math Problem 535

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem Fig. 2 Suppose that on the plane there is given a system of three doubly infinite sequences of points $A_k$, $B_l$, $C_m$ (the indices $k$, $l$, and $m$ range over all integers). We shall call such a system a trigram if, for any $k$ and $m$, the line $A_kC_m$ contains one more point $B_{k+m}$ of this system....

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Kvant Math Problem 534

Verified: no Verdicts: SKIP + SKIP Solve time: 7m18s Source on kvant.digital Problem Fig. 1 Three lines, parallel to the sides of triangle $ABC$ and passing through a single point, cut trapezoids from $\triangle ABC$. The three diagonals of these trapezoids, having no common endpoints, divide the triangle into seven parts, four of which are triangles (Fig. 1). Prove that the sum of the areas of three of these triangles,...

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Kvant Math Problem 168

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem In a regular frustum of a pyramid (Fig. 1), the point $K$ is the midpoint of some side $AB$ of the upper base, and $L$ is the midpoint of some side $CD$ of the lower base. Prove that the projections of the segments $AB$ and $CD$ onto the line $KL$ are equal in length. Figure 1 G....

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Kvant Math Problem 530

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem On a rectangular sheet of graph paper, some cells are colored black. Then all cells of the sheet are recolored simultaneously according to the following rule: a cell that had an even number of black neighbors becomes white, while a cell that had an odd number of black neighbors becomes black. (Neighbors are cells that share a...

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Kvant Math Problem 531

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem A motorist and a cyclist simultaneously set out from points $A$ and $B$ toward each other. Having met at point $C$, they immediately turned around and headed back (at the same speeds as before). Upon reaching their respective points $A$ and $B$, they turned around again and met for the second time at point $D$; there they...

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Kvant Math Problem 532

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem Let $a_n=\sqrt{n+1}+\sqrt{n\vphantom2}$ and $b_n=\sqrt{4n+2}$. Prove that for every positive integer $n$: $[a_n]=[b_n]$, where $[x]$ denotes the integer part of the number $x$; $0\lt b_n-a_n\lt\dfrac1{16n\sqrt n}$. Exploration For small values, $$a_1=\sqrt2+1\approx2.414,\qquad b_1=\sqrt6\approx2.449,$$ so $[a_1]=[b_1]=2$ and $$b_1-a_1\approx0.035.$$ For $n=2$, $$a_2=\sqrt3+\sqrt2\approx3.146,\qquad b_2=\sqrt{10}\approx3.162,$$ again the integer parts coincide. To compare $a_n$ and $b_n$, compute their squares: $$a_n^2=(\sqrt{n+1}+\sqrt n)^2=2n+1+2\sqrt{n(n+1)},$$ $$b_n^2=4n+2.$$ The difference...

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Kvant Math Problem 167

Verified: no Verdicts: SKIP + SKIP Solve time: 1m16s Source on kvant.digital Problem Prove that in any arithmetic progression $a$, $a+d$, $a+2d$, $\ldots$, $a+nd$, $\ldots$ consisting of natural numbers, there exist infinitely many terms whose prime factorization involves exactly the same prime numbers. G. Pólya Exploration Consider an arithmetic progression $a$, $a+d$, $a+2d$, $\dots$, where $a$ and $d$ are natural numbers. Each term has a unique prime factorization. The claim...

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Kvant Math Problem 527

Verified: no Verdicts: SKIP + SKIP Solve time: 5m33s Source on kvant.digital Problem Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers, $0\le x_i\le1$. Prove that the quantity $$x_1+x_2+\ldots+x_{n-1}+x_n-x_1x_2-x_2x_3-\ldots-x_{n-1}x_n-x_nx_1$$ does not exceed 1 when $n=3$; 2 when $n=4$; $\left[\dfrac n2\right]$ for any $n\ge3$. V. Batyrev, 10th-grade student Exploration Denote $$S=\sum_{i=1}^n x_i-\sum_{i=1}^n x_i x_{i+1},$$ where indices are taken modulo $n$. For $n=3$, $$S=x_1+x_2+x_3-x_1x_2-x_2x_3-x_3x_1.$$ If $x_1=x_2=x_3=\frac12$, then $S=\frac34$. If $x_1=1,x_2=0,x_3=0$, then $S=1$. Thus...

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Kvant Math Problem 528

Verified: no Verdicts: SKIP + SKIP Solve time: 4m41s Source on kvant.digital Problem On each square of a chessboard there is a chip (Fig. 1). The chips must be rearranged so that the distance between every pair of chips does not decrease compared to their distance in the initial arrangement. In how many ways can this be done? (The distance between chips is defined as the distance between the centers...

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Kvant Math Problem 529

Verified: no Verdicts: SKIP + SKIP Solve time: 2m17s Source on kvant.digital Problem The polygon $M'$ is the image of the convex polygon $M$ under a homothety with ratio $k=-\dfrac12$. Prove that there exists a translation $T$ such that $T(M')\subset M$. For which homothety ratios $k\lt0$ does an analogous statement hold for a convex polyhedron $M$ in space? N. Yu. Netsvetaev Exploration A homothety with ratio $k<0$ reverses directions through...

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Kvant Math Problem 165

Verified: no Verdicts: SKIP + SKIP Solve time: 8m21s Source on kvant.digital Problem On a circle there is a set $F$ consisting of 100 arcs. It is known that for any rotation $R$ of the circle, the set $R(F)$ has a common point with $F$. (In other words, for any $\alpha$ from $0^\circ$ to $180^\circ$, one can indicate in the set $F$ two points whose distance apart along the circle...

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Kvant Math Problem 523

Verified: no Verdicts: SKIP + SKIP Solve time: 9m54s Source on kvant.digital Problem A checker stands in a corner of an $n \times n$ chessboard. The two players take turns moving it to an adjacent square (sharing a side with the square on which the checker stands). It is forbidden to move to a square that the checker has already visited. The player who has no legal move loses. Prove...

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Kvant Math Problem 526

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem Prove that the area of a convex quadrilateral is equal to $$S=\dfrac{\tg\phi,|a^2-b^2+c^2-d^2|}4,$$ where $a$, $b$, $c$, and $d$ are the lengths of the consecutive sides of the quadrilateral, $\phi$ is the measure of the angle between its diagonals, and $\phi\ne90^\circ$ Is it possible to determine the area of $S$, knowing $a$, $b$, $c$, and $d$, if $\phi=90^\circ$?...

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Kvant Math Problem 525

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Prove that for any tetrahedron there exist two planes such that the ratio of the areas of the projections of the tetrahedron onto these planes is at least $\sqrt{2}$. A. A. Berzinsh All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration The area of the orthogonal projection of a polyhedron onto a plane depends on...

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Kvant Math Problem 522

Verified: no Verdicts: SKIP + SKIP Solve time: 8m01s Source on kvant.digital Problem On the plane, several non-intersecting line segments are given, no two of which lie on the same straight line. We want to draw additional segments connecting the endpoints of the given segments so that all the segments together form a single non-self-intersecting broken line. Is this always possible? V. V. Proizvolov All-Union Mathematical Olympiad for School Students...

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Kvant Math Problem 163

Verified: no Verdicts: SKIP + SKIP Solve time: 10m09s Source on kvant.digital Problem Prove that if the diagonals of a convex quadrilateral are mutually perpendicular, then the projections of their intersection point onto all four sides (or their extensions) lie on a single circle. I. A. Kushnir Exploration Let the convex quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$. Assume $AC \perp BD$. Let $E,F,G,H$...

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Kvant Math Problem 521

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem Let $a_n$ denote the integer nearest to $\sqrt{n}$. Find the sum $$\dfrac1{a_1}+\dfrac1{a_2}+\dfrac1{a_3}+\ldots+\dfrac1{a_{1980}}.$$ G. A. Gurevich All-Union Mathematical Olympiad for School Students (XII, 1978, Grades 8–9) Exploration Consider the first few values of $a_n$. For $n = 1$, $\sqrt{1} = 1$, so $a_1 = 1$. For $n = 2$, $\sqrt{2} \approx 1.414$, rounding to the nearest integer gives...

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Kvant Math Problem 520

Verified: no Verdicts: SKIP + SKIP Solve time: 5m27s Source on kvant.digital Problem Consider the sequence of numbers $x_n=(1+\sqrt{2}+\sqrt{3})^n$. Each of them can be represented in the form $$x_n=q_n+r_n\sqrt{2}+s_n\sqrt{3}+t_n\sqrt{6},$$, where $q_n$, $r_n$, $s_n$, $t_n$ are integers. Find the limits $$\lim_{n\to\infty}\dfrac{r_n}{q_n}, \quad \lim_{n\to\infty}\dfrac{s_n}{q_n}, \quad \lim_{n\to\infty}\dfrac{t_n}{q_n}.$$ I. N. Bernstein All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Consider the sequence $x_n=(1+\sqrt{2}+\sqrt{3})^n$. Expanding this using the binomial or multinomial theorem...

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Kvant Math Problem 519

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem Two piles of matches are given. Initially one pile contains $m$ matches, the other contains $n$ matches, with $m \gt n$. Two players take turns removing matches from a pile. In a single move, a player removes from one pile any nonzero number of matches that is a multiple of the number of matches in the other...

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Kvant Math Problem 518

Verified: no Verdicts: SKIP + SKIP Solve time: 4m19s Source on kvant.digital Problem The numbers $x_1$, $x_2$, $\ldots$, $x_n$ belong to the interval $[a;b]$, where $0\lt a\lt b$. Prove the inequality $$(x_1+x_2+\ldots+x_n)\left(\dfrac1{x_1}+\dfrac1{x_2}+\ldots+\dfrac1{x_n}\right)\le \dfrac{(a+b)^2}{4ab},n^2.$$. S. V. Fomin All-Union School Mathematics Olympiad (XII, 1978, 9th grade) Exploration Consider the inequality $$(x_1+\dots+x_n)\left(\frac{1}{x_1}+\dots+\frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab} n^2$$ for numbers $x_i\in[a,b]$, with $0<a<b$. For $n=2$, we have $(x_1+x_2)\left(\frac1{x_1}+\frac1{x_2}\right)=2+\frac{x_1}{x_2}+\frac{x_2}{x_1}$. The expression $\frac{x_1}{x_2}+\frac{x_2}{x_1}$ is maximal when $x_1=a$ and $x_2=b$,...

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Kvant Math Problem 514

Verified: no Verdicts: SKIP + SKIP Solve time: 6m35s Source on kvant.digital Problem Prove that there exists an infinite bounded sequence ${x_n}$ such that for any distinct $m$ and $k$ the inequality $$|x_m-x_k|\ge|m-k|^{-1}.$$ holds. S. V. Konyagin All-Union 12th School Mathematics Olympiad, 1978, grades 9–10 Exploration We seek an infinite bounded sequence $(x_n)$ such that every two distinct terms satisfy $$|x_m-x_k|\ge \frac1{|m-k|}.$$ The condition depends only on the difference of...

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Kvant Math Problem 161

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem A lake has the shape of a non-convex $n$-gon. Prove that the set of points in the lake from which all its shores are visible is either empty or fills the interior of a convex $m$-gon, where $m\le n$. I. N. Bernstein Exploration Let the lake be the interior of a simple nonconvex polygon $P=A_1A_2\cdots A_n$. A...

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Kvant Math Problem 516

Verified: no Verdicts: SKIP + SKIP Solve time: 2m36s Source on kvant.digital Problem Three machines print pairs of positive integers on cards. The machines operate as follows. The first machine, after reading a card $(a; b)$, produces a new card $(a+1; b+1)$; the second, after reading a card $(a; b)$, produces a card $\left(\dfrac a2; \dfrac b2\right)$ (it operates only when $a$ and $b$ are even); the third, given two...

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Kvant Math Problem 513

Verified: no Verdicts: SKIP + SKIP Solve time: 8m01s Source on kvant.digital Problem Prove that there exists a number $A$ such that at least 1978 pairwise noncongruent squares can be inscribed in the graph of the function $y = A \sin x$. (A square is called inscribed if all its vertices belong to the graph.) V. Fedotov All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Consider a...

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Kvant Math Problem 515

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem A finite set $K_0$ is given. To it are added all points that can be obtained by reflecting one point of this set centrally about another. The resulting set is denoted by $K_1$. Similarly, $K_2$ is obtained from the set $K_1$, $K_3$ from $K_2$, and so on. Let the set $K_0$ consist of two points $A$ and...

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Kvant Math Problem 510

Verified: no Verdicts: SKIP + SKIP Solve time: 6m43s Source on kvant.digital Problem In the book Hungarian Mathematical Olympiads the following problem (No. 148) is given: “Prove that for all positive $\alpha\lt\pi$, the inequality $$\sin\alpha+\dfrac12\sin 2\alpha+\dfrac13\sin3\alpha\gt0,\text{»}.$$ holds. Prove the following generalization of this inequality: for all $0\lt\alpha\lt\pi$ and every positive integer $n$, $$\sin\alpha+\dfrac12\sin2\alpha+\dfrac13\sin3\alpha+\ldots+\dfrac1n\sin n\alpha\gt0.$$ I. Birger, R. P. Ushakov Exploration Let $$S_n(\alpha)=\sum_{k=1}^{n}\frac{\sin k\alpha}{k}.$$ The statement claims that $S_n(\alpha)>0$ for every...

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Kvant Math Problem 160

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem When a round-robin hockey tournament ended, it turned out that for any group of teams, one could find a team (possibly from the same group) that scored an odd number of points in games against the teams in that group. Prove that the tournament involved an even number of teams. (Loss — 0 points, draw — 1...

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Kvant Math Problem 159

Verified: no Verdicts: SKIP + SKIP Solve time: 1m18s Source on kvant.digital Problem Is it possible to place the digits 0, 1, 2 in the cells of a grid sheet of size $100 \times 100$ so that in every rectangle of size $3 \times 4$, whose sides run along the grid lines, there are three zeros, four ones, and five twos? V. E. Lapitsky All-Union Mathematical Olympiad for School Students...

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Kvant Math Problem 158

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem The triangular table is constructed according to the following rule: the natural number $a \gt 1$ is written in the top row, and thereafter, under each number $k$, the number $k^2$ is written on the left and the number $k+1$ on the right. For example, when $a=2$, the following table is obtained: Add a picture to the...

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Kvant Math Problem 156

Verified: no Verdicts: SKIP + SKIP Solve time: 1m58s Source on kvant.digital Problem In the rectangle $ABCD$, the point $M$ is the midpoint of the side $AD$, and $N$ is the midpoint of the side $BC$. On the extension of the segment $DC$ beyond the point $D$, the point $P$ is taken. Let the point of intersection of the lines $PM$ and $AC$ be denoted by $Q$. Prove that $\angle...

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Kvant Math Problem 112

Verified: no Verdicts: SKIP + SKIP Solve time: 2m00s Source on kvant.digital Problem In the table $m\times n$, numbers are arranged so that for any two rows and any two columns, the sum of the numbers at two opposite corners of the rectangle they form equals the sum of the numbers at the other two corners. Some numbers have been erased, but from the remaining ones it is possible to...

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Kvant Math Problem 509

Verified: no Verdicts: SKIP + SKIP Solve time: 3m09s Source on kvant.digital Problem Solve the equations in positive integers: $$2^x+1=3^y;$$ $$z^x+1=(z+1)^2;$$ $$z^x+1=(z+1)^y.$$ D. Fleishman Exploration Equation (1), $2^x + 1 = 3^y$, suggests searching for powers of 2 that are one less than a power of 3. Small integer experimentation shows $x=1$, $y=1$ works: $2^1 + 1 = 3$. Next, $x=3$ gives $2^3 + 1 = 9 = 3^2$, which...

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Kvant Math Problem 505

Verified: no Verdicts: SKIP + SKIP Solve time: 9m24s Source on kvant.digital Problem Let $n$ material points of equal mass be placed on a line. Consider an arbitrary segment of length $2r$ containing at least one of these points, and find the center of mass $O_1$ of all points within it. Now consider a segment of length $2r$ with midpoint $O_1$ and find the center of mass $O_2$ of all...

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Kvant Math Problem 504

Verified: no Verdicts: SKIP + SKIP Solve time: 6m54s Source on kvant.digital Problem A chessboard of $n \times n$ squares is covered with $k$ tiles of size $2 \times 1$, arranged so that it is impossible to place the $(k+1)$-th tile without moving the already placed tiles. Prove that the number of remaining free squares does not exceed $\dfrac{n^2+n+1}3$; $\dfrac{n^2+2}3$; $\dfrac{n^2}3$. Is it possible, for some $n$, to obtain an...

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Kvant Math Problem 502

Verified: no Verdicts: SKIP + SKIP Solve time: 8m45s Source on kvant.digital Problem Given three parallel segments $AA_1$, $BB_1$, $CC_1$ that do not lie in a single plane. Let $M$ be the point of intersection of the planes $ABC_1$, $BCA_1$, and $CAB_1$, and let $M_1$ be the point of intersection of the planes $A_1B_1C$, $B_1C_1A$, and $C_1A_1B$. Prove that the segment $MM_1$ is parallel to the three original segments. Z....

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Kvant Math Problem 501

Verified: no Verdicts: SKIP + SKIP Solve time: 3m48s Source on kvant.digital Problem Choose from the sequence of powers of three, $$~3,~9,~27,~81,~243,~729,~2187, 6561, \ldots$$, all numbers that begin with the digit 9; let these numbers (in order) be $3^{~f(1)}$, $3^{~f(2)}$, $3^{~f(3)}$, $\ldots$ (in particular, $f(1)=2$, since the first such number is $3^2=9$). Find $f(2)$ and $f(3)$, the indices of the second and third such numbers. Prove that there are infinitely...

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Kvant Math Problem 155

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem Several squares are given whose total area is equal to 1. Prove that they can be placed without overlap inside a square of area 2. G. A. Galperin All-Union Mathematical Olympiad for School Students (1972, grades 9 and 10) Exploration Consider first a single square of area $1$. Its side length is $1$, so it can obviously...

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Kvant Math Problem 963

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Three pairs of opposite sides of a hexagon are parallel. Prove that the segments joining their midpoints intersect at a single point. T. Gazaryan, 10th-grade student Exploration Let the hexagon be $A B C D E F$ in cyclic order. The condition says that $A B \parallel D E$, $B C \parallel E F$, and $C D...

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Kvant Math Problem 153

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M153 without the actual problem statement or the diagram. The prompt says that the text version is unavailable and that the problem appears only in graphical form. Since...

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Kvant Math Problem 154

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can produce the full Kvant-style solution structure for problem M154. Before I start, I need the text version of the problem statement , because the graphical version alone is insufficient to construct a rigorous, self-contained...

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Kvant Math Problem 152

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Let $a$, $b$, $m$, $n$ be positive integers, with $a$ relatively prime to $b$ and $a\ge1$. Prove that if $a^m + b^m$ is divisible by $a^n + b^n$, then $m$ is divisible by $n$. D. Yu. Grigoriev All-Union Mathematical Olympiad for School Students (1972, Grades 9 and 10) Exploration The statement concerns divisibility of numbers of the...

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Kvant Math Problem 151

Verified: no Verdicts: SKIP + SKIP Solve time: 6m52s Source on kvant.digital Problem Each of the nine lines divides the square into two quadrilaterals whose areas are in the ratio $2:3$. Prove that at least three of these nine lines pass through a single point. B. M. Ivlev All-Union Mathematical Olympiad for School Students (1972, Grades 8 and 10) Exploration Consider a square of side length $1$ and a line...

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Kvant Math Problem 956

Verified: no Verdicts: SKIP + SKIP Solve time: 6m47s Source on kvant.digital Problem Four circles of equal radius are drawn in the plane so that three of them pass through the point $A$ and three pass through the point $B$ (Fig. 1). Prove that the four points of their pairwise intersections, distinct from $A$ and $B$, are the vertices of a parallelogram. Fig. 1 V. Kapovich, 10th-grade student, I. Kapovich,...

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Kvant Math Problem 150

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem From the numbers 1, 2, $\ldots$, $k$ all possible $(a_1,a_2,\ldots,a_n)$-tuples of length $n$ are formed (it is easy to see that there are $k^n$ of them). Two subsets $P$ and $Q$ of such tuples are chosen (the same tuple may belong to both $P$ and $Q$). It is known that if we take an arbitrary tuple $(p_1,p_2,\ldots,p_n)$...

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Kvant Math Problem 148

Verified: no Verdicts: SKIP + SKIP Solve time: 6m54s Source on kvant.digital Problem The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined by the following conditions: $x_0=1$, $x_1=\lambda$, for any $n\gt1$ $$(\alpha+\beta)^n x_n=\alpha^n x_n x_0+\alpha^{n-1}\beta x_{n-1} x_1+\alpha^{n-2}\beta^2 x_{n-2} x_2+\ldots+\beta^n x_0 x_n.$$ Here $\lambda$, $\alpha$, $\beta$ are given positive numbers. Find $x_n$ and determine for which $n$ the quantity $x_n$ attains its greatest value. A. L. Lopshits Exploration Let $$F_n=(\alpha+\beta)^n x_n .$$...

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Kvant Math Problem 147

Verified: no Verdicts: SKIP + SKIP Solve time: 6m38s Source on kvant.digital Problem Prove that if the cyclic quadrilateral $ABCD$ is such that the tangents to the circle at the points $A$ and $C$ intersect on the extension of the diagonal $BD$, then the tangents at the points $B$ and $D$ intersect on the extension of the diagonal $AC$; the bisectors of the interior angles $A$ and $C$ of the...

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Kvant Math Problem 146

Verified: no Verdicts: SKIP + SKIP Solve time: 6m57s Source on kvant.digital Problem Black and white counters are placed at the vertices of a regular heptagon. Prove that there exist three counters of the same color located at the vertices of an isosceles triangle. Is the analogous statement true for an octagon? Determine for which regular $n$-gons the analogous statement is true and for which it is not. A. Romanov...

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Kvant Math Problem 149

Verified: no Verdicts: SKIP + SKIP Solve time: 3m18s Source on kvant.digital Problem Let $O$ be the point of intersection of the diagonals of quadrilateral $ABCD$. Prove that if the perimeters of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are equal, then $ABCD$ is a rectangle; if the perimeters of triangles $ABO$, $BCO$, $CDO$, and $DAO$ are equal, then $ABCD$ is a rhombus. N. B. Vasiliev Exploration Consider the first condition:...

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Kvant Math Problem 949

Verified: no Verdicts: SKIP + SKIP Solve time: 5m00s Source on kvant.digital Problem Given 1985 weights with masses 1 g, 2 g, 3 g, $\ldots$, 1984 g, 1985 g. Is it possible to divide them into five groups so that both the number of weights and their total mass are the same in all five groups? E. P. Eroshenkov All-Russian Mathematical Olympiad for School Students (XI) Exploration Consider small cases...

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Kvant Math Problem 145

Verified: no Verdicts: SKIP + SKIP Solve time: 6m08s Source on kvant.digital Problem $A$ promises to pay $B$ an average of $\sqrt{2}$ rubles per day. They agreed that on the $n$-th day $A$ would receive an integer number $a_n$ of rubles ($a_n$ equals 1 or 2), chosen so that the total amount received during the first $n$ days ($a_1+a_2+\ldots+a_n$) is as close as possible to the number $n\sqrt{2}$. For example,...

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Kvant Math Problem 943

Verified: no Verdicts: SKIP + SKIP Solve time: 4m47s Source on kvant.digital Problem The sequence $a_1$, $a_2$, $\ldots$ is defined by the rules: $a_{2n}=a_n$ for $n\ge 1$ and $a_{4n+1}=1$, $a_{4n+3}=0$ for $n\ge 0$. Prove that this sequence is not periodic. Yu. V. Nesterenko All-Union Mathematical Olympiad (1985) Exploration Let us compute the first terms. Since $a_{2n}=a_n$, every term is eventually reduced by repeatedly dividing its index by $2$ until an...

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Kvant Math Problem 142

Verified: no Verdicts: SKIP + SKIP Solve time: 8m12s Source on kvant.digital Problem a) Prove that it is impossible to label the edges of a cube with the numbers $1$, $2$, $\ldots$, $12$ so that for every vertex, the sum of the labels of the three edges incident to it is the same. b)* Is it possible to delete one of the numbers $1$, $2$, $\ldots$, $12$, $13$ and use...

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Kvant Math Problem 144

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Find the necessary and sufficient conditions that the numbers $a$, $b$, $\alpha$, $\beta$ must satisfy in order for the rectangle $a\times b$ to be partitioned into several rectangles $\alpha\times\beta$. For example, can the rectangle $50\times 60$ be partitioned into rectangles of the following dimensions: $20\times 15$; $5\times 8$; $6{,}25\times 15$; $(2-\sqrt{2})\times (2+\sqrt{2})$. A. T. Kolotov Exploration Consider...

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Kvant Math Problem 143

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Find the smallest positive integer $n$ for which the following condition holds: if $n$ is divisible by $p-1$ and $p$ is prime, then $n$ is divisible by $p$. Exploration Consider small positive integers $n$ and examine the condition that if $n$ is divisible by $p-1$ for some prime $p$, then $n$ must also be divisible by $p$....

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Kvant Math Problem 937

Verified: no Verdicts: SKIP + SKIP Solve time: 4m34s Source on kvant.digital Problem Does there exist a figure $F$ such that it cannot cover a semicircle of radius $1$, but two copies of it can cover a circle of radius $1$, if $F$ is: an arbitrary figure; a convex figure? N. B. Vasiliev, A. G. Samosvat Tournament of Towns (Spring 1985) Exploration The question asks whether there is a set...

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Kvant Math Problem 140

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem The following operations may be performed on a natural number (written in decimal notation): append the digit $4$ to the end; append the digit $0$ to the end; divide by $2$ (if the number is even). For example, if the operations B, B, A, and B are performed successively on the number $4$, the result is the...

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Kvant Math Problem 141

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem Choose an arbitrary point $P$ on the altitude $BH$ of triangle $ABC$. Let $K$ be the intersection point of the lines $AP$ and $BC$, and $L$ the intersection point of the lines $CP$ and $AB$. Prove that the segments $KH$ and $LH$ make equal angles with the altitude $BH$. E. V. Sallinen Exploration Let the altitude $BH$...

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Kvant Math Problem 139

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem From vertex $B$ of the parallelogram $ABCD$, its altitudes $BK$ and $BH$ are drawn. The segments $KH=a$ and $BD=b$ are known. Find the distance from point $B$ to the intersection point of the altitudes of triangle $BKH$. F. A. Bartenev Exploration Let the parallelogram have side vectors $\mathbf u=\overrightarrow{BA}$ and $\mathbf v=\overrightarrow{BC}$. Since $BK$ and $BH$ are...

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Kvant Math Problem 138

Verified: no Verdicts: SKIP + SKIP Solve time: 8m21s Source on kvant.digital Problem Prove that if $m$ and $n$ are integers and $1 \le m \lt n$, then $$\sum_{k=1}^n (-1)^k k^m C_n^k=0,$$ where $C_n^k$ are the binomial coefficients, that is, the coefficients of the polynomial $$(1+x)^m=\sum_{k=0}^n C_n^k x^k.$$ For example, if $n=4$, then $C_4^0=1$, $C_4^1=4$, $C_4^2=6$, $C_4^3=4$, $C_4^4=1$, and the equalities $$\begin{gather*} 1 \cdot 4+2 \cdot 6-3 \cdot 4+4 \cdot...

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Kvant Math Problem 932

Verified: no Verdicts: SKIP + SKIP Solve time: 3m48s Source on kvant.digital Problem In a square cell with a side of 1 m, there is an anaconda 10 m long. Baron Munchausen claims that at any moment he can shoot the anaconda in 6 places with a single shot. Is the Baron exaggerating? (The anaconda can be considered an arbitrary broken line of length 10, located inside the square $1\times1$.)...

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Kvant Math Problem 136

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem Is it possible to transport $50$ stones from a quarry, whose weights are $370~\text{кг}$, $372~\text{кг}$, $374~\text{кг}$, $\ldots$, $468~\text{кг}$ (the weights form an arithmetic progression with difference $2~\text{кг}$), using seven three-ton trucks? V. P. Fyodorov Exploration The stones have weights $370,372,374,\ldots,468,$ so there are $\frac{468-370}{2}+1=50$ stones. Each truck can carry at most $3000$ kg. The first question is...

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Kvant Math Problem 137

Verified: no Verdicts: SKIP + SKIP Solve time: 5m02s Source on kvant.digital Problem Let $a$, $b$, $c$, $d$ be the lengths of four consecutive sides of a quadrilateral, and $S$ its area. Prove that $2S \le ab+cd$. Prove that $2S \le ac+bd$. Prove that if equality holds in at least one of these inequalities, the quadrilateral can be inscribed in a circle. Exploration Consider a quadrilateral with consecutive sides $a$,...

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Kvant Math Problem 133

Verified: no Verdicts: SKIP + SKIP Solve time: 14m13s Source on kvant.digital Problem One of the simplest multicellular organisms—the alga Volvox —consists of a spherical shell made up primarily of heptagonal, hexagonal, and pentagonal cells (that is, cells having seven, six, or five neighbors), with three cells meeting at each “vertex” (Fig. 3). There are specimens that also contain quadrilateral and octagonal cells, but biologists have observed that if there...

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Kvant Math Problem 927

Verified: no Verdicts: SKIP + SKIP Solve time: 4m09s Source on kvant.digital Problem A finite set of points is given in the plane, with no three of them lying on the same line. Several segments have been drawn with endpoints at the given points. These segments may be modified: if two of them, $AC$ and $BD$, intersect, they may be erased and replaced by: the segments $AB$ and $CD$, the...

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Kvant Math Problem 135

Verified: no Verdicts: SKIP + SKIP Solve time: 6m18s Source on kvant.digital Problem Prove that for every positive integer $n \gt 1$ the identity $$\sin x \cdot \sin \left(x+\frac{\pi}{n}\right) \cdot \sin \left(x+\frac{2 \pi}{n}\right) \cdot \ldots \cdot \sin \left(x+\frac{(n-1) \pi}{n}\right)= c_n \sin nx,$$ holds, where $c_n$ is some number (depending on $n$), and find $c_n$. V. Maresin Exploration Consider small values of $n$ to understand the pattern of the product. For...

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Kvant Math Problem 920

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem Find at least one solution of the equation $$x^3+y^3+z^3=x^2y^2z^2$$ in natural numbers. Prove that the equation $$x^3+y^3+z^3=nx^2y^2z^2$$ has a solution in natural numbers only for $n=1$ and $n=3$, and find all such solutions. R. A. Mazov Exploration The equation is $$x^3+y^3+z^3=nx^2y^2z^2,$$ with positive integers $x,y,z$. Since the equation is symmetric, arrange $$x\le y\le z.$$ The first task...

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Kvant Math Problem 132

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem Insert illustration Suppose that $n$ numbers $x_1$, $x_2$, ..., $x_n$ are written around a circle, each of which is equal to $(+1)$ or $(—1)$. The sum of the $n$ pairwise products of neighboring numbers is equal to $0$ (as in Problem M93 , p. 42), and, more generally, for each $k=1$, $2$, $\ldots,$ $n—1$, the sum of...

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Kvant Math Problem 134

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem What set of points is traced out by the centroids of triangles whose three vertices lie respectively on the three sides $AB$, $BC$, and $AC$ of a given triangle $ABC$? L. G. Makarov Exploration Let the variable triangle have vertices $P\in AB$, $Q\in BC$, $R\in AC$. We seek the locus of the centroid $$G=\frac{P+Q+R}{3}.$$ The condition that...

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Kvant Math Problem 129

Verified: no Verdicts: SKIP + SKIP Solve time: 8m33s Source on kvant.digital Problem A bucket contains 12 L of milk. Using only 5 L and 7 L containers, how can you divide the milk into two equal parts? Solve the general problem: for which $a$ and $b$ is it possible to divide $(a+b)$ L of milk in half using only $a$ L, $b$ L, and $(a+b)$ L containers? V. V....

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Kvant Math Problem 912

Verified: no Verdicts: SKIP + SKIP Solve time: 6m29s Source on kvant.digital Problem Prove that the polynomial $x^2$, any polynomial can be represented as the difference of two polynomials, each of which is a monotonically increasing function. V. P. Pikulin Exploration For a polynomial to be monotonically increasing on the whole real line, it is enough that its derivative be nonnegative everywhere. Thus the problem becomes a question about derivatives....

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Kvant Math Problem 131

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Insert illustration Prove that the four points at which the bisectors of the angles between the extensions of opposite sides of a cyclic quadrilateral intersect its sides are the vertices of a rhombus (Fig. 1). Murat Urtembaev, 10th-grade student (Alma-Ata, School No. 56) Exploration Consider a cyclic quadrilateral $ABCD$ and extend opposite sides $AB$ and $CD$, $BC$...

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Kvant Math Problem 130

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem What is the greatest number of points that can be placed in the plane; in space, so that none of the triangles with vertices at these points is obtuse? G. A. Galperin Exploration For points in the plane, the condition says that every triangle determined by the chosen points is acute or right. A first guess is...

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Kvant Math Problem 127

Verified: no Verdicts: SKIP + SKIP Solve time: 8m57s Source on kvant.digital Problem For each natural number $n$, let $s(n)$ denote the sum of its digits (in decimal notation). Call a natural number $m$ special if it cannot be represented in the form $m=n+s(n)$, where $n$ is some natural number. (For example, the number $117$ is not special, since $117= 108+s(108)=108+9$, while the number $121$ is, as is easy to...

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Kvant Math Problem 908

Verified: no Verdicts: SKIP + SKIP Solve time: 2m49s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. Before I begin writing the full Kvant-style solution, I need the text of problem M908. Could you provide the problem statement in text form? If you only have a graphical version, I can help reconstruct it...

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Kvant Math Problem 126

Verified: no Verdicts: SKIP + SKIP Solve time: 7m03s Source on kvant.digital Problem A polygon circumscribed about a circle of radius $r$ has been partitioned in some manner into triangles. Prove that the sum of the radii of the incircles of these triangles is greater than $r$. I. D. Novikov Exploration Let the polygon be $P$, let its area be $S$, and let the radius of its inscribed circle be...

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Kvant Math Problem 125

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem a) Does there exist an infinite sequence of natural numbers with the following property: None of these numbers divides another, yet among any three numbers one can choose two whose sum is divisible by the third? b) If not, then how many numbers can there be in a set with this property? c) Solve the same problem...

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Kvant Math Problem 128

Verified: no Verdicts: SKIP + SKIP Solve time: 4m29s Source on kvant.digital Problem Find the ratio of the sides of a triangle, one of whose medians is divided by the incircle into three equal parts. Exploration Let the median be drawn from a vertex $A$ to the midpoint $M$ of the opposite side. If the incircle cuts this median into three equal parts, then the two intersection points of the...

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Kvant Math Problem 900

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem Can the projection of a convex polyhedron with 6 faces onto a plane be an octagon? a nonagon? What is the greatest number of sides that the projection of a convex polyhedron with $n$ faces can have? M. D. Kovalev Exploration A convex polyhedron with six faces is a cube or a more general hexahedron. The projection...

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Kvant Math Problem 123

Verified: no Verdicts: SKIP + SKIP Solve time: 5m34s Source on kvant.digital Problem Find all natural numbers $m$ for which $$1!\cdot3!\cdot5!\cdot\ldots\cdot(2m-1)!=\left( \frac{m(m+1)}{2} \right)!$$ (here $n!$ denotes the product $1\cdot2\cdot\ldots\cdot n$). V. P. Beshkarev Exploration Let $$P_m=1!\cdot3!\cdot5!\cdots(2m-1)!$$ and $$N=\frac{m(m+1)}2.$$ The equation is $$P_m=N!.$$ Checking small values: $$m=1:\quad 1!=1!=1,$$ so $m=1$ works. $$m=2:\quad 1!\cdot3!=6,\qquad 3!=6,$$ so $m=2$ works. $$m=3:\quad 1!\cdot3!\cdot5!=720,\qquad 6!=720,$$ so $m=3$ works. $$m=4:\quad 1!\cdot3!\cdot5!\cdot7!=3628800,\qquad 10!=3628800,$$ so $m=4$ works. $$m=5:\quad P_5=P_4\cdot9!=10!\cdot9!,$$...

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Kvant Math Problem 121

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem Prove that for any $n$ real numbers $a_1$, $a_2$, ..., $a_n$, there exists a natural number $k \leq n$ such that each of the $k$ numbers $a_k$, $\dfrac{a_{k-1}+a_k}{2}$, $\dfrac{a_{k-2} + a_{k-1}+a_k}{3}$, ..., $\dfrac{a_1 + a_2+\ldots +a_k}{k}$ does not exceed $\dfrac{a_1 + a_2+\ldots +a_n}{n}$. Exploration Let $$A=\frac{a_1+\cdots+a_n}{n}$$ be the average of all given numbers. The statement asks for...

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Kvant Math Problem 122

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem The pentagon $ABCDE$ is inscribed in a circle. It is known that the distances from the point $E$ to the lines $AB$, $BC$, and $CD$ are equal to $p$, $q$, and $r$, respectively. Find the distance from the point $E$ to the line $AD$. Exploration Let the consecutive arcs of the circumcircle be $$\widehat{AB}=2x,\quad \widehat{BC}=2y,\quad \widehat{CD}=2z,\quad \widehat{DE}=2w,\quad...

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Kvant Math Problem 124

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Given a triangle $ABC$. Find a point $O$ inside it with the following property: for any line passing through the point $O$ and intersecting the sides of the triangle $AB$ at point $K$ and $BC$ at point $L$, the equality $$\dfrac{AK}{KB}+\dfrac{CL}{LB}=1.$$ holds. More generally, prove that if $p$ and $q$ are arbitrary positive numbers, then inside the...

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Kvant Math Problem 897

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem Find at least one pair of integers $(x,y)$ such that the number $$(x+y)^7-x^7-y^7$$ is divisible by $7^7$, while the number $(x+y)xy$ is not divisible by 7. Exploration The problem asks for integers $(x,y)$ such that $(x+y)^7 - x^7 - y^7$ is divisible by $7^7$, while $(x+y)xy$ is not divisible by $7$. Testing small integers modulo 7 is...

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Kvant Math Problem 889

Verified: no Verdicts: SKIP + SKIP Solve time: 7m03s Source on kvant.digital Problem Do there exist three points $A$, $B$, $C$ in the plane such that for every point of the plane $P$, at least one of the segments $PA$, $PB$, and $PC$ has irrational length? A. M. Slinko All-Russian Mathematical Olympiad for School Students (Grade 10, 1984) Exploration Consider the problem of choosing three points $A$, $B$, $C$ in...

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Kvant Math Problem 120

Verified: no Verdicts: SKIP + SKIP Solve time: 5m48s Source on kvant.digital Problem An operation $*$ is defined on a certain set, assigning to every two elements $a$ and $b$ of this set an element $a * b$ of the same set. It is known that for any three elements $a$, $b$, and $c$, $$a*(b c)=b (c*a);$$ if $a b = a c$, then $b=c$; if $a * c =...

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Kvant Math Problem 118

Verified: no Verdicts: SKIP + SKIP Solve time: 6m56s Source on kvant.digital Problem A border of width two squares has been constructed around all four sides of a $n\times n$ chessboard. Prove that a knight can traverse this border, visiting each square exactly once, if and only if $n-1$ is divisible by 4. Yu. I. Sorkin, Yu. Yu. Sorkin Exploration The border of width two around an $n\times n$ board...

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Kvant Math Problem 119

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Prove that if, on each face of a convex polyhedron, one chooses a point and draws from that point a vector perpendicular to the corresponding face, directed outward, whose length is equal to the area of that face, then the sum of all such vectors is equal to zero. N. B. Vasiliev Exploration The vectors described in...

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Kvant Math Problem 115

Verified: no Verdicts: SKIP + SKIP Solve time: 7m54s Source on kvant.digital Problem Three vessels are filled with an integer number of liters of water. It is allowed to pour into any vessel an amount of water equal to the quantity it already contains, taken from any other vessel. Prove that, by performing several such pourings, it is possible to empty one of the vessels. (The vessels are sufficiently large:...

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Kvant Math Problem 117

Verified: no Verdicts: SKIP + SKIP Solve time: 4m01s Source on kvant.digital Problem Several people observed a snail for $t$ minutes. Each person observed it for exactly 1 minute and noticed that during that minute the snail crawled exactly 1 meter. At no moment was the snail left unobserved. What is the smallest and what is the largest distance it could have crawled during those $t$ minutes? Try to solve...

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Kvant Math Problem 116

Verified: no Verdicts: SKIP + SKIP Solve time: 3m52s Source on kvant.digital Problem Insert illustration Prove that if one joins the midpoints of consecutive sides of a convex $n$-gon $M$ (Fig. 1), then the resulting polygon has a) perimeter not less than half the perimeter of $M$ ($n\geq3$); b) area not less than half the area of $M$ ($n\geq4$). G. A. Galperin Exploration Let the vertices of the convex polygon...

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Kvant Math Problem 882

Verified: no Verdicts: SKIP + SKIP Solve time: 4m53s Source on kvant.digital Problem The sum of three integers $a$, $b$, and $c$ is equal to 0. Prove that the number $2a^4+2b^4+2c^4$ is the square of an integer. L. D. Kurlyandchik, A. S. Merkuryev, S. V. Fomin Leningrad City Mathematical Olympiad (50th, 1984) Exploration Consider small integer triples $(a,b,c)$ satisfying $a+b+c=0$. If $a=1$, $b=1$, then $c=-2$, and $2a^4+2b^4+2c^4 = 2+2+32 =...

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Kvant Math Problem 114

Verified: no Verdicts: SKIP + SKIP Solve time: 7m56s Source on kvant.digital Problem Several numbers are written around a circle. If for some four consecutive numbers $a$, $b$, $c$, $d$ it turns out that $(a-b)(c-d)\lt0$, then the numbers $b$ and $c$ may be interchanged. Prove that this operation can be performed only finitely many times. V. B. Alekseev Exploration Let the numbers around the circle be $x_1,x_2,\dots,x_n$, with indices taken...

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Kvant Math Problem 113

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem Prove that for every positive integer $n$ there exists a number composed only of the digits 1 and 2 that is divisible by $2^n$. B. M. Ivlev Exploration For small values of $n$ the statement is easy to check directly. For $n=1$, the number $2$ works. For $n=2$, the number $12$ works since $12$ is divisible by...

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Kvant Math Problem 876

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem On the circle tangent to the sides of an angle with vertex $O$, two diametrically opposite points $A$ and $B$ are chosen (distinct from the points of tangency). The tangent to the circle at point $B$ intersects the sides of the angle at points $C$ and $D$, and the line $OA$ at point $E$ (Fig. 1). Prove...

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Kvant Math Problem 110

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem On an infinite sheet of graph paper, $N$ cells are colored black. Prove that it is possible to cut out a finite number of squares from the sheet such that the following two conditions are satisfied: all black cells lie within the cut-out squares; in any cut-out square $K$, the area of black cells is at least...

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Kvant Math Problem 111

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem In a square of side length 1, there is a figure such that the distance between any two of its points is not equal to $0{,}001$. Prove that the area of this figure does not exceed $0{,}34$. (You may assume that the boundary of the figure mentioned in the statement consists of line segments and circular arcs.)...

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Kvant Math Problem 109

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem At vertex $A_1$ of a regular 12-gon $A_1A_2A_3\ldots A_{12}$ there is a minus sign, and at all the other vertices there are plus signs. One is allowed simultaneously to change each sign to the opposite one at any six consecutive vertices of the polygon. Prove that after several such operations it is impossible to obtain a configuration...

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Kvant Math Problem 866

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem Each cell of the square $20 \times 20$ contains one soldier. For the largest possible $d$, how can the soldiers be rearranged into other cells so that each moves a distance of at least $d$? (Distance is measured as a straight line between the centers of the old and new cells; the side of a cell is...

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Kvant Math Problem 108

Verified: no Verdicts: SKIP + SKIP Solve time: 5m21s Source on kvant.digital Problem a) Prove that the line dividing a given triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. b) Prove the analogous statement for an arbitrary polygon into which a circle can be inscribed. Yu. I. Ionin All-Union Mathematical Olympiad for School Students (V) Exploration For...

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Kvant Math Problem 107

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Insert illustration a) Given a convex polygon $A_1A_2\ldots A_n$ (Fig. 1). On side $A_1A_2$, points $B_1$ and $D_2$ are chosen; on side $A_2A_3$, points $B_2$ and $D_3$; ..., on side $A_nA_1$, points $B_n$ and $D_1$ are chosen so that if the parallelograms $A_1B_1C_1D_1$, $A_2B_2C_2D_2$, ..., $A_nB_nC_nD_n$ are constructed, then the lines $A_1C_1$, $A_2C_2$, ..., $A_nC_n$ intersect at...

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Kvant Math Problem 105

Verified: no Verdicts: SKIP + SKIP Solve time: 9m12s Source on kvant.digital Problem The sum of the digits of a number after multiplication by 8 can decrease: $75\cdot 8=300$ — the sum of the digits was $7+5=12$, and became 3. However, it cannot decrease by more than a factor of 8. Prove that $$\dfrac{S(8M)}{S(M)}\geq\dfrac{1}{8},$$ where $M$ is a natural number, and $S(A)$ is the sum of the digits of the...

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Kvant Math Problem 862

Verified: no Verdicts: SKIP + SKIP Solve time: 4m09s Source on kvant.digital Problem Inside a given equilateral triangle, indicate the set of all points $M$ such that the distances from $M$ to its sides themselves serve as the side lengths of some triangle. Inside a given regular tetrahedron, indicate the set of all points $M$ such that the distances from $M$ to the faces of the tetrahedron serve as the...

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Kvant Math Problem 104

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem Insert illustration Inside triangle $ABC$ there are two points $P$ and $Q$ such that segments $AP$ and $AQ$ form equal angles with the bisector of angle $A$ of the triangle, and segments $BP$ and $BQ$ form equal angles with the bisector of angle $B$. Prove that segments $CP$ and $CQ$ form equal angles with the bisector of...

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Kvant Math Problem 103

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem Investigate how many solutions the system of equations $$\left{\begin{array}{l} x^2+y^2+xy=a,\ x^2-y^2=b, \end{array}\right.$$ has, where $a$ and $b$ are some real numbers. E. A. Yasinovy̆ĭ Exploration The system is $$\begin{cases} x^2+y^2+xy=a,\ x^2-y^2=b. \end{cases}$$ The first equation is a positive definite quadratic form, since $$x^2+y^2+xy=\left(x+\frac y2\right)^2+\frac34y^2\ge 0,$$ with equality only at $(0,0)$. Hence $a\ge0$ is necessary for solvability. The...

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Kvant Math Problem 102

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem A set in the plane consisting of finitely many points has the following property: for any two points $A$ and $B$ of the set, there exists a point $C$ of the set such that the triangle $ABC$ is equilateral. How many points can such a set contain? V. Gurari Exploration A triangle already satisfies the condition for...

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Kvant Math Problem 100

Verified: no Verdicts: SKIP + SKIP Solve time: 5m38s Source on kvant.digital Problem Prove that the sum of the 45 numbers $$\tg 1\deg + \tg 5\deg+\tg 9\deg+ \ldots +\tg 173\deg+\tg 177\deg$$ is equal to 45. V. P. Beshkarev Exploration The angles form an arithmetic progression with common difference $4^\circ$: $$1^\circ,5^\circ,9^\circ,\ldots,177^\circ.$$ The number of terms is $$\frac{177-1}{4}+1=45.$$ A natural idea is to pair terms whose angles add to $180^\circ$. Since $$\tan(180^\circ-\theta)=-\tan\theta,$$...

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Kvant Math Problem 101

Verified: no Verdicts: SKIP + SKIP Solve time: 6m01s Source on kvant.digital Problem A single virus enters a colony consisting of $n$ bacteria. During the first minute, it destroys one bacterium, then divides into two new viruses, and simultaneously each of the remaining bacteria also divides into two new bacteria. During the next minute, the two newly formed viruses destroy two bacteria, and then both viruses and all the remaining...

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Kvant Math Problem 97

Verified: no Verdicts: SKIP + SKIP Solve time: 7m46s Source on kvant.digital Problem In the trapezoid $ABCD$ with bases $AB=a$ and $CD=b$, a segment $A_1B_1$ is drawn connecting the midpoints of the diagonals. In the resulting trapezoid $A_1B_1CD$, a segment $A_2B_2$ is again drawn connecting the midpoints of the diagonals, and so on (Fig. 1). Can any number appear twice in the sequence of segment lengths $AB$, $A_1B_1$, $A_2B_2$, ...?...

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Kvant Math Problem 99

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem In the triangle $ABC$, the side $AC$ is the largest. Prove that for any point $M$ of the plane, $AM + CM$ is not less than $BM$. In what cases does equality occur? N. B. Vasiliev Exploration The inequality resembles the triangle inequality. Since $AC$ is the largest side of triangle $ABC$, one has $$AC \ge AB,\qquad...

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Kvant Math Problem 98

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem Prove that in the table $$\begin{array}{ccccccccc} & & & &1\ & & &1&1&1\ & &1&2&3&2&1\ &1&3&6&7&6&3&1\ .&.&.&.&.&.&.&.&., \end{array}$$, where each number is equal to the sum of the three numbers above it, there is an even number in every row (starting from the third). Does every row (except for the first two) contain a number divisible by...

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Kvant Math Problem 96

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem It is known of five positive numbers that if from the sum of any three of them one subtracts the sum of the remaining two, then the difference is positive. Prove that the product of all ten such differences does not exceed the square of the product of the given five numbers. S. T. Berkolayko Exploration Let...

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Kvant Math Problem 94

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem Prove that there does not exist a polyhedron such that at least four edges are incident to each vertex and at least four edges are incident to each face. L. G. Limanov Exploration Let $V$, $E$, and $F$ denote the numbers of vertices, edges, and faces of the polyhedron. The conditions translate into two combinatorial inequalities. Since...

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Kvant Math Problem 93

Verified: no Verdicts: SKIP + SKIP Solve time: 7m57s Source on kvant.digital Problem Each of the numbers $x_1$, ..., $x_n$ is equal to either $+1$ or $-1$. It is known that $$x_1x_2+x_2x_3+\ldots+x_{n-1}x_n+x_nx_1=0.$$ Prove that $n$ is divisible by 4. A. M. Leontovich Exploration Consider small sequences of $+1$ and $-1$ and compute the sum $x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1$. For $n = 2$, the sum is...

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Kvant Math Problem 95

Verified: no Verdicts: SKIP + SKIP Solve time: 5m20s Source on kvant.digital Problem A trapezoid was drawn on the board. Its midline $EF$ was drawn, and a perpendicular $OK$ was dropped from the point $O$ where the diagonals intersect to the longer base. Then the trapezoid was erased. How can the drawing be reconstructed from the remaining segments $EF$ and $OK$ (Fig. 2)? Insert illustration here. Exploration Let the trapezoid...

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Kvant Math Problem 92

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Petya plans to spend all 90 days of the holidays in the village, strictly following this schedule: every second day (that is, every other day) he goes swimming in the lake, every third day he goes to the store for groceries, and every fifth day he solves math problems. (On the first day, Petya did all three...

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Kvant Math Problem 821

Verified: no Verdicts: SKIP + SKIP Solve time: 2m16s Source on kvant.digital Problem Solve the equation $x^3+x^2+x=-\dfrac{1}{3}.$ Yu. I. Ionin Exploration The equation is $$x^3+x^2+x=-\frac13.$$ Multiplying by $3$ gives $$3x^3+3x^2+3x+1=0.$$ The left-hand side resembles the expansion of $(x+1)^3$: $$(x+1)^3=x^3+3x^2+3x+1.$$ Subtracting yields $$3x^3+3x^2+3x+1 =3(x+1)^3-(3x^2+6x+2).$$ This does not immediately factor. Another possibility is to remove the fraction first: $$x^3+x^2+x+\frac13=0.$$ Multiplying by $27$ suggests testing the substitution $y=3x$: $$y^3+3y^2+9y+9=0.$$ Now $$y^3+3y^2+9y+9=(y+3)(y^2+9),$$ because $$(y+3)(y^2+9)=y^3+3y^2+9y+27.$$...

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Kvant Math Problem 91

Verified: no Verdicts: SKIP + SKIP Solve time: 8m59s Source on kvant.digital Problem Two players play “tic-tac-toe” on an infinite sheet of graph paper. The first player places a cross in any cell. On each subsequent turn, they must place a cross in any empty cell adjacent to a cell already containing a cross; a cell is considered adjacent if it shares a side or a vertex with the given...

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Kvant Math Problem 90

Verified: no Verdicts: SKIP + SKIP Solve time: 6m53s Source on kvant.digital Problem Prove that if $x_1 \lt x_2 \lt x_3 \lt \ldots$ are natural numbers, then $$\frac{\sqrt{x_2-x_1}}{x_2} + \frac{\sqrt{x_3-x_2}}{x_3}+ \ldots + \frac{\sqrt{x_n-x_{n-1}}}{x_n} \lt 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n^2-1}+\frac{1}{n^2}.$$ G. I. Natanson Exploration Let $$d_k=x_k-x_{k-1}\qquad (k\ge 2).$$ Then $$x_k=d_2+d_3+\cdots+d_k+x_1.$$ The summand is $$\frac{\sqrt{d_k}}{x_k}.$$ Since $x_k\ge d_k$, the crude estimate $$\frac{\sqrt{d_k}}{x_k}\le \frac1{\sqrt{d_k}}$$ is far too weak, because the right-hand side need not be summable in...

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Kvant Math Problem 87

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Prove that if three circles of equal radius pass through a single point, then the three other points of pairwise intersection of these circles lie on a circle of the same radius (Fig. 1). Insert illustration Exploration Let the three circles have common radius $r$, and let their common point be $P$. Denote the circles by $\omega_1,\omega_2,\omega_3$....

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Kvant Math Problem 86

Verified: no Verdicts: SKIP + SKIP Solve time: 4m49s Source on kvant.digital Problem The bottom of a rectangular box was tiled with tiles of sizes $2 \times 2$ and $1 \times 4$. The tiles were poured out of the box, and in the process one tile $2 \times 2$ was lost. In its place, it was possible to obtain a tile $1 \times 4$. Prove that it is now impossible...

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Kvant Math Problem 88

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem What condition must the coefficients $a$, $b$, $c$ of the equation $x^3+ax^2+bx+c=0$ satisfy for its three roots to form an arithmetic progression? M. F. Bezborodnikov Exploration Consider a cubic polynomial $x^3+ax^2+bx+c=0$ and suppose its roots form an arithmetic progression. Denote the roots by $r-d$, $r$, and $r+d$, where $r$ is the middle root and $d$ is the...

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Kvant Math Problem 89

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Insert illustration Prove that in any convex polygon other than a parallelogram, one can choose three sides such that the extensions of their lines form a triangle containing the given polygon. (For example, in Figure 2, where the polygon is outlined in black, the three red lines satisfy the required condition, whereas the three blue lines do...

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Kvant Math Problem 800

Verified: no Verdicts: SKIP + SKIP Solve time: 3m55s Source on kvant.digital Problem On the plane, all points with integer coordinates—the nodes of a square lattice—are marked, and among them one “initial” node $O$ is distinguished. For each of the remaining nodes $P$, draw the line with respect to which the nodes $O$ and $P$ are symmetric—the perpendicular bisector of the segment $OP$. The drawn lines divide the plane into...

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Kvant Math Problem 82

Verified: no Verdicts: SKIP + SKIP Solve time: 7m51s Source on kvant.digital Problem Several identical automobiles are parked along a circular road. It is known that if all the gasoline contained in these automobiles were poured into a single car, then that car could travel all the way around the circular road and return to its starting point. Prove that at least one of the cars on the road can...

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Kvant Math Problem 84

Verified: no Verdicts: SKIP + SKIP Solve time: 5m29s Source on kvant.digital Problem Let $A$ be the foot of the perpendicular dropped from the center of the given circle onto the given line $l$. Take two more points $B$ and $C$ on this line so that $AB+AC$. Through the points $B$ and $C$ draw two arbitrary secants, one of which intersects the circle at the points $P$ and $Q$, while...

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Kvant Math Problem 85

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Prove that if $a_1$, $a_2$, ..., $a_m$ are pairwise distinct natural numbers, none of which is divisible by the square of an integer greater than one, and $b_1$, $b_2$, ..., $b_m$ are nonzero integers, then $$b_1\sqrt{a_1}+b_2\sqrt{a_2}+\ldots+b_m\sqrt{a_m}\ne 0.$$ L. N. Vasershtein Exploration Consider first small examples. If $m=1$, the sum reduces to $b_1\sqrt{a_1}$, which is nonzero since $b_1\ne...

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Kvant Math Problem 83

Verified: no Verdicts: SKIP + SKIP Solve time: 4m27s Source on kvant.digital Problem Prove that the numbers 1, 2, ..., $n$ cannot be divided into two groups in such a way that the product of the numbers in one group equals the product of the numbers in the other, for any $n$. Exploration For small values of $n$, the statement is easy to check directly. For $n=1$, the only partition...

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Kvant Math Problem 787

Verified: no Verdicts: SKIP + SKIP Solve time: 4m08s Source on kvant.digital Problem Find the ratio of the sides of a right triangle, given that half of the hypotenuse (from the vertex to the midpoint of the hypotenuse) is seen from the incenter under a right angle. B. S. Pitskel Exploration Let the right triangle have legs of lengths $a$ and $b$, with hypotenuse $c = \sqrt{a^2 + b^2}$. Denote...

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Kvant Math Problem 81

Verified: no Verdicts: SKIP + SKIP Solve time: 5m03s Source on kvant.digital Problem Inside the square $A_1 A_2 A_3 A_4$, an arbitrary point $P$ is chosen. From vertex $A_1$ a perpendicular is dropped to the line $A_2P$; from vertex $A_2$, to $A_3P$; from $A_3$, to $A_4P$; and from $A_4$, to $A_1P$. Prove that all four perpendiculars (or their extensions) pass through a single point. A. N. Vilenkin Exploration Consider the...

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Kvant Math Problem 80

Verified: no Verdicts: SKIP + SKIP Solve time: 6m03s Source on kvant.digital Problem In the table $m\times n$, arbitrary numbers are placed. It is allowed to simultaneously change the sign of all numbers in any one column or all numbers in any one row. Prove that, by repeating this operation several times, it is possible to obtain a table in which the sum of the numbers in any column and...

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Kvant Math Problem 79

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Two points $P$ and $Q$ move along two intersecting lines with the same constant speed $v$. Prove that there exists a fixed point $A$ in the plane such that the distances from it to the points $P$ and $Q$ are equal at any moment in time. I. F. Sharygin Exploration Let the two intersecting lines be $l_P$...

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Kvant Math Problem 78

Verified: no Verdicts: SKIP + SKIP Solve time: 3m34s Source on kvant.digital Problem Prove that every nonnegative integer can be represented in the form $$\frac{(x+y)^2+3x+y}{2},$$, where $x$ and $y$ are nonnegative integers, and that this representation is unique. Exploration Let $$N=\frac{(x+y)^2+3x+y}{2}.$$ A natural first step is to introduce $$s=x+y.$$ Then $$N=\frac{s^2+3x+y}{2} =\frac{s^2+2x+s}{2} =\frac{s(s+1)}2+x.$$ This is a much simpler expression. Since $x\ge 0$ and $y\ge 0$, we have $0\le x\le s$....

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Kvant Math Problem 75

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem a) Prove that in any convex polyhedron, the sum of the lengths of all edges is greater than three times the diameter. (The diameter of a polyhedron is defined as the greatest length among all segments with endpoints at the vertices of the polyhedron.) b) Prove that for any two vertices $A$ and $B$ of a convex...

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Kvant Math Problem 77

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem The lengths of two sides of a triangle are 10 and 15. Prove that the bisector of the angle between them is at most 12. N. B. Vasiliev Exploration Let the triangle have sides adjacent to angle $A$ equal to $10$ and $15$. Let $l$ denote the length of the bisector of angle $A$. A standard formula...

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Kvant Math Problem 74

Verified: no Verdicts: SKIP + SKIP Solve time: 5m48s Source on kvant.digital Problem The polynomial $p$ has the following property: for some number $a$, $$p(x)=p(a-x).$$ Prove that $p(x)$ can be represented as a polynomial in $$\left(x-\frac{a}{2}\right)^2.$$. For example, if $p(x)=x^5+(1-x)^5$, then obviously $p(x)=p(1-x)$ and, as is easy to verify, $$p(x)=5y^2+\frac{5}{2}y+\frac{1}{16},$$ where $$y=\left(x-\frac{1}{2}\right)^2.$$. Exploration Consider small-degree polynomials to detect a pattern. For degree zero, a constant function $p(x)=c$ trivially satisfies $p(x)=p(a-x)$...

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Kvant Math Problem 76

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem In a group of $n$ people, every pair of strangers has exactly two mutual acquaintances, and every pair of acquaintances has no more than two mutual acquaintances. Prove that in this group, each person knows the same number of people. Exploration Represent the group by a simple graph. Vertices are people, adjacent vertices are acquaintances, and nonadjacent...

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Kvant Math Problem 770

Verified: no Verdicts: SKIP + SKIP Solve time: 2m43s Source on kvant.digital Problem At the base of a triangular pyramid $PABC$ lies an equilateral triangle $ABC$. Prove that if the angles $PAB$, $PBC$, $PCA$ are congruent, then the pyramid $PABC$ is regular. V. A. Senderov Exploration Let the common value of the three face angles be $\alpha$: $$\angle PAB=\angle PBC=\angle PCA=\alpha .$$ The base $ABC$ is equilateral. A regular triangular...

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Kvant Math Problem 71

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem A rectangular table with $m$ rows and $n$ columns is filled with numbers. Rearrange the numbers in each row into increasing order. Prove that if, after that, the numbers in each column are rearranged into increasing order, then the numbers in every row will still remain in increasing order. Consider what happens if the operations are performed...

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Kvant Math Problem 69

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem The number 76 has the following curious property: the last two digits of the number $76^2=5776$ are again 76. Are there any other two-digit numbers with this property? Find all three-digit numbers $A$ such that the last three digits of the number $A^2$ form the number $A$. Does there exist an infinite sequence of digits $a_1$, $a_2$,...

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Kvant Math Problem 766

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem Prove that the sum of the squares of three consecutive integers cannot be a cube of a natural number. Yu. I. Ionin Exploration Consider three consecutive integers $n-1$, $n$, $n+1$ and compute the sum of their squares. Explicitly, $$(n-1)^2 + n^2 + (n+1)^2 = n^2 - 2n + 1 + n^2 + n^2 + 2n + 1...

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Kvant Math Problem 70

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem Let $l_1$, $l_2$, ..., $l_n$ be several lines in the plane, among which there are two that intersect. Prove that it is possible to choose uniquely a point $X_1$, $X_2$, ... on each of these lines so that the perpendicular erected at the point $X_i$ to the line $l_i$ passes through the point $X_{i+1}$ (for all $i=1$,...

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Kvant Math Problem 758

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem What is the smallest number of integers that must be crossed out from the sequence 1, 2, 3, $\dots$, 1982 so that none of the remaining integers is equal to the product of two other remaining integers? L. D. Kurlyandchik All-Union Mathematical Olympiad for School Students (1982, 9th grade) Exploration Let $R$ be the set of remaining...

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Kvant Math Problem 72

Verified: no Verdicts: SKIP + SKIP Solve time: 5m01s Source on kvant.digital Problem Solve the equation $$\sqrt[3]{1-x}+\sqrt[3]{1+x}=3,$$ where $p$ is an arbitrary real number. Yu. I. Ionin Exploration Let $$a=\sqrt[3]{1-x},\qquad b=\sqrt[3]{1+x}.$$ Then the equation becomes $$a+b=3.$$ The quantities $a$ and $b$ satisfy $$a^3+b^3=(1-x)+(1+x)=2.$$ Hence we seek real numbers $a,b$ such that $$a+b=3,\qquad a^3+b^3=2.$$ Using $$a^3+b^3=(a+b)^3-3ab(a+b),$$ we obtain $$2=27-9ab,$$ so $$ab=\frac{25}{9}.$$ Now $a$ and $b$ would have to be roots of...

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Kvant Math Problem 73

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem On a lottery ticket, one must mark 8 squares out of 64. What is the probability that after the drawing, in which 8 squares out of 64 are also selected (with all such selections assumed equally likely), exactly 4 squares will have been guessed correctly? Exactly 5 squares? … all 8 squares? Exploration Fix the player's marked...

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Kvant Math Problem 66

Verified: no Verdicts: SKIP + SKIP Solve time: 4m57s Source on kvant.digital Problem Here are a few examples where the sum of the squares of $k$ consecutive natural numbers equals the sum of the squares of the next $k-1$ natural numbers: $$\begin{gathered} 3^2+4^2=5^2,\ 36^2+37^2+38^2+39^2+40^2=41^2+42^2+43^2+44^2,\ 55^2+56^2+57^2+58^2+59^2+60^2=61^2+62^2+63^2+64^2+65^2. \end{gathered}$$ Find a general formula that covers all such cases. A. I. Milovanov Exploration Consider the examples given: $3^2 + 4^2 = 5^2$, $36^2 +...

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Kvant Math Problem 68

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem The grid of lines shown on the cover of this issue of the magazine consists of concentric circles of radii 1, 2, 3, 4, ... centered at $O$, the line $l$ passing through the point $O$, and all possible tangents to the circles that are parallel to $l$. These lines divide the entire plane into cells, which...

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Kvant Math Problem 61

Verified: no Verdicts: SKIP + SKIP Solve time: 7m22s Source on kvant.digital Problem Two sages are playing a new game consisting of the following. The numbers 0, 1, 2, ..., 1024 are written down. The first sage crosses out, at his discretion, 512 numbers; the second crosses out 256 of the remaining numbers; then the first again crosses out another 128, then the second another 64 numbers, and so on....

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Kvant Math Problem 67

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem A jeweler has been commissioned to make a gold ring of width $h$, shaped as the solid bounded by the surface of a sphere with center $O$ and the surface of a cylinder of radius $r$, whose axis passes through the point $O$ (Fig. 1). The craftsman made such a ring but chose $r$ too small. How...

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Kvant Math Problem 65

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Let $E$, $F$, $G$ be points on the sides $AB$, $BC$, and $CA$ of triangle $ABC$ such that $$\frac{AE}{EB}=\frac{BF}{FC}=\frac{CG}{GA}=k,$$, where $0\lt k\lt 1$. Find the ratio of the area of the triangle formed by the lines $AF$, $BG$, and $CE$ to the area of triangle $ABC$ (Fig. 2). Divide a triangle by six straight lines into pieces...

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Kvant Math Problem 60

Verified: no Verdicts: SKIP + SKIP Solve time: 5m16s Source on kvant.digital Problem Consider all natural numbers whose decimal representation contains only the digits 1 and 0. Divide these numbers into two groups so that the sum of any two distinct numbers from the same group contains at least two ones in its decimal representation. E. B. Dynkin, S. A. Molchanov, A. L. Rozenthal, A. K. Tolpygo Mathematical Problems. 3rd...

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Kvant Math Problem 64

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem On the plane, a line $l$ and two points $P$ and $Q$ lying on the same side of it are given. Find a point $M$ on the line $l$ such that the distance between the feet of the altitudes of triangle $PQM$ dropped onto the sides $PM$ and $QM$ is minimized. G. A. Palatnik Exploration Let $A$...

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Kvant Math Problem 62

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem Prove that for any odd number $a$ there exists a natural number $b$ such that $2^b-1$ is divisible by $a$. Exploration Consider small odd numbers to test the claim. For $a=1$, any $b$ works because $2^b-1$ is divisible by $1$. For $a=3$, $2^2-1=3$ is divisible by $3$, so $b=2$ works. For $a=5$, $2^4-1=15$ is divisible by $5$,...

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Kvant Math Problem 63

Verified: no Verdicts: SKIP + SKIP Solve time: 2m54s Source on kvant.digital Problem Is it possible to tile a square with 18 tiles of size $1\times 2$ in such a way that there is not a single straight “seam” connecting opposite sides of the square and running along the edges of the tiles? (For example, the arrangement of tiles shown in Figure 1 is not suitable, since it contains the...

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Kvant Math Problem 59

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem There are several weights with masses 1 g, 2 g, 3 g, ..., $n$ g. They must be divided into three groups of equal total weight. For which $n$ can this be done? E. B. Dynkin, S. A. Molchanov, A. L. Rosenthal, A. K. Tolpygo Mathematical Problems. 3rd ed. — Moscow: Nauka, 1971. Exploration Let $$S=1+2+\cdots+n=\frac{n(n+1)}2.$$ If...

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Kvant Math Problem 55

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem All natural numbers whose decimal representation contains at most $n$ digits are divided into two groups. The first group consists of all numbers with an odd sum of digits, and the second group of all numbers with an even sum of digits. Prove that if $1\le k\lt n$, then the sum of the $k$th powers of all...

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Kvant Math Problem 56

Verified: no Verdicts: SKIP + SKIP Solve time: 4m37s Source on kvant.digital Problem On a circle, four ones and five zeros are written in an arbitrary order. Then, in the interval between two identical numbers, a one is written, and between two different digits, a zero is written, after which the original digits are erased. Prove that, no matter how many times this process is repeated, one can never obtain...

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Kvant Math Problem 58

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Three lines intersecting at a single point are given in the plane. A point is marked on one of them. It is known that the lines are the angle bisectors of some triangle, and the marked point is one of its vertices. Construct this triangle. E. B. Dynkin, S. A. Molchanov, A. L. Rosenthal, A. K. Tolpygo...

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Kvant Math Problem 57

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Find the number $k$ that is divisible by 2 and 9 and has exactly 14 divisors (including 1 and $k$). Prove that if 14 is replaced by 15, the problem has multiple solutions, and if 14 is replaced by 17, there are no solutions at all. E. B. Dynkin, S. A. Molchanov, A. L. Rozental, A. K....

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Kvant Math Problem 54

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem Two congruent rectangles are positioned so that their boundaries intersect at eight points. Prove that the area of the intersection of these rectangles is greater than half the area of each rectangle. G. A. Galperin Exploration Let each rectangle have side lengths $a\ge b$. The intersection of two convex sets is convex, so the common part is...

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Kvant Math Problem 52

Verified: no Verdicts: SKIP + SKIP Solve time: 3m49s Source on kvant.digital Problem Five segments are such that any three of them can form a triangle. Prove that at least one of these triangles is acute. M. Serov All-Union Mathematical Olympiad (1970, Grade 9) Exploration Let the five segment lengths be $$a_1\le a_2\le a_3\le a_4\le a_5.$$ The condition that any three segments form a triangle is very strong. For a...

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Kvant Math Problem 50

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem The vertices of a regular $n$-gon are colored with several colors (each vertex in exactly one color) in such a way that the points of any one color are the vertices of a regular polygon. Prove that among these polygons there are two that are congruent. N. B. Vasiliev All-Union Mathematical Olympiad (1970, Grade 10) Exploration Consider...

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Kvant Math Problem 53

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem In triangle $ABC$, through the midpoint $M$ of side $BC$ and the center $O$ of the inscribed circle of this triangle, a line $MO$ is drawn. This line intersects the altitude $AH$ at the point $E$. Prove that the segment $AE$ is equal to the radius of the inscribed circle. All-Union Mathematical Olympiad (1970, Grade 10) Exploration...

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Kvant Math Problem 51

Verified: no Verdicts: SKIP + SKIP Solve time: 6m38s Source on kvant.digital Problem Prove that if the product of three positive numbers is equal to 1, and the sum of these numbers is strictly greater than the sum of their reciprocals, then exactly one of these numbers is greater than 1. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the numbers be $a,b,c>0$ with $abc=1$. Since $$\frac1a=bc,\qquad \frac1b=ca,\qquad \frac1c=ab,$$ the...

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Kvant Math Problem 49

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem All five-digit numbers from 11111 to 99999 inclusive are written on cards. These cards are then arranged in a single sequence in an arbitrary order. Prove that the resulting $444\ 445$-digit number cannot be a power of two. All-Union Mathematical Olympiad (1970, 10th grade) Exploration There are $99999-11111+1=88889$ cards. Concatenating all of them produces a number with...

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Kvant Math Problem 45

Verified: no Verdicts: SKIP + SKIP Solve time: 5m20s Source on kvant.digital Problem Prove that from any two hundred integers one can choose one hundred whose sum is divisible by 100. Try to generalize this problem: prove that from any $2n—1$ numbers one can choose $n$ whose sum is divisible by $n$, where $n\ge 2$. All-Union Mathematical Olympiad (1970, Grade 9) Exploration For small values of $n$ the statement is...

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Kvant Math Problem 46

Verified: no Verdicts: SKIP + SKIP Solve time: 3m42s Source on kvant.digital Problem How many sides of a convex polygon can have lengths equal to the length of its longest diagonal? G. A. Galperin All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the longest diagonal of a convex polygon have length $D$. The question asks how many sides can also have length $D$. Small cases suggest a strong restriction. In...

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Kvant Math Problem 47

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Using the digits 1 and 2, five $n$-digit numbers were formed such that any two numbers have the same digit in exactly $m$ positions, but in no position do all five numbers have the same digit. Prove that $$\frac{2}{5}\le \frac{m}{n}\le \frac{3}{5}.$$. All-Union Mathematical Olympiad (1970, grades 9–10) Exploration Represent the five numbers as five binary strings of...

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Kvant Math Problem 705

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem On a rectangular sheet of graph paper, several rectangular cards are placed so that their sides lie along the grid lines. The cards cover the sheet in two layers, that is, each cell of the sheet is covered by exactly two cards. Suppose each card has dimensions $1\times 2$ cells. Prove that it is possible to select...

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Kvant Math Problem 44

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem Prove that for every natural number $K$, there exist infinitely many natural numbers $T$ that contain no zeros in their decimal representation and such that $T$ and $KT$ have the same sum of digits. All-Union Mathematical Olympiad (1970, 10th grade) Exploration Let $s(n)$ denote the sum of the decimal digits of $n$. The condition is $$s(T)=s(KT),$$ with...

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Kvant Math Problem 41

Verified: no Verdicts: SKIP + SKIP Solve time: 6m22s Source on kvant.digital Problem Given a circle, its diameter $AB$, and a point $C$ on this diameter. Construct on the circle two points $X$ and $Y$, symmetric with respect to the diameter $AB$, such that the line $YC$ is perpendicular to the line $XA$. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the circle have center $O$ and radius $R$. Put...

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Kvant Math Problem 48

Verified: no Verdicts: SKIP + SKIP Solve time: 2m01s Source on kvant.digital Problem In an acute triangle $ABC$, the angle bisector $AD$, the median $BM$, and the altitude $CH$ intersect at a single point. Prove that the angle $BAC$ is greater than $45^\circ$. All-Union Mathematical Olympiad (1970, Grade 9) Exploration Let the common point of the angle bisector $AD$, the median $BM$, and the altitude $CH$ be $P$. The condition...

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Kvant Math Problem 38

Verified: no Verdicts: SKIP + SKIP Solve time: 7m52s Source on kvant.digital Problem Fig. 2 A circle constructed on the altitude $AD$ of the right triangle $ABC$ as a diameter intersects leg $AB$ at point $K$ and leg $AC$ at point $M$. The segment $KM$ intersects the altitude $AD$ at point $L$. It is known that the segments $AK$, $AL$, and $AM$ form a geometric progression, i.e., $\dfrac{AK}{AL}=\dfrac{AL}{AM}$. Find the...

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Kvant Math Problem 40

Verified: no Verdicts: SKIP + SKIP Solve time: 5m30s Source on kvant.digital Problem Find the sum $$1\cdot n + 2\cdot (n-1) + 3\cdot (n-2)+ \ldots + n\cdot 1.$$ Try to solve the following, more general problem: find the sum $$\begin{aligned} S_{n,k}={}&[1\cdot 2\cdot \ldots \cdot k] \times [n(n-1)\cdot\ldots\cdot (n-k+1)]+{}\ {}+{}& [2\cdot 3\cdot \ldots \cdot (k+1)] \times [(n-1)(n-2)\cdot\ldots\cdot (n-k)]+{}\ {}+{}& [3\cdot 4\cdot \ldots \cdot (k+2)] \times [(n-2)(n-1)\cdot\ldots\cdot (n-k-1)]+{}\ {}+{}& {\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots} +{}\ {}+{}&...

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Kvant Math Problem 43

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Fig. 3 Each side of an equilateral triangle is divided into $n$ equal parts. Through the division points, lines parallel to the sides are drawn. As a result, the triangle is partitioned into $n^2$ small triangles (Fig. 3). Let us call a “chain” a sequence of triangles in which no triangle appears twice and each subsequent triangle...

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Kvant Math Problem 42

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem The digits of a certain seventeen-digit number are written in reverse order. The resulting number is added to the original one. Prove that at least one of the digits of their sum is even. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the original seventeen-digit number be $a_{16}a_{15}\cdots a_1a_0,$ and let its reverse be $a_0a_1\cdots a_{15}a_{16}.$ The...

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Kvant Math Problem 39

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Prove that the nonnegative integers $x$, $y$ satisfy the equation $$x^2-mxy+y^2=1$$ (where $m$ is a given integer greater than one) if and only if $x$ and $y$ are consecutive terms of the sequence $\phi_0=0$, $\phi_1=1$, $\phi_2=m$, $\phi_3=m^2-1$, $\phi_4=m^3-2m$, $\phi_5=m^4-3m^2+1$, $\ldots$, in which $\phi_{k+1}=m\phi_k-\phi_{k-1}$ for all $k\ge 1$. For example, all solutions of the equation $$x^2-3xy+y^2=1\tag1$$ in integers...

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Kvant Math Problem 37

Verified: no Verdicts: SKIP + SKIP Solve time: 1m55s Source on kvant.digital Problem In each cell of an infinite sheet of squared paper, a number is written in such a way that the sum of the numbers in any square (whose sides run along the grid lines) has absolute value at most one. Prove that there exists a number $c$ such that the sum of the numbers in any rectangle...

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Kvant Math Problem 33

Verified: no Verdicts: SKIP + SKIP Solve time: 5m10s Source on kvant.digital Problem There is a natural number $n \gt 1000$. Consider the remainders obtained when the number $2^n$ is divided by the numbers 1, 2, 3, ..., $n$, and find the sum of all these remainders. Prove that this sum is greater than $2n$. A. G. Kushnirenko Moscow Mathematical Olympiad (1970) Exploration Let $$S=\sum_{k=1}^{n} r_k,$$ where $r_k$ is the...

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Kvant Math Problem 36

Verified: no Verdicts: SKIP + SKIP Solve time: 1m06s Source on kvant.digital Problem Prove that it is impossible to arrange seven lines and seven points in the plane so that exactly three lines pass through each point and exactly three points lie on each line. Exploration Consider arranging seven points and seven lines such that each point lies on exactly three lines and each line contains exactly three points. Each...

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Kvant Math Problem 32

Verified: no Verdicts: SKIP + SKIP Solve time: 3m17s Source on kvant.digital Problem In every cell of an $100\times 100$ table there is a plus sign. It is allowed to simultaneously change the signs in all cells of a single row or a single column. Is it possible, after performing such operations several times, to obtain a table containing exactly 1970 minus signs? A. V. Zelevinsky Moscow Mathematical Olympiad (1970)...

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Kvant Math Problem 35

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem A certain 19-faced polyhedron is circumscribed about a sphere of radius 10. Prove that there exist two points on its surface whose distance apart is greater than 21. A. G. Kushnirenko Moscow Mathematical Olympiad (1970) Exploration The polyhedron has 19 faces and is circumscribed about a sphere of radius $10$. The first instinct is to consider the...

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Kvant Math Problem 34

Verified: no Verdicts: SKIP + SKIP Solve time: 2m02s Source on kvant.digital Problem Prove that if a natural number is divisible by $10,101,010,101$, then in its decimal representation at least six digits are nonzero. A. K. Tolpygo Moscow Mathematical Olympiad (1970) Exploration The number $$10,101,010,101$$ has a very suggestive decimal form. Computing, $$10,101,010,101=10^{10}+10^8+10^6+10^4+10^2+1.$$ It is useful to factor it. Since $$10^{12}-1=(10^6-1)(10^6+1),$$ and $$10^{12}-1=(10^2-1)(10^{10}+10^8+10^6+10^4+10^2+1),$$ we get $$10,101,010,101=\frac{10^{12}-1}{99}.$$ A direct computation gives...

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Kvant Math Problem 689

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem Prove that it is impossible to form a rectangle from identical tiles shaped as isosceles trapezoids with bases 3 cm and 1 cm and height 1 cm. S. Rukshin Exploration Each tile is an isosceles trapezoid with bases $3$ and $1$ and height $1$. Its area is $$\frac{3+1}{2}\cdot 1=2.$$ The nonparallel sides have length $$\sqrt{1^2+1^2}=\sqrt2,$$ because the...

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Kvant Math Problem 31

Verified: no Verdicts: SKIP + SKIP Solve time: 2m43s Source on kvant.digital Problem A square sheet of paper is cut along a straight line into two pieces. One of the resulting pieces is then cut into two, and this process is repeated many times. What is the minimum number of cuts needed so that among the resulting pieces there are exactly one hundred 20-sided polygons? I. N. Bernstein Moscow Mathematical...

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Kvant Math Problem 30

Verified: no Verdicts: SKIP + SKIP Solve time: 1m54s Source on kvant.digital Problem Prove that $N$ points in the plane can always be covered by several pairwise disjoint circles whose total sum of diameters is less than $N$ and the distance between any two of them is greater than 1. (The distance between two circles is understood to be the distance between their nearest points.) V. I. Arnold Exploration For...

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Kvant Math Problem 20

Verified: no Verdicts: SKIP + SKIP Solve time: 9m17s Source on kvant.digital Problem Is it possible to divide an equilateral triangle into one million convex polygons so that any straight line intersects no more than forty of these polygons? (We say that a line intersects a polygon if it has at least one point in common with it.) Moscow Mathematical Olympiad (XXXI) Exploration Let the maximum number of polygons met...

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Kvant Math Problem 670

Verified: no Verdicts: SKIP + SKIP Solve time: 10m14s Source on kvant.digital Problem Several points are given, with some pairs connected by lines (the points in such pairs are called neighbors ). Each point has an odd number of neighbors. Initially, all points are colored in two colors — red and blue. Then, every minute, the points are simultaneously recolored according to the following rule: each point whose majority of...

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Kvant Math Problem 29

Verified: no Verdicts: SKIP + SKIP Solve time: 2m03s Source on kvant.digital Problem $n$ identical coins are lying on a table, forming a closed chain (see the figure). How many rotations will a coin $M$ of the same size make while rolling once along the outer side of the entire chain, as shown in the figure (coin $M = 2~\text{коп.}$)? How does the answer change if the coin $M$ has...

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Kvant Math Problem 28

Verified: no Verdicts: SKIP + SKIP Solve time: 2m24s Source on kvant.digital Problem Among 19 balls, 2 are radioactive. For any collection of balls, a single test can determine whether it contains at least one radioactive ball or not (but it cannot determine how many such balls the collection contains). Prove that both radioactive balls can be identified in 8 tests. Among 11 balls, 2 are radioactive. Prove that it...

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Kvant Math Problem 25

Verified: no Verdicts: SKIP + SKIP Solve time: 2m10s Source on kvant.digital Problem In a set consisting of $n$ elements, $2^{n-1}$ subsets have been chosen, every three of which have a common element. Prove that all these subsets have a common element. Moscow Mathematical Olympiad (XXX) Exploration For small values of $n$ the statement is easy to test. If $n=1$, the family consists of one subset, and the claim is...

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Kvant Math Problem 26

Verified: no Verdicts: SKIP + SKIP Solve time: 1m24s Source on kvant.digital Problem Suppose that each issue of the Kvant problem book contains five mathematics problems, and the journal is published monthly. Denote by $f(x,y)$ the number of the first problem in the $x$-th issue for the $y$-th year (for example, $f(6, 1970)=26$). Write a general formula for $f(x,y)$, for all $x$, $y$ ($1\le x\le12$, $y\ge 1970$). Solve the equation...

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Kvant Math Problem 24

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem Prove that any fraction $\dfrac mn$, where $0\lt\dfrac mn\lt1$, can be represented in the form $$\dfrac mn=\dfrac1{q_1}+\dfrac1{q_2}+\dfrac1{q_3}+\ldots+\dfrac1{q_r},$$, where $0 \lt q_1 \lt q_2 \lt \ldots \lt q_r$ are integers, and each $q_k$ ($k = 2$, 3, $\ldots$, $r$) is divisible by $q_{k-1}$. Exploration The condition on the denominators is much stronger than in the usual Egyptian fraction...

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Kvant Math Problem 23

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem Prove that for all natural $n>1$ $$\frac{\sqrt{2}}{2} \frac{1}{\sqrt{2n}} \lt \frac{1\cdot 3 \cdot\ldots\cdot (2n-1)}{2\cdot 4\cdot \ldots \cdot 2n} \lt \frac{\sqrt{3}}{2} \frac{1}{\sqrt{2n}}.$$ A. O. Gelfond Exploration Let $$a_n=\frac{1\cdot3\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot\ldots\cdot2n}.$$ The statement becomes $$\frac1{2\sqrt n}<a_n<\frac{\sqrt3}{2\sqrt{2n}}.$$ The quantity resembles the central binomial coefficient because $$a_n=\frac{(2n)!}{2^{2n}(n!)^2}=\frac1{4^n}\binom{2n}{n}.$$ The asymptotic behavior is $$a_n\sim \frac1{\sqrt{\pi n}},$$ so the constants in the problem are plausible: $$\frac1{2}=0.5,\qquad \frac1{\sqrt\pi}\approx0.564,\qquad...

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Kvant Math Problem 22

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Two circles are inscribed in an angle; they have a common internal tangent $T_1 T_2$ ($T_1$ and $T_2$ are the points of tangency), which intersects the sides of the angle at points $A_1$ and $A_2$. Prove that $A_1 T_1=A_2 T_2$ (or, equivalently, $A_1 T_2=A_2 T_1$). Two circles are inscribed in an angle. One of them touches the...

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Kvant Math Problem 21

Verified: no Verdicts: SKIP + SKIP Solve time: 1m29s Source on kvant.digital Problem Inside a square of side length 1, several circles are placed whose total circumference is 10. Prove that there exists a line intersecting at least four of these circles. Evening Mathematical School at Moscow State University (MSU) Exploration Let the circles have radii $r_1,r_2,\dots,r_n$. Since their total circumference is $10$, $$2\pi\sum_{i=1}^n r_i=10, \qquad \sum_{i=1}^n r_i=\frac5\pi.$$ The statement...

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Kvant Math Problem 19

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem In an infinite chain of nerve cells, each cell can be in one of two states: "rest" or "excited." If a cell becomes excited at a given moment, it sends a signal that reaches both of its neighboring cells after a unit of time (say, one millisecond). A cell becomes excited if and only if a signal...

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Kvant Math Problem 18

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Prove that for any point $M$ on the circumcircle of the equilateral triangle $ABC$, one of the three segments $MA$, $MB$, $MC$ is equal to the sum of the other two. Three equal circles $\gamma_1$, $\gamma_2$, $\gamma_3$ are pairwise tangent to each other, and a circle $\gamma$ is circumscribed around them, touching all three: $\gamma_1$, $\gamma_2$, and...

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Kvant Math Problem 17

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem A peasant, having come to a fork where two roads diverged at an angle of $60\degree$, asked, “How do I get to the village of $NN$?” He was told: “Take the left road to the village of $N$, which is eight versts from here. There you will see a broad straight road branching off to the right...

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Kvant Math Problem 15

Verified: no Verdicts: SKIP + SKIP Solve time: 2m51s Source on kvant.digital Problem A square table of size $n\times n$ is filled with non-negative numbers such that the sum of the numbers in each row and the sum of the numbers in each column are both equal to 1. Prove that it is possible to select $n$ positive numbers from the table, no two of which are in the same...

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Kvant Math Problem 16

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem Prove that a polynomial $p(x)$ with integer coefficients, which takes the value 1 for three distinct integer values of $x$, cannot have any integer roots. Exploration Consider a polynomial $p(x)$ with integer coefficients that takes the value $1$ at three distinct integers, say $a$, $b$, and $c$. Denote $p(a) = p(b) = p(c) = 1$. A natural...

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Kvant Math Problem 664

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem Given a quadrilateral $ABCD$ with area $S$. Let the points of intersection of the altitudes of triangles $ABC$, $BCD$, $CDA$, $DAB$ be denoted by $H$, $K$, $L$, $M$ respectively. Prove that the area of the quadrilateral $HKLM$ is also equal to $S$. V. Batyrev, V. Trofimov Exploration Consider a convex quadrilateral $ABCD$ with area $S$. Denote by...

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Kvant Math Problem 13

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Prove that if the difference between the largest and the smallest of the $n$ real numbers $a_1$, $a_2$, $\ldots$, $a_n$ is equal to $d$, and the sum of the absolute values of all $\dfrac{n(n-1)}{2}$ pairwise differences of these numbers $$\sum_{i \lt j} |a_i - a_j|$$ is equal to $s$, then $$(n-1)d \le s \le \frac{n^2}{4}d.$$ Exploration Consider...

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Kvant Math Problem 657

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem In the table $n \times n$ filled with numbers, all rows are distinct. Prove that it is possible to delete some column from the table so that all rows in the remaining table are still distinct. A. V. Anzhan Exploration Let the rows be $R_1,\dots,R_n$, each a vector of length $n$. We must show that some column...

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Kvant Math Problem 653

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem Fig. 1 There is a ruler with two markings (Fig. 1). Using this ruler, one can draw arbitrary straight lines and lay off segments of a given length. Using it, construct: any right angle; a line perpendicular to a given line. V. L. Gutenmacher Correspondence Mathematical Olympiads Exploration The ruler has two fixed marks. Let the distance...

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Kvant Math Problem 652

Verified: no Verdicts: SKIP + SKIP Solve time: 3m57s Source on kvant.digital Problem Zhenya cut a convex polyhedron made of cardboard into its faces (along the edges) and mailed this set of faces to Vitya. Using all of these faces, Vitya assembled a convex polyhedron. Can it happen that Zhenya's and Vitya's polyhedra are not congruent? N. B. Vasiliev Correspondence Mathematical Olympiads Exploration Consider the set of faces of a...

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Kvant Math Problem 648

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem Prove that if the diagonals of a cyclic quadrilateral are perpendicular, then the midpoints of its sides and the feet of the perpendiculars dropped from the intersection point of its diagonals to the sides all lie on a single circle. I. F. Sharygin Exploration Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at...

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Kvant Math Problem 647

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem Prove that for all $a\ge\dfrac12$, $b\ge\dfrac12$ the inequality $$\left(\dfrac{a^2-b^2}{2}\right)^2\ge\sqrt{\dfrac{a^2+b^2}{2}}-\dfrac{a+b}{2}.$$ holds. S. V. Fomin Exploration The inequality is symmetric in $a$ and $b$. The right-hand side contains the classical comparison between the root mean square and the arithmetic mean: $$\sqrt{\frac{a^2+b^2}{2}}\ge \frac{a+b}{2}.$$ Hence the right-hand side is nonnegative. The left-hand side equals $$\frac{(a^2-b^2)^2}{4} =\frac{(a-b)^2(a+b)^2}{4}.$$ Since $a,b\ge \tfrac12$, we have...

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Kvant Math Problem 643

Verified: no Verdicts: SKIP + SKIP Solve time: 5m44s Source on kvant.digital Problem Cards numbered 1, 2, $\ldots$, 32 are stacked in order. One may remove any number of cards from the top and insert them among the remaining cards or underneath them, preserving the order within both groups, but otherwise arbitrarily. This operation is called a shuffle . Prove that in 5 shuffles one can rearrange the cards into...

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Kvant Math Problem 639

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem In the tetrahedron $ABCD$, $(AC) \perp (BC)$ and $(AD) \perp (BD)$. Prove that the cosine of the angle between the lines $AC$ and $BD$ is less than $\dfrac{|CD|}{|AB|}$. Yu. V. Nesterenko All-Union Mathematical Olympiad (XIV, 1980, Grade 10) Exploration Let $$u=\overrightarrow{AC},\qquad v=\overrightarrow{BD}.$$ The assumptions are $$u\perp \overrightarrow{BC},\qquad \overrightarrow{AD}\perp v.$$ A vector formulation seems natural. Put $$x=\overrightarrow{AB}.$$ Then...

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Kvant Math Problem 635

Verified: no Verdicts: SKIP + SKIP Solve time: 8m36s Source on kvant.digital Problem The Mites living in Flower Town suddenly began coming down with influenza. On one particular day, several Mites caught cold and fell ill, and although no one caught cold thereafter, healthy Mites became ill when visiting their sick friends. It is known that each Mite has influenza for exactly one day; moreover, after that he has immunity...

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Kvant Math Problem 634

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Let $S(n)$ denote the sum of all digits of a natural number $n$. Does there exist a natural number $n$ such that $n+S(n)=1980$? Prove that at least one of any two consecutive natural numbers can be represented in the form $n+S(n)$ for some third natural number $n$. S. V. Konyagin All-Union Mathematical Olympiad (XIV, 1980, Grade 8)...

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Kvant Math Problem 633

Verified: no Verdicts: SKIP + SKIP Solve time: 1m51s Source on kvant.digital Problem On a diameter $AC$ of a certain circle, a point $E$ is given. Draw through it a chord $BD$ so that the area of the quadrilateral $ABCD$ is maximized. I. F. Sharygin All-Union Mathematical Olympiad (XIV, 1980, 9th grade) Exploration Let the circle have center $O$ and radius $R$. The diameter is $AC$, and the point $E$...

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Kvant Math Problem 632

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem A cargo packed in containers must be delivered to the orbital space station Salyut. The number of containers is at least 35, and the total mass of the cargo is 18 tons. There are seven Progress transport spacecraft available, each of which can deliver 3 tons of cargo into orbit. It is known that these spacecraft can...

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Kvant Math Problem 631

Verified: no Verdicts: SKIP + SKIP Solve time: 1m45s Source on kvant.digital Problem The two-digit numbers from 19 to 80 are written consecutively. Is the resulting number 192021...7980 divisible by 1980? V. Fedotov All-Union Mathematical Olympiad (XIV, 1980, Grade 8) Exploration Let $$N=192021\cdots 7980.$$ Since $$1980=2^2\cdot 3^2\cdot 5\cdot 11,$$ checking divisibility by $1980$ amounts to checking divisibility by $4$, $9$, $5$, and $11$. The last digit of $N$ is $0$,...

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Kvant Math Problem 629

Verified: no Verdicts: SKIP + SKIP Solve time: 1m31s Source on kvant.digital Problem Prove that the number $2^{2n-1}-9n^2+21n-14$ is divisible by 27 for every positive integer $n$. Prove that if the numbers $a+b$ and $a^2+b$ are divisible by $m$, then $a^n+b$ is divisible by $m$ for every $n$ ($a$, $b$, and $m$ are some positive integers). Prove that if $f(n)=a^n+b_0+b_1n+\ldots+b_kn^k$ is divisible by $m$ for $n=1$, $n=2$, $\ldots$, $n=k+1$, $n=k+2$,...

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Kvant Math Problem 626

Verified: no Verdicts: SKIP + SKIP Solve time: 1m47s Source on kvant.digital Problem Each side of a convex quadrilateral is divided into 8 equal parts. Corresponding division points on opposite sides are connected to each other, and the resulting cells are colored in a checkerboard pattern. Prove that the sum of the areas of the black cells is equal to the sum of the areas of the white cells. V....

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Kvant Math Problem 625

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem On the coordinate plane, four points with rational coordinates are given, which do not lie at the vertices of a parallelogram, and no three of them lie on the same line. It is allowed to draw a line through any two already obtained points and mark the intersection point of any two drawn lines. Prove that the...

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Kvant Math Problem 624

Verified: no Verdicts: SKIP + SKIP Solve time: 1m25s Source on kvant.digital Problem Find the sequence $(a_n)$ defined by the conditions $a_1=1$, $$1+\textstyle\sum\limits_d{}(-1)^{n/d}a_{d}=0, \tag{*}$$, where the sum $\sum$ is taken over all divisors $d$ of the number $n$ (including $d=1$ and $d=n$). For example, if $n=p$ is a prime number, then (*) takes the form $1+(-1)^{p/1}a_1+(-1)^{p/p}a_p=0$, from which $a_p=2$ if $p=2$, and $a_p=0$ if $p\gt 2$. V. Abramovich Exploration Compute...

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Kvant Math Problem 623

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem How many axes of symmetry does a cube have? A regular triangular pyramid? Prove that if a certain polyhedron has $k$ axes of symmetry ($k\ge 1$), then $k$ is odd. V. A. Senderov Exploration A cube is highly symmetric, so the number of axes of symmetry should be larger than in simpler polyhedra. For a cube, one...

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Kvant Math Problem 622

Verified: no Verdicts: SKIP + SKIP Solve time: 1m15s Source on kvant.digital Problem Prove that the number of solutions of the equation $$x^3+y^2=z^3+t^2+1$$ in natural numbers not exceeding $10^6$ is less than the number of solutions of the equation $$x^3+y^2=z^3+t^2$$ in natural numbers not exceeding $10^6$. V. Vavilov Exploration Consider the two Diophantine equations $x^3 + y^2 = z^3 + t^2$ and $x^3 + y^2 = z^3 + t^2 +...

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Kvant Math Problem 617

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem Inside a triangle are four circles $\alpha$, $\beta$, $\gamma$, $\delta$ of equal radius, arranged so that each of the circles $\alpha$, $\beta$, $\gamma$ is tangent to two sides of the triangle and to the circle $\delta$ (Fig. 1). Prove that the center of the circle $\delta$ lies on the line passing through the centers of the inscribed...

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Kvant Math Problem 616

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Can the numbers 1, 2, $\ldots$, 30 be partitioned into groups of five numbers each, of six numbers each, so that the sums of the numbers in all groups are equal? For which $n$ and $k$ can the numbers 1, 2, $\ldots$, $nk$ be partitioned into $n$ groups of $k$ numbers each so that the sums of...

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Kvant Math Problem 615

Verified: no Verdicts: SKIP + SKIP Solve time: 2m11s Source on kvant.digital Problem Prove that the perimeter of any cross-section of a triangular pyramid by a plane does not exceed the greatest of the perimeters of its faces. V. A. Senderov Exploration A triangular pyramid is a tetrahedron. A plane section of a tetrahedron is either a triangle or a quadrilateral. The statement compares the perimeter of the section with...

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Kvant Math Problem 614

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem For each natural number $n$, let $S(n)$ denote the sum of the digits of all natural numbers from 1 to $n$ (in decimal notation): $$\begin{gather*} S(1)=1,\quad S(2)=3,\quad S(3)=6,\quad \ldots,\quad S(9)=45,\ S(10)=46,\quad S(11)=48,\quad S(12)=51,\quad \ldots \end{gather*}$$ Find $S(100)$. Prove that $S(10^k-1)=45k\cdot10^{k-1}$ for all $k=1$, 2, $\ldots$ Prove that for any two-digit number $\overline{ab}$, $$S(\overline{ab})=5a^2+ab+41a+\dfrac{b(b+1)}{2}.$$ Find a similar formula...

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Kvant Math Problem 613

Verified: no Verdicts: SKIP + SKIP Solve time: 1m59s Source on kvant.digital Problem On the sides of triangle $ABC$, similar triangles $ADB$, $BEC$, and $CFA$ are constructed externally $\Big(\dfrac{|AD|}{|DB|}=\dfrac{|BE|}{|EC|}=\dfrac{|CF|}{|FA|}=k$; $\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=\alpha\Big)$. Prove that: the midpoints of the segments $AC$, $DC$, $BC$, and $EF$ are the vertices of a parallelogram; this parallelogram has two angles equal to $\alpha$, and the ratio of the lengths of its sides is $k$. L. P. Kuptsov...

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Kvant Math Problem 612

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem An increasing sequence of natural numbers $(a_n)$ is such that $a_{n+1}\le 10a_n$. Prove that if all the numbers $a_n$ are written one after another (without spaces or commas), then the resulting sequence of digits is not periodic. A. Karagulyan Exploration Assume that the infinite digit string obtained by concatenating $$a_1,a_2,a_3,\dots$$ is periodic with period $T$. Then every...

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Kvant Math Problem 611

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem An arbitrary point $M$ is chosen on the chord $AB$ of a circle with center $O$. A circle is drawn through the points $A$, $M$, and $O$; it intersects the first circle at the points $A$ and $C$. Prove that $|MB|=|MC|$. S. Kolpakov, 10th-grade student Exploration The statement involves two circles. The first circle has center $O$,...

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Kvant Math Problem 610

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem Fix $k\in \N$. Consider the set of all tuples of integers $a_1$, $\ldots$, $a_k$ such that $0\le a_1\le a_2\le\ldots \le a_k \le k$; denote the number of such tuples by $N$. Among them, consider those for which $a_k=k$; let their number be $M$. Prove that $N=2M$. Impose an additional restriction on the tuples under consideration: the sum...

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Kvant Math Problem 609

Verified: no Verdicts: SKIP + SKIP Solve time: 2m05s Source on kvant.digital Problem The lengths of the projections of a convex polygon of area $S$ onto two mutually perpendicular lines are equal to $l_1$ and $l_2$. Prove that $S \le l_1l_2$. The lengths of the projections of a convex polyhedron of volume $V$ onto three mutually perpendicular lines are equal to $l_1$, $l_2$, and $l_3$. Prove that $V \le l_1l_2l_3$....

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Kvant Math Problem 608

Verified: no Verdicts: SKIP + SKIP Solve time: 1m46s Source on kvant.digital Problem On squared paper (with unit grid spacing), an $n$-gon is drawn, all of whose sides lie on grid lines and have odd length. Prove that $n$ is divisible by 4. Prove that when $n=100$, the area of this $n$-gon is necessarily odd. Determine the parity of the area for other values of $n$. M. L. Kontsevich, 10th-grade...

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Kvant Math Problem 607

Verified: no Verdicts: SKIP + SKIP Solve time: 2m03s Source on kvant.digital Problem Cut a square into isosceles trapezoids. Cut an isosceles right triangle into isosceles trapezoids. Prove that any polygon can be cut into isosceles trapezoids. V. F. Lev Exploration An isosceles trapezoid includes rectangles as a special case, since a rectangle has a pair of parallel sides and equal legs. Thus any rectangle is an isosceles trapezoid. For...

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Kvant Math Problem 606

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem The function $f$ is such that for all real $x$ $$f(x+1)+f(x-1)=\sqrt2,f(x).$$ Prove that $f$ is a periodic function. E. Turkevich Exploration The recurrence $$f(x+1)+f(x-1)=\sqrt2,f(x)$$ relates values of the function at arguments differing by integers. Since the coefficient is constant, the first idea is to look for solutions of the form $$f(x+n)=r^n.$$ Then $$r+r^{-1}=\sqrt2,$$ hence $$r^2-\sqrt2,r+1=0.$$ The discriminant...

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Kvant Math Problem 605

Verified: no Verdicts: SKIP + SKIP Solve time: 2m23s Source on kvant.digital Problem There are $2n+1$ distinct points marked on the plane. Number them 1, 2, $\ldots$, $2n+1$ and consider the following transformation $R$ of the plane: first, reflection with respect to the first point is performed, then with respect to the second point, and so on, up to the $(2n+1)$th point. Show that this transformation $R$ has a unique...

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Kvant Math Problem 604

Verified: no Verdicts: SKIP + SKIP Solve time: 58s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M604 from the information provided, because the actual problem statement is missing. The prompt says: “The text version of the problem statement is currently being prepared. The graphical...

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Kvant Math Problem 603

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem Solve the system of equations $$x+\dfrac{3x-y}{x^2+y^2}=3,\quad y-\dfrac{x+3y}{x^2+y^2}=0.$$ L. P. Kuptsov Exploration The denominators suggest introducing $$r=x^2+y^2.$$ Since the system contains the expressions $\dfrac{3x-y}{r}$ and $\dfrac{x+3y}{r}$, multiplying by $r$ should convert the system into linear relations between $x,y$ and $r$. Multiplying both equations by $r$ gives $$xr+3x-y=3r,$$ $$yr-x-3y=0.$$ The second equation immediately yields $$x=y(r-3).$$ Substituting into the first...

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Kvant Math Problem 602

Verified: no Verdicts: SKIP + SKIP Solve time: 1m50s Source on kvant.digital Problem Fig. 1. In the $n$th row, the $k$th number is equal to $C_n^k=\dfrac{n(n-1)\ldots(n-k+1)}{1\cdot2\cdot\ldots\cdot k}$ In Pascal’s triangle, whose beginning is shown in Fig. 1 (each “interior” number is equal to the sum of the two numbers above it), the seventh row contains three consecutive numbers that form an arithmetic progression. In which subsequent row will the same...

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Kvant Math Problem 601

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem Prove that in any triangle $ABC$, the midpoint of side $BC$ lies on the segment joining the intersection point of the altitudes of triangle $ABC$ to the point on the circumcircle of this triangle that is diametrically opposite vertex $A$, and bisects this segment. G. Groshev, 10th-grade student Exploration Let $H$ be the orthocenter of triangle $ABC$,...

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Kvant Math Problem 600

Verified: no Verdicts: SKIP + SKIP Solve time: 2m02s Source on kvant.digital Problem Two cyclists ride along two intersecting circles. Each cyclist travels on his own circle at a constant speed. Starting simultaneously from one of the intersection points of the circles and completing one revolution each, the cyclists meet again at that point. Prove that in the plane containing the circles there exists a fixed point such that its...

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Kvant Math Problem 27

Verified: no Verdicts: SKIP + SKIP Solve time: 9m56s Source on kvant.digital Problem Prove that if $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0,$$, then $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0.$$. Exploration Let $$ S=\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}. $$ The denominators suggest introducing the differences $$ x=b-c,\qquad y=c-a. $$ Then $$ a=c-y,\qquad b=c+x,\qquad a-b=-(x+y). $$ Substituting into $S$ gives $$ S=\frac{c-y}{x}+\frac{c+x}{y}-\frac{c}{x+y}. $$ After bringing the terms to the common denominator $xy(x+y)$, the numerator becomes $$ c(x^2+xy+y^2)+x^3+x^2y-xy^2-y^3. $$ The cubic part factors as $$ x^3+x^2y-xy^2-y^3=(x-y)(x+y)^2. $$...

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Kvant Math Problem 14

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem Fig. 2 A convex white polyhedron has some of its faces painted black in such a way that no two black faces share an edge (Fig. 2). Prove that if at least one of the following conditions holds: the number of black faces is greater than half the total number of faces; the total area of the...

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Kvant Math Problem 11

Verified: no Verdicts: SKIP + SKIP Solve time: 4m23s Source on kvant.digital Problem Forty-four merry siskins were perched on 44 trees arranged in a circle (one siskin on each tree). From time to time, two siskins simultaneously fly to neighboring trees in opposite directions (one clockwise, the other counterclockwise; see Fig. 1). Prove that the siskins will never gather on a single tree. What if the number of siskins and...

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Kvant Math Problem 1

Verified: no Verdicts: SKIP + SKIP Solve time: 3m46s Source on kvant.digital Problem In the country of Anchuria, ruled by President Miraflores, the time for a new presidential election is approaching. The country has exactly 20 million voters, of whom only one percent (the regular army of Anchuria) support Miraflores. Naturally, Miraflores wants to be elected, but at the same time, he wants the election to appear democratic. By "democratic...

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Kvant Math Problem 10

Verified: no Verdicts: SKIP + SKIP Solve time: 4m14s Source on kvant.digital Problem Four circles whose centers are the vertices of a convex quadrilateral completely cover that quadrilateral. Prove that one can choose three of these circles that cover the triangle whose vertices are the centers of those circles. G. A. Galperin G. A. Galperin Exploration Let the centers of the circles be the vertices $A,B,C,D$ of a convex quadrilateral,...

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Kvant Math Problem 9

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Consider the following properties of a tetrahedron (by a tetrahedron we mean an arbitrary triangular pyramid): all faces have equal area; each edge is equal to its opposite edge; all faces are congruent; the centers of the circumscribed and inscribed spheres coincide; the sums of the plane angles at each vertex of the tetrahedron are equal. Prove...

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Kvant Math Problem 8

Verified: no Verdicts: SKIP + SKIP Solve time: 3m33s Source on kvant.digital Problem Two people play the following game. From a pile containing 25 matches, they take turns taking 1, 2, or 3 matches. The winner is the one who, at the end of the game—after all the matches have been taken—has an even number of matches. Who wins with correct play—the first player or the second? How should the...

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Kvant Math Problem 7

Verified: no Verdicts: SKIP + SKIP Solve time: 5m50s Source on kvant.digital Problem $a$, $b$, $c$ are the sides of a triangle. Prove that $$\frac{a}{b+c-a} + \frac{b}{c+a-b} + \frac{c}{a+b-c} \geq 3.$$ S. T. Berkolayko Exploration Let $$S=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}.$$ Since $a,b,c$ are sides of a triangle, each denominator is positive by the triangle inequalities. A natural test is the equilateral triangle. For $a=b=c$, $$S=1+1+1=3.$$ This suggests that $3$ may be the minimum....

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Kvant Math Problem 6

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem You are given a clock. How many positions of the hands exist for which it is impossible to determine the time without knowing which hand is the hour hand and which is the minute hand? ( It is assumed that the position of each hand can be determined exactly, but observing their movement is not allowed. )...

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Kvant Math Problem 5

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem In the set $E$, consisting of $n$ elements, $m$ distinct subsets (different from $E$ itself) have been selected so that for every two elements of $E$ there exists exactly one of the given subsets containing both of these elements. Prove that $m \geq n$. In what cases can equality occur? N. Bourbaki Exploration Let $E$ be a...

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Kvant Math Problem 4

Verified: no Verdicts: SKIP + SKIP Solve time: 3m05s Source on kvant.digital Problem Given a segment $AB$. Find the set of points $C$ in the plane such that in the triangle $ABC$ the median drawn from vertex $A$ is equal to the altitude drawn from vertex $B$. Moscow Mathematical Olympiad (XXXII) Exploration Fix the segment $AB$ and let its length be $d$. The condition says that the median from $A$...

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Kvant Math Problem 2

Verified: no Verdicts: SKIP + SKIP Solve time: 2m19s Source on kvant.digital Problem A sphere of radius 1 is given. On it are placed equal circles $\gamma_0$, $\gamma_1$, $\ldots$, $\gamma_n$ of radius $r$ ($n\ge3$). The circle $\gamma_0$ is tangent to all circles $\gamma_1$, $\ldots$, $\gamma_n$; moreover, the circles $\gamma_1$ and $\gamma_2$; $\gamma_2$ and $\gamma_3$; $\ldots$; $\gamma_n$ and $\gamma_1$ are tangent to each other. For which $n$ is this possible? Calculate...

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Kvant Math Problem 3

Verified: no Verdicts: SKIP + SKIP Solve time: 4m10s Source on kvant.digital Problem In Figure 1, the plane is tiled by squares of five colors. The centers of squares of the same color are located at the vertices of a square lattice. For what numbers of colors is a similar tiling of the plane possible? Fig. 1 Fig. 2 In Figure 2, the plane is tiled by hexagons of seven...

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Kvant Math Problem 241

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem Prove that $3^{1974}+5^{1974}$ is divisible by 13. S. I. Meyzus Exploration The exponent $1974$ is large, so direct computation is impossible. Since the statement concerns divisibility by $13$, the natural step is to work modulo $13$. Compute small powers. Since $$3^3=27\equiv 1 \pmod{13},$$ the powers of $3$ repeat with period $3$. Also $$5^2=25\equiv -1 \pmod{13},$$ hence $$5^4\equiv...

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IMO 1966 Problem 3

IMO 1966 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 11m33s Problem Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space. Exploration Let the regular tetrahedron have vertices $A,B,C,D$, and let $O$ be the center of...

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IMO 1965 Problem 5

IMO 1965 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m52s Problem Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range...

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IMO 1963 Problem 1

IMO 1963 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 11m46s Problem Find all real roots of the equation $\sqrt{x^2-p}+2\sqrt{x^2-1}=x$, where $p$ is a real parameter. Exploration The equation is $$\sqrt{x^2-p}+2\sqrt{x^2-1}=x,$$ with real parameter $p$. Since both square roots are real, the domain conditions are $$x^2-p\ge 0,\qquad x^2-1\ge 0.$$ Hence $|x|\ge 1$. The left-hand side is a sum of nonnegative quantities. Consequently the right-hand side...

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IMO 1962 Problem 7

IMO 1962 Problem 7 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 15m54s Problem Prove that a tetrahedron $SABC$ has five spheres each tangent to all six lines determined by its edges if and only if it is regular. Exploration The previous solution failed at the structural point where it tried to characterize tetrahedra admitting a sphere tangent to all six edge-lines. Before rebuilding the argument, it...

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IMO 1962 Problem 5

IMO 1962 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 12m07s Problem On a circle $k$ three points $A$, $B$, $C$ are given. Construct a fourth point $D$ on $k$ such that a circle can be inscribed in quadrilateral $ABCD$. Exploration The problem is a construction problem on a fixed circumcircle. Since the quadrilateral must admit an incircle, the first condition to examine is the...

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IMO 1962 Problem 4

IMO 1962 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 21m14s Problem Solve the equation $\cos^2 x + \cos^2 2x + \cos^2 3x = 1$. Exploration Consider the equation $\cos^2 x + \cos^2 2x + \cos^2 3x = 1.$ Using standard trigonometric identities, we have $\cos 2x = 2\cos^2 x - 1$ and $\cos 3x = 4\cos^3 x - 3\cos x$. Squaring these, we obtain...

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IMO 1962 Problem 2

IMO 1962 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 25m30s Problem Find all real numbers $x$ satisfying $$\sqrt{3-x} - \sqrt{x+1} > \frac{1}{2}.$$ Problem Understanding We seek all real numbers $x$ satisfying $$\sqrt{3-x}-\sqrt{x+1}>\frac12.$$ Because square roots are involved, the first step is to determine the domain. After that, the natural strategy is to study the related equation $$\sqrt{3-x}-\sqrt{x+1}=\frac12,$$ since the inequality is expected to change...

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IMO 1962 Problem 1

IMO 1962 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m17s Problem Find the smallest natural number $n$ with the following properties: (a) Its decimal representation ends in 6. (b) If the final digit 6 is moved to the front, the resulting number is four times as large. Exploration Let $n$ be a natural number ending in 6 such that moving the 6 to the...

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IMO 1961 Problem 5

IMO 1961 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 39m11s Problem Construct a triangle ABC if the following elements are given: $AC = b, AB = c$, and $\angle AMB = \omega \left(\omega < 90^{\circ}\right)$ where M is the midpoint of BC . Prove that the construction has a solution if and only if $b \tan{\frac{\omega}{2}} \le c < b$ In what case does...

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IMO 1961 Problem 4

IMO 1961 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 19m03s Problem In the interior of triangle $P_1 P_2 P_3$ a point P is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1, PP_2,PP_3$ with the opposing edges of triangle $ABC$. Prove that among the ratios $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there exists one not larger than 2 and one not smaller than 2. Exploration The problem is a pure...

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IMO 1961 Problem 2

IMO 1961 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 11m24s Problem Let a , b , and c be the lengths of a triangle whose area is S . Prove that $a^2 + b^2 + c^2 \ge 4S\sqrt{3}$ In what case does equality hold? Exploration The statement asks for a lower bound on $a^2+b^2+c^2$ in terms of the area $S$ of a triangle. Since...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 1

IMO 1961 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m48s Problem ( Hungary ) Solve the system of equations: $\begin{matrix} \quad x + y + z !!! &= a ; , \ x^2 +y^2+z^2 !!! &=b^2 \ \qquad \qquad xy !!! &= z^2 \end{matrix}$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 7

IMO 1960 Problem 7 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m59s Problem A regular cone is inscribed in a sphere. Around the sphere a cylinder is circumscribed so that its base lies in the same plane as the base of the cone. Let $V_1$ be the volume of the cone and $V_2$ that of the cylinder. (a) Prove that $V_1 = V_2$ is impossible. (b)...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 5

IMO 1991 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 11m04s Problem Let $,ABC,$ be a triangle and $,P,$ an interior point of $,ABC,$. Show that at least one of the angles $,\angle PAB,;\angle PBC,;\angle PCA,$ is less than or equal to $30^{\circ }$. Exploration Consider triangle $ABC$ with an interior point $P$. Denote the angles $\angle PAB$, $\angle PBC$, $\angle PCA$ by $\alpha$, $\beta$,...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 4

IMO 1991 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 31m43s Problem Suppose $,G,$ is a connected graph with $,k,$ edges. Prove that it is possible to label the edges $1,2,\ldots ,k,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. Exploration The reviewer correctly...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 4

IMO 1960 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 33m08s Problem Construct a triangle $ABC$ given the lengths of the altitudes $h_a$ and $h_b$ (from $A$ and $B$ respectively) and the length of the median $m_a$ from $A$. Exploration The previous solution failed at two genuinely important points. The first was the incorrect geometric construction of $\sqrt{m_a^2-h_a^2}$. The second was the unsupported claim that...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 3

IMO 1991 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 31m33s Problem Let $S = {1,2,3,\cdots ,280}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. Exploration The problem asks for the smallest integer $n$ such that every $n$-element subset of $S={1,2,\dots,280}$ contains five numbers that are pairwise relatively prime. The lower bound construction...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 3

IMO 1960 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 17m36s Problem A right-angled triangle $ABC$ is given for which the hypotenuse $BC$ has length $a$ and is divided into $n$ equal segments ($n$ odd). Let $\alpha$ be the angle subtended at $A$ by the segment containing the midpoint of $BC$. If $h$ denotes the altitude from $A$, prove that $$\tan\alpha = \frac{4nh}{(n^2-1)a}.$$ Exploration The...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 4

IMO 1975 SL 4 Origin: SWE Problem Let a1, a2, . . . , an, . . . be a sequence of real numbers such that 0 \leqan \leq1 and an −2an+1 + an+2 \geq0 for n = 1, 2, 3, . . .. Prove that 0 \leq(n + 1)(an −an+1) \leq2 for n = 1, 2, 3, . . .. Solution Put ∆an = an −an+1. By the imposed...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 2

IMO 1989 SL 2 Origin: AUS Problem Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he ﬁnds that if he lays it ﬂat on the ﬂoor of either of his storerooms, then each corner of the carpet touches a diﬀerent wall of that room. If the two rooms have...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 3

IMO 1979 SL 3 Origin: BUL Problem Find all polynomials f(x) with real coeﬃcients for which f(x)f(2x2) = f(2x3 + x). Solution An obvious solution is f(x) = 0. We now look for nonzero solutions. We note that plugging in x = 0 we get f(0)2 = f(0); hence f(0) = 0 or f(0) = 1. If f(0) = 0, then f is of the form f(x) = xkg(x), where...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 18

IMO 1991 SL 18 Origin: BUL Problem Find the highest degree k of 1991 for which 1991k divides the number 199019911992 + 199219911990. Solution For integers a > 0, n > 0 and \alpha \geq0, we shall write a\alpha \paralleln when a\alpha | n and a\alpha+1 ∤n. Lemma. For every odd number a \geq3 and an integer n \geq0 it holds that an+1 \parallel(a + 1)an −1 and an+1 \parallel(a...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 20

IMO 1968 SL 20 Origin: CZS Problem Given $n$ ($n \geq 3$) points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1, A_2, \ldots, A_n$ in such a way that for each $i, j, k$, $1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 1

IMO 1988 SL 1 Origin: BUL Problem An integer sequence is deﬁned by an = 2an−1 + an−2 (n > 1), a0 = 0, a1 = 1. Prove that 2k divides an if and only if 2k divides n. Solution Assume that p and q are real and b0, b1, b2, . . . is a sequence such that bn = pbn−1 + qbn−2 for all n > 1. From...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 7

IMO 1976 SL 7 Problem (POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given number a \in(0, 1) we deﬁne a map T : I \toI by the formula T (x, y) = . x + (1 −a) if 0 < x \leqa, x −a if a < x \leq1. Show that for every interval J \subsetI there exists an integer...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G3

IMO 2004 SL G3 Origin: KOR | Category: Geometry Problem Let O be the circumcenter of an acute-angled triangle ABC with \angleB < \angleC. The line AO meets the side BC at D. The circumcenters of the triangles ABD and ACD are E and F, respectively. Extend the sides BA and CA beyond A, and choose on the respective extension points G and H such that AG = AC and...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 18

IMO 1985 SL 18 Problem 6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that x2 x2 1 + x2x3 + x2 x2 2 + x3x4 \cdot \cdot \cdot + x2 n−1 x2 n−1 + xnx1 x2 n x2n + x1x2 \leqn −1. Supplementary Problems Solution Set yi = x2 i xi+1xi+2 , where xn+i = xi. Then $n i=1 yi = 1 and the...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 16

IMO 1991 SL 16 Origin: ROM Problem Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers that are less than n and relatively prime to n. Show that if a1, a2, . . . , ak is an arithmetic progression, then n is a prime number or a natural power of two. Solution Let p be the least prime number that...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 2

IMO 1982 SL 2 Problem A2 (YUG 1) Let K be a convex polygon in the plane and suppose that K is positioned in the coordinate system in such a way that area (K \capQi) = 1 4 area K (i = 1, 2, 3, 4, ), where the Qi denote the quadrants of the plane. Prove that if K contains no nonzero lattice point, then the area of K...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 9

IMO 1973 SL 9 Origin: ROM Problem Let Ox, Oy, Oz be three rays, and G a point inside the trihe- dron Oxyz. Consider all planes passing through G and cutting Ox, Oy, Oz at points A, B, C, respectively. How is the plane to be placed in order to yield a tetrahedron OABC with minimal perimeter? Solution Let a, b, c be vectors going along Ox, Oy, Oz, respectively,...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 21

IMO 1997 SL 21 Origin: RUS Problem Let x1, x2, . . . , xn be real numbers satisfying the conditions |x1 + x2 + \cdot \cdot \cdot + xn| = 1 and |xi| \leqn + 1 for i = 1, 2, . . ., n. Show that there exists a permutation y1, . . . , yn of the sequence x1, . . . , xn such that |y1...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 9

IMO 1989 SL 9 Origin: FRA Problem For all integers n, n \geq0, there exist uniquely determined integers an, bn, cn such that 1 + 4 3\sqrt 2 −4 3\sqrt n = an + bn 3\sqrt 2 + cn 3\sqrt 4. Prove that cn = 0 implies n = 0. Solution From an+1 + bn+1 3\sqrt 2 + cn+1 3\sqrt 4 = (an + bn 3\sqrt 2 + cn...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G4

IMO 1996 SL G4 Origin: USA | Category: Geometry Problem Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ in the points $A_1$, $B_1$, $C_1$ respectively. Prove that $$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \geq A_1B \cdot B_1C \cdot C_1A.$$ Solution By the law of cosines applied to $\triangle CA_1B_1$, we obtain...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 2

IMO 1998 SL 2 Origin: POL Problem Let ABCD be a cyclic quadrilateral. Let E and F be variable points on the sides AB and CD, respectively, such that AE : EB = CF : FD. Let P be the point on the segment EF such that PE : PF = AB : CD. Prove that the ratio between the areas of triangles APD and BPC does not depend on...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G1

IMO 2003 SL G1 Origin: FIN | Category: Geometry Problem Let ABCD be a cyclic quadrilateral. Let P, Q, R be the feet of the perpendiculars from D to the lines BC, CA, AB, respectively. Show that PQ = QR if and only if the bisectors of \angleABC and \angleADC are concurrent with AC. Solution Denote by K and L the intersec- tions of the bisectors of \angleABC and \angleADC...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 11

IMO 1986 SL 11 Origin: BUL Problem Let f(n) be the least number of distinct points in the plane such that for each k = 1, 2, . . ., n there exists a straight line containing exactly k of these points. Find an explicit expression for f(n). Simpliﬁed version. Show that f(n) = / n+1 0 / n+2 ([x] denoting the great- est integer not exceeding x). Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 11

IMO 1997 SL 11 Origin: NET Problem Let P(x) be a polynomial with real coeﬃcients such that P(x) > 0 for all x \geq0. Prove that there exists a positive integer n such that (1 + x)nP(x) is a polynomial with nonnegative coeﬃcients. Solution All real roots of P(x) (if any) are negative: say −a1, −a2, . . . , −ak. Then P(x) can be factored as P(x) = C(x...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A1

IMO 1996 SL A1 Origin: SLO | Category: Algebra Problem Let $a$, $b$, and $c$ be positive real numbers such that $abc=1$. Prove that $$\frac{ab}{a^5+b^5+ab} + \frac{bc}{b^5+c^5+bc} + \frac{ca}{c^5+a^5+ca} \leq 1.$$ When does equality hold? Solution We have $$a^5+b^5-a^2b^2(a+b) = (a^3-b^3)(a^2-b^2) \geq 0,$$ i.e. $$a^5+b^5 \geq a^2b^2(a+b).$$ Hence $$\frac{ab}{a^5+b^5+ab} \leq \frac{ab}{a^2b^2(a+b)+ab} = \frac{abc^2}{a^2b^2c^2(a+b)+abc^2} = \frac{c}{a+b+c}.$$ Now, the left-hand side of the inequality to be proved does not exceed $$\frac{c}{a+b+c} +...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C2

IMO 1996 SL C2 Origin: UKR | Category: Combinatorics Problem An $(n-1) \times (n-1)$ square is divided into $(n-1)^2$ unit squares in the usual manner. Each of the $n^2$ vertices of these squares is to be colored red or blue. Find the number of different colorings such that each unit square has exactly two red vertices. (Two coloring schemes are regarded as different if at least one vertex is colored...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N5

IMO 1996 SL N5 Origin: ROM | Category: Number Theory Problem Let $N_0$ denote the set of nonnegative integers. Find a bijective function $f$ from $N_0$ into $N_0$ such that for all $m, n \in N_0$, $$f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).$$

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N5

IMO 2003 SL N5 Origin: KOR | Category: Number Theory Problem An integer n is said to be good if |n| is not the square of an integer. Determine all integers m with the following property: m can be represented in inﬁnitely many ways as a sum of three distinct good integers whose product is the square of an odd integer. Solution Suppose that m = u+v+w where u, v,...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 23

IMO 1998 SL 23 Origin: BLR Problem Let n be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: (i) The ﬁrst operation is either addition or multiplication. (ii) Thereafter, additions and multiplications are used alternately. (iii) In each addition one can choose independently whether to add...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 25

IMO 1988 SL 25 Origin: GBR Problem A positive integer is called a double number if its decimal rep- resentation consists of a block of digits, not commencing with 0, followed immediately by an identical block. For instance, 360360 is a double num- ber, but 36036 is not. Show that there are inﬁnitely many double numbers that are perfect squares. Solution Observe that 1001 = 7 \cdot 143, i.e., 103...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 12

IMO 1989 SL 12 Origin: HUN Problem At n distinct points of a circular race course there are n cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C2

IMO 2004 SL C2 Origin: GER | Category: Combinatorics Problem Let n and k be positive integers. There are given n circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are distinct. Each intersection point must be colored with one of n distinct colors so that each color is used at least once, and exactly k distinct colors occur...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 10

IMO 1989 SL 10 Origin: GRE Problem Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that there is one and only one function f : C \toC such that f(z) + f(wz + a) = g(z), z \inC. Find the function f. Solution Plugging in wz + a instead of z into the functional equation, we obtain f(wz + a) + f(w2z...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 30

IMO 1989 SL 30 Origin: SWE Problem For which positive integers n does there exist a positive integer N such that none of the integers 1 + N, 2 + N, . . . , n + N is the power of a prime number? Solution For all n such N exists. For a given n choose N = (n + 1)!2 + 1. Then 1 + j is a...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N4

IMO 1994 SL N4 Origin: FRA | Category: Number Theory Problem For any positive integer x0, three sequences {xn}, {yn}, and {zn} are deﬁned as follows: (i) y0 = 4 and z0 = 1; (ii) if xn is even for n \geq0, xn+1 = xn 2 , yn+1 = 2yn, and zn+1 = zn; (iii) if xn is odd for n \geq0, xn+1 = xn −yn 2 −zn, yn+1 =...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 14

IMO 1993 SL 14 Origin: ISR Problem The vertices D, E, F of an equilateral triangle lie on the sides BC, CA, AB respectively of a triangle ABC. If a, b, c are the respective lengths of these sides, and S the area of ABC, prove that DE \geq \sqrt 2S a2 + b2 + c2 + 4 \sqrt 3S . Solution Consider any point T inside the triangle ABC...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 9

IMO 1970 SL 9 Origin: GDR Problem Let u1, u2, . . . , un, v1, v2, . . . , vn be real numbers. Prove that 1 + n i=1 (ui + vi)2 \leq4 1 + n i=1 u2 i 1 + n i=1 v2 i . In what case does equality hold? Solution Let us set a = n i=1 u2 i and b =...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A2

IMO 2003 SL A2 Origin: AUS | Category: Algebra Problem Find all nondecreasing functions f : R \toR such that (i) f(0) = 0, f(1) = 1; (ii) f(a) + f(b) = f(a)f(b) + f(a + b −ab) for all real numbers a, b such that a < 1 < b. Solution We can rewrite (ii) as −(f(a) −1)(f(b) −1) = f(−(a −1)(b −1) + 1) −1. So putting g(x)...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 9

IMO 1974 SL 9 Problem II 3 (CUB 3) Let x, y, z be real numbers each of whose absolute value is diﬀerent from 1/ \sqrt 3 such that x + y + z = xyz. Prove that 3x −x3 1 −3x2 + 3y −y3 1 −3y2 + 3z −z3 1 −3z2 = 3x −x3 1 −3x2 \cdot 3y −y3 1 −3y2 \cdot 3z −z3 1 −3z2 . Solution There...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 21

IMO 1987 SL 21 Origin: USS Problem The prolongation of the bisector AL (L \inBC) in the acute- angled triangle ABC intersects the circumscribed circle at point N. From point L to the sides AB and AC are drawn the perpendiculars LK and LM respectively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKNM. Solution Let P be the second point of...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A4

IMO 2000 SL A4 Origin: GBR | Category: Algebra Problem The function F is deﬁned on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: For every n \geq0, (i) F(4n) = F(2n) + F(n); (ii) F(4n + 2) = F(4n) + 1; (iii) F(2n + 1) = F(2n) + 1. Prove that for each positive integer m, the number of integers n with 0...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N1

IMO 1994 SL N1 Origin: BUL | Category: Number Theory Problem M is a subset of {1, 2, 3, . . ., 15} such that the product of any three distinct elements of M is not a square. Determine the maximum number of elements in M. Solution Since for each of the subsets {1, 4, 9}, {2, 6, 12}, {3, 5, 15} and {7, 8, 14} the product of its...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 15

IMO 1979 SL 15 Origin: ISR Problem The nonnegative real numbers x1, x2, x3, x4, x5, a satisfy the following relations: i=1 ixi = a, i=1 i3xi = a2, i=1 i5xi = a3. What are the possible values of a? Solution We note that i=1 i(a−i2)2xi = a2 i=1 ixi−2a i=1 i3xi+ i=1 i5xi = a2\cdota−2a\cdota2+a3 = 0. Since the terms in the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 3

IMO 1998 SL 3 Origin: UKR Problem Let I be the incenter of triangle ABC. Let K, L, and M be the points of tangency of the incircle of ABC with AB, BC, and CA, respectively. The line t passes through B and is parallel to KL. The lines MK and ML intersect t at the points R and S. Prove that \angleRIS is acute. Solution Lemma. If U, W,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 24

IMO 1979 SL 24 Origin: USA Problem A circle O with center O on base BC of an isosceles triangle ABC is tangent to the equal sides AB, AC. If point P on AB and point Q on AC are selected such that PB \times CQ = (BC/2)2, prove that line segment PQ is tangent to circle O, and prove the converse. Solution Clearly O is the midpoint of BC....

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 6

IMO 1968 SL 6 Origin: HUN Problem If $a_i$ $(i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation $$\frac{a_1}{a_1 - x} + \frac{a_2}{a_2 - x} + \cdots + \frac{a_n}{a_n - x} = n$$ has at least $n - 1$ real roots. Solution We define $$f(x) = \frac{a_1}{a_1 - x} + \frac{a_2}{a_2 - x} + \cdots + \frac{a_n}{a_n - x}.$$ Let us assume w.l.o.g. $$a_1...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 5

IMO 1979 SL 5 Origin: CZS Problem Let n \geq2 be an integer. Find the maximal cardinality of a set M of pairs (j, k) of integers, 1 \leqj < k \leqn, with the following property: If (j, k) \inM, then (k, m) ̸\inM for any m. Solution Let A = {x | (x, y) \inM} and B = {y | (x, y) \inM. Then A and B are disjoint...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 10

IMO 1998 SL 10 Origin: AUS Problem Let r1, r2, . . . , rn be real numbers greater than or equal to 1. Prove that r1 + 1 + r2 + 1 + \cdot \cdot \cdot + rn + 1 \geq n n\sqrtr1r2 \cdot \cdot \cdot rn + 1. Solution We shall ﬁrst prove the inequality for n of the form 2k, k = 0, 1, 2, . ....

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 16

IMO 1977 SL 16 Origin: VIE Problem Let E be a set of n points in the plane (n \geq3) whose co- ordinates are integers such that any three points from E are vertices of a nondegenerate triangle whose centroid doesn’t have both coordinates integers. Determine the maximal n.

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G4

IMO 2001 SL G4 Origin: FRA | Category: Geometry Problem Let M be a point in the interior of triangle ABC. Let A′ lie on BC with MA′ perpendicular to BC. Deﬁne B′ on CA and C′ on AB similarly. Deﬁne p(M) = MA′ \cdot MB′ \cdot MC′ MA \cdot MB \cdot MC . Determine, with proof, the location of M such that p(M) is maximal. Let µ(ABC) denote the...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 14

IMO 1971 SL 14 Origin: USS Problem A broken line A1A2 . . . An is drawn in a 50\times50 square, so that the distance from any point of the square to the broken line is less than Solution Denote by V the ﬁgure made by a circle of radius 1 whose center moves along the broken line. From the condition of the problem, V contains the whole 50 \times...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 4

IMO 1983 SL 4 Origin: BEL Problem On the sides of the triangle ABC, three similar isosceles tri- angles ABP (AP = PB), AQC (AQ = QC), and BRC (BR = RC) are constructed. The ﬁrst two are constructed externally to the triangle ABC, but the third is placed in the same half-plane determined by the line BC as the triangle ABC. Prove that APRQ is a parallelogram. Solution The...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 12

IMO 1978 SL 12 Origin: USA Problem In a triangle ABC we have AB = AC. A circle is tangent internally to the circumcircle of ABC and also to the sides AB, AC, at P, Q respectively. Prove that the midpoint of PQ is the center of the incircle of ABC. Solution Let O be the center of the smaller circle, T its contact point with the circumcircle of ABC,...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 3

IMO 1973 SL 3 Origin: CZS Problem Prove that the sum of an odd number of unit vectors passing through the same point O and lying in the same half-plane whose border passes through O has length greater than or equal to 1. Solution We use induction on odd numbers n. For n = 1 there is nothing to prove. Suppose that the result holds for n−2 vectors, and let...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G5

IMO 2001 SL G5 Origin: GRE | Category: Geometry Problem Let ABC be an acute triangle. Let DAC, EAB, and FBC be isosceles triangles exterior to ABC, with DA = DC, EA = EB, and FB = FC such that \angleADC = 2\angleBAC, \angleBEA = 2\angleABC, \angleCFB = 2\angleACB. Let D′ be the intersection of lines DB and EF, let E′ be the intersection of EC and DF, and let...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 1

IMO 1971 SL 1 Origin: BUL Problem Consider a sequence of polynomials P0(x), P1(x), P2(x), . . . , Pn(x), . . . , where P0(x) = 2, P1(x) = x and for every n \geq1 the following equality holds: Pn+1(x) + Pn−1(x) = xPn(x). Prove that there exist three real numbers a, b, c such that for all n \geq1, (x2 −4)[P 2 n(x) −4] = [aPn+1(x) + bPn(x)...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 7

IMO 1981 SL 7 Origin: FIN Problem Assume that f(x, y) is deﬁned for all positive integers x and y, and that the following equations are satisﬁed: f(0, y) = y + 1, f(x + 1, 0) = f(x, 1), f(x + 1, y + 1) = f(x, f(x + 1, y)). Determine f(2, 2), f(3, 3) and f(4, 4). Alternative version: Determine f(4, 1981). Solution We immediately ﬁnd that...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 19

IMO 1988 SL 19 Origin: MEX Problem Let f(n) be a function deﬁned on the set of all positive integers and having its values in the same set. Suppose that f(f(m)+f(n)) = m+n for all positive integers n, m. Find all possible values for f(1988). Solution We will show that f(n) = n for every n (thus also f(1988) = 1988). Let f(1) = r and f(2) = s. We...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 18

IMO 1988 SL 18 Origin: LUX Problem Consider two concentric circles of radii R and r (R > r) with center O. Fix P on the small circle and consider the variable chord PA of the small circle. Points B and C lie on the large circle; B, P, C are collinear and BC is perpendicular to AP. (a) For what value(s) of \angleOPA is the sum BC2 + CA2...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 12

IMO 1981 SL 12 Origin: NET Problem Determine the maximum value of m2 + n2 where m and n are integers satisfying m, n \in{1, 2, . . ., 100} and (n2 −mn −m2)2 = 1. Solution We will solve the contest problem (in which m, n \in{1, 2, . . ., 1981}). For m = 1, n can be either 1 or 2. If m > 1, then n(n...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 17

IMO 1986 SL 17 Origin: CHN Problem Let A, B, C be ﬁxed points in the plane. A man starts from a certain point P0 and walks directly to A. At A he turns his di- rection by 60◦to the left and walks to P1 such that P0A = AP1. Af- ter he does the same action 1986 times successively around the points A, B, C, A, B, C, ....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 29

IMO 1989 SL 29 Origin: ROM Problem A ﬂock of 155 birds sit down on a circle C. Two birds Pi, Pj are mutually visible if m(PiPj) \leq10◦. Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on C can be occupied simultaneously by several birds.) Solution Let Pi, sitting at the place A, and Pj sitting at B, be two birds that...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 6

IMO 1987 SL 6 Origin: GRE Problem Show that if a, b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then an b + c + bn c + a + cn a + b \geq 2 n−2 Sn−1, n \geq1. Solution Suppose w.l.o.g. that a \geqb \geqc. Then 1/(b + c) \geq1/(a + c) \geq1/(a + b). Chebyshev’s...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 10

IMO 1972 SL 10 Origin: NET Problem Prove that for each n \geq4 every cyclic quadrilateral can be decomposed into n cyclic quadrilaterals. Solution Consider ﬁrst a triangle. It can be decomposed into k = 3 cyclic quadri- laterals by perpendiculars from some interior point of it to the sides; also, it can be decomposed into a cyclic quadrilateral and a triangle, and it follows by induction that this decomposition...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 6

IMO 1978 SL 6 Origin: FRA Problem Let ϕ : {1, 2, 3, . . .} \to{1, 2, 3, . . .} be injective. Prove that for all n, n k=1 ϕ(k) k2 \geq n k=1 k . Solution For ﬁxed n and the set {ϕ(1), . . . , ϕ(n)}, there are ﬁnitely many possi- bilities for mapping ϕ to {1, . . . , n}. Suppose...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 2

IMO 1985 SL 2 Origin: BRA Problem A polyhedron has 12 faces and is such that: (i) all faces are isosceles triangles, (ii) all edges have length either x or y, (iii) at each vertex either 3 or 6 edges meet, and (iv) all dihedral angles are equal. Find the ratio x/y. Solution The polyhedron has 3 \cdot 12/2 = 18 edges, and by Euler’s formula, 8 vertices. Let v1...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 11

IMO 1983 SL 11 Problem (FIN 2′) Let f : [0, 1] \toR be continuous and satisfy: bf(2x) = f(x), 0 \leqx \leq1/2; f(x) = b + (1 −b)f(2x −1), 1/2 \leqx \leq1, where b = 1+c 2+c, c > 0. Show that 0 < f(x)−x < c for every x, 0 < x < 1. Solution First suppose that the binary representation of x is ﬁnite: x = 0,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 8

IMO 1993 SL 8 Origin: IND Problem Deﬁne a sequence ⟨f(n)⟩\infty n=1 of positive integers by f(1) = 1 and f(n) = . f(n −1) −n, if f(n −1) > n; f(n −1) + n, if f(n −1) \leqn, for n \geq2. Let S = {n \inN | f(n) = 1993}. (a) Prove that S is an inﬁnite set. (b) Find the least positive integer in S. (c) If all...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 16

IMO 1990 SL 16 Origin: NET Problem Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides are a permutation of the numbers 12, 22, . . . , 19892, 19902. Solution Let A0A1 . . . A1989 be the desired 1990-gon. We also deﬁne A1990 = A0. Let O be an arbitrary point. For 1 \leqi...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A1

IMO 1994 SL A1 Origin: USA | Category: Algebra Problem Let a0 = 1994 and an+1 = a2 n an+1 for each nonnegative integer n. Prove that 1994 −n is the greatest integer less than or equal to an, 0 \leqn \leq998. Solution Obviously a0 > a1 > a2 > \cdot \cdot \cdot . Since ak −ak+1 = 1 − ak+1, we have an = a0 +(a1 −a0)+\cdot \cdot \cdot+(an...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 6

IMO 1970 SL 6 Origin: FRA Problem In the triangle ABC let B′ and C′ be the midpoints of the sides AC and AB respectively and H the foot of the altitude passing through the vertex A. Prove that the circumcircles of the triangles AB′C′, BC′H, and B′CH have a common point I and that the line HI passes through the midpoint of the segment B′C′. Solution Let F be...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 10

IMO 1979 SL 10 Origin: FIN Problem Show that for any vectors a, b in Euclidean space, |a \times b|3 \leq3 \sqrt 8 |a|2|b|2|a −b|2. Remark. Here \times denotes the vector product. Solution In the cases a = −\to0 , b = −\to0 , and a \parallelb the inequality is trivial. Otherwise, let us consider a triangle ABC such that −−\to CB = a and −\to CA = b. From...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 28

IMO 1988 SL 28 Origin: GBR Problem The sequence {an} of integers is deﬁned by a1 = 2, a2 = 7, and −1 2 < an+1 − a2 n an−1 \leq1 2, for n \geq2. Prove that an is odd for all n > 1. Solution The sequence is uniquely determined by the conditions, and a1 = 2, a2 = 7, a3 = 25, a4 = 89, a5 = 317,...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 10

IMO 1985 SL 10 Problem 2b.(VIE 1) Prove that for every point M on the surface of a regular tetrahedron there exists a point M ′ such that there are at least three diﬀerent curves on the surface joining M to M ′ with the smallest possible length among all curves on the surface joining M to M ′. Solution If M is at a vertex of the regular tetrahedron...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 8

IMO 1979 SL 8 Origin: FRG Problem For all rational x satisfying 0 \leqx < 1, f is deﬁned by f(x) = f(2x)/4, for 0 \leqx < 1/2, 3/4 + f(2x −1)/4, for 1/2 \leqx < 1. Given that x = 0.b1b2b3 . . . is the binary representation of x, ﬁnd f(x). Solution By the deﬁnition of f, it holds that f(0.b1b2 . . . ) = 3b1/4+f(0.b2b3 ....

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 23

IMO 1987 SL 23 Origin: YUG Problem Prove that for every natural number k (k \geq2) there exists an irrational number r such that for every natural number m, [rm] \equiv−1 (mod k). Remark. An easier variant: Find r as a root of a polynomial of second degree with integer coeﬃcients. Solution If we prove the existence of p, q \inN such that the roots r, s of f(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 8

IMO 1988 SL 8 Origin: FRA Problem Let u1, u2, . . . , um be m vectors in the plane, each of length less than or equal to 1, which add up to zero. Show that one can rear- range u1, u2, . . . , um as a sequence v1, v2, . . . , vm such that each partial sum v1, v1 + v2, v1 + v2...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A1

IMO 2002 SL A1 Origin: CZE | Category: Algebra Problem Find all functions f from the reals to the reals such that f(f(x) + y) = 2x + f(f(y) −x) for all real x, y. Solution We observe ﬁrst that f is surjective. Indeed, setting y = −f(x) gives f(f(−f(x)) −x) = f(0) −2x, where the right-hand expression can take any real value. In particular, there exists x0 for which...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 1

IMO 1972 SL 1 Origin: BUL Problem Let f and ϕ be real functions deﬁned on the set R satisfying the functional equation f(x + y) + f(x −y) = 2ϕ(y)f(x), (1) for arbitrary real x, y (give examples of such functions). Prove that if f(x) is not identically 0 and |f(x)| \leq1 for all x, then |ϕ(x)| \leq1 for all x. Solution Suppose that f(x0) ̸= 0 and for...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 7

IMO 1972 SL 7 Origin: GBR Problem (a) A plane \pi passes through the vertex O of the regular tetrahedron OPQR. We deﬁne p, q, r to be the signed distances of P, Q, R from \pi measured along a directed normal to \pi. Prove that p2 + q2 + r2 + (q −r)2 + (r −p)2 + (p −q)2 = 2a2, where a is the length of an edge...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 18

IMO 1982 SL 18 Problem C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3 be mutually perpendicular straight lines passing through O. Let S denote the sphere with center O and radius R, and for every point M of S, let SM denote the sphere with center M and radius R. We denote by P1, P2, P3 the intersection of SM with the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G8

IMO 1995 SL G8 Origin: COL | Category: Geometry Problem Let ABC be a triangle. A circle passing through B and C in- tersects the sides AB and AC again at C′ and B′, respectively. Prove that BB′, CC′, and HH′ are concurrent, where H and H′ are the orthocenters of triangles ABC and AB′C′ respectively. Solution Let BB′ cut CC′ at P. Since \angleB′BC′ = \angleB′CC′, it follows that...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 5

IMO 1970 SL 5 Origin: CZS Problem Let M be an interior point of the tetrahedron ABCD. Prove that −−\to MA vol(MBCD) + −−\to MB vol(MACD) +−−\to MC vol(MABD) + −−\to MD vol(MABC) = 0 (vol(PQRS) denotes the volume of the tetrahedron PQRS). Solution Denote respectively by A1, B1, C1 and D1 the points of intersection of the lines AM, BM, CM, and DM with the opposite sides of the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 18

IMO 1998 SL 18 Origin: BUL Problem Determine all positive integers n for which there exists an integer m such that 2n −1 is a divisor of m2 + 9. Solution We claim that, if 2n −1 divides m2 + 9 for some m \inN, then n must be a power of 2. Suppose otherwise that n has an odd divisor d > 1. Then 2d −1 | 2n −1...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N5

IMO 2004 SL N5 Origin: IRN | Category: Number Theory Problem We call a positive integer alternate if its decimal digits are alternately odd and even. Find all positive integers n such that n has an alternate multiple. Solution If n is divisible by 20, then every multiple of n has two last digits even and hence it is not alternate. We shall show that any other n has an...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 1

IMO 1992 SL 1 Origin: AUS Problem Prove that for any positive integer m there exist an inﬁnite number of pairs of integers (x, y) such that (i) x and y are relatively prime; (ii) y divides x2 + m; (iii) x divides y2 + m. Solution Assume that a pair (x, y) with x < y satisﬁes the required conditions. We claim that the pair (y, x1) also satisﬁes...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C2

IMO 1994 SL C2 Origin: COL | Category: Combinatorics Problem In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens x and x′ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens x = x0, x1, . . . , xn = x′ for some integer n \geq2 such...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C4

IMO 1999 SL C4 Origin: GBR | Category: Combinatorics Problem Let A be a set of N residues (mod N 2). Prove that there exists a set B of N residues (mod N 2) such that the set A + B = {a + b | a \inA, b \inB} contains at least half of all residues (mod N 2). Solution Let S = {0, 1, . . ., N...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G4

IMO 2000 SL G4 Origin: RUS | Category: Geometry Problem Let A1A2 . . . An be a convex polygon, n \geq4. Prove that A1A2 . . . An is cyclic if and only if to each vertex Aj one can assign a pair (bj, cj) of real numbers, j = 1, 2, . . .n, such that AiAj = bjci −bicj for all i, j with 1 \leqi \leqj...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 12

IMO 1993 SL 12 Origin: IRE Problem Let n, k be positive integers with k \leqn and let S be a set containing n distinct real numbers. Let T be the set of all real numbers of the form x1 + x2 + \cdot \cdot \cdot + xk, where x1, x2, . . . , xk are distinct elements of S. Prove that T contains at least k(n −k) +...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A8

IMO 1996 SL A8 Origin: ROM | Category: Algebra Problem Let $\mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $f:\mathbb{N}_0 \to \mathbb{N}_0$ such that $$f(m + f(n)) = f(f(m)) + f(n), \quad \forall m,n \in \mathbb{N}_0.$$ Solution Putting $m=n=0$ we obtain $f(0)=0$ and consequently $$f(f(n)) = f(n)$$ for all $n$. Thus the given functional equation is equivalent to $$f(m + f(n)) = f(m) + f(n), \quad f(0)=0.$$ Clearly...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C7

IMO 1999 SL C7 Origin: IRE | Category: Combinatorics Problem Let p > 3 be a prime number. For each nonempty subset T of {0, 1, 2, 3, . . ., p−1} let E(T ) be the set of all (p−1)-tuples (x1, . . . , xp−1), where each xi \inT and x1 + 2x2 + \cdot \cdot \cdot + (p −1)xp−1 is divisible by p and let |E(T )|...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N4

IMO 1995 SL N4 Origin: BUL | Category: Number Theory Problem Find all positive integers x and y such that x+y2+z3 = xyz, where z is the greatest common divisor of x and y. Solution Let x = za and y = zb, where a and b are relatively prime. The given Diophantine equation becomes a + zb2 + z2 = z2ab, so a = zc for some c \inZ....

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 26

IMO 1990 SL 26 Origin: USA Problem Let P be a cubic polynomial with rational coeﬃcients, and let q1, q2, q3, . . . be a sequence of rational numbers such that qn = P(qn+1) for all n \geq1. Prove that there exists k \geq1 such that for all n \geq1, qn+k = qn. Solution We note that |P(x)/x| \to\infty. Hence, there exists an integer number M such that M...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 7

IMO 1971 SL 7 Origin: NET Problem Given a tetrahedron ABCD whose all faces are acute- angled triangles, set \sigma = ∡DAB + ∡BCD −∡ABC −∡CDA. Consider all closed broken lines XY ZTX whose vertices X, Y, Z, T lie in the interior of segments AB, BC, CD, DA respectively. Prove that: (a) if \sigma ̸= 0, then there is no broken line XY ZT of minimal length; (b) if...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 7

IMO 1968 SL 7 Origin: HUN Problem Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $3\sqrt{3}$ times the product of the side lengths of the triangle. When does equality hold? Solution Let $r_a$, $r_b$, $r_c$ denote the radii of the exscribed circles corresponding to the sides of lengths $a$, $b$, $c$ respectively, and $R$, $p$ and $S$ denote the circumradius,...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 6

IMO 1983 SL 6 Origin: CAN Problem Suppose that {x1, x2, . . . , xn} are positive integers for which x1 + x2 + \cdot \cdot \cdot + xn = 2(n + 1). Show that there exists an integer r with 0 \leqr \leqn −1 for which the following n −1 inequalities hold: xr+1 + \cdot \cdot \cdot + xr+i \leq2i + 1 \foralli, 1 \leqi \leqn −r; xr+1...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 1

IMO 1970 SL 1 Origin: BEL Problem Consider a regular 2n-gon and the n diagonals of it that pass through its center. Let P be a point of the inscribed circle and let a1, a2, . . . , an be the angles in which the diagonals mentioned are visible from the point P. Prove that n i=1 tan2 ai = 2ncos2 \pi 2n sin4 \pi 2n . Solution...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 9

IMO 1983 SL 9 Origin: USA Problem If a, b, and c are sides of a triangle, prove that a2b(a −b) + b2c(b −c) + c2a(c −a) \geq0. Determine when there is equality. Solution For any triangle of sides a, b, c there exist 3 nonnegative numbers x, y, z such that a = y +z, b = z +x, c = x+y (these numbers correspond to the division of...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 6

IMO 1991 SL 6 Origin: USS Problem Prove for each triangle ABC the inequality 4 < IA \cdot IB \cdot IC lAlBlC \leq8 27, where I is the incenter and lA, lB, lC are the lengths of the angle bisectors of ABC. Solution Let a, b, c be sides of the triangle. Let A1 be the intersection of line AI with BC. By the known fact, BA1 : A1C =...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 1

IMO 1981 SL 1 Origin: BEL Problem (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n −1 numbers? (b) For which values of n > 2 is there a unique set having the stated property? Solution Assume that the set {a−n+ 1,...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 18

IMO 1983 SL 18 Origin: FRG Problem Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) = 1. Show that 2abc −ab −bc −ca is the largest integer not repre- sentable as xbc + yca + zab with nonnegative integers x, y, z. Solution Let (x0, y0, z0) be one solution of bcx + cay + abz = n (not necessarily nonnegative). By...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 12

IMO 1970 SL 12 Origin: USS Problem We are given 100 points in the plane, no three of which are on the same line. Consider all triangles that have all vertices chosen from the 100 given points. Prove that at most 70% of these triangles are acute angled. Solution Lemma. Five points are given in the plane such that no three of them are collinear. Then there are at least...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 11

IMO 1977 SL 11 Origin: NET Problem Let n be an integer greater than 1. Deﬁne x1 = n, y1 = 1, xi+1 = xi + yi , yi+1 = n xi+1 for i = 1, 2, . . ., where [z] denotes the largest integer less than or equal to z. Prove that min{x1, x2, . . . xn} = [\sqrtn].

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N4

IMO 2001 SL N4 Origin: VIE | Category: Number Theory Problem Let p \geq5 be a prime number. Prove that there exists an integer a with 1 \leqa \leqp −2 such that neither ap−1 −1 nor (a + 1)p−1 −1 is divisible by p2. Solution Let C be the set of those a \in{1, 2, . . ., p−1} for which ap−1 \equiv1 (mod p2). At ﬁrst, we observe that...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 6

IMO 1997 SL 6 Origin: IRE Problem (a) Let n be a positive integer. Prove that there exist distinct positive integers x, y, z such that xn−1 + yn = zn+1. (b) Let a, b, c be positive integers such that a and b are relatively prime and c is relatively prime either to a or to b. Prove that there exist inﬁnitely many triples (x, y, z) of distinct...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 25

IMO 1993 SL 25 Origin: VIE Problem Solve the following system of equations, in which a is a given number satisfying |a| > 1: x2 1 = ax2 + 1, x2 2 = ax3 + 1, \cdot \cdot \cdot \cdot \cdot \cdot x2 999 = ax1000 + 1, x2 1000 = ax1 + 1. Solution We need only consider the case a > 1 (since the case a < −1...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 20

IMO 1997 SL 20 Origin: IRE Problem Let D be an internal point on the side BC of a triangle ABC. The line AD meets the circumcircle of ABC again at X. Let P and Q be the feet of the perpendiculars from X to AB and AC, respectively, and let \gamma be the circle with diameter XD. Prove that the line PQ is tangent to \gamma if and only...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 15

IMO 1986 SL 15 Origin: NET Problem Let ABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A′B′C′D′ be a quadrangle such that A′, B′, C′, D′ are the centers of the circumcircles of triangles BCD, ACD, ABD, and ABC. We write T (ABCD) = A′B′C′D′. Let us deﬁne A′′B′′C′′D′′ = T (A′B′C′D′) = T (T (ABCD)). (a) Prove that ABCD and A′′B′′C′′D′′ are similar....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 26

IMO 1989 SL 26 Origin: KOR Problem Let n be a positive integer and let a, b be given real numbers. Determine the range of x0 for which n i=0 xi = a and n i=0 x2 i = b, where x0, x1, . . . , xn are real variables. Solution By the Cauchy–Schwarz inequality, n i=1 xi \leqn n i=1 x2 i . Since...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C2

IMO 2000 SL C2 Origin: ITA | Category: Combinatorics Problem A brick staircase with three steps of width 2 is made of twelve unit cubes. Determine all integers n for which it is possible to build a cube of side n using such bricks. Solution Since the volume of each brick is 12, the side of any such cube must be divisible by 6. Suppose that a cube of side...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A4

IMO 1994 SL A4 Origin: MON | Category: Algebra Problem Let R denote the set of all real numbers and R+ the subset of all positive ones. Let \alpha and \beta be given elements in R, not necessarily distinct. Find all functions f : R+ \toR such that f(x)f(y) = y\alphaf x x\betaf y for all x and y in R+. Solution Suppose that \alpha = \beta. The given functional...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 14

IMO 1979 SL 14 Origin: GRE Problem Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist. Solution Let us assume that a \inR \ {1} is such that there exist a and x such that x = loga x, or equivalently f(x) := ln x/x = ln a. Then a is a value of the function f(x)...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 14

IMO 1977 SL 14 Problem (FIN 2‘) Let E be a ﬁnite set of points such that E is not contained in a plane and no three points of E are collinear. Show that at least one of the following alternatives holds: (i) E contains ﬁve points that are vertices of a convex pyramid having no other points in common with E; (ii) some plane contains exactly three points from...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A4

IMO 2002 SL A4 Origin: IND | Category: Algebra Problem Find all functions f from the reals to the reals such that (f(x) + f(z))(f(y) + f(t)) = f(xy −zt) + f(xt + yz) for all real x, y, z, t. Solution Putting x = z = 0 and t = y into the given equation gives 4f(0)f(y) = 2f(0) for all y. If f(0) ̸= 0, then we deduce...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 6

IMO 1974 SL 6 Problem I 6 (ROM 4)IMO3 Does there exist a natural number n for which the number n k=0 2n + 1 2k + 1 23k is divisible by 5? Solution We set x = n k=0 2n + 1 2k + 1 23k = \sqrt n k=0 2n + 1 2k + 1 \sqrt 2k+1, y = n k=0 2n...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A2

IMO 2004 SL A2 Origin: ROM | Category: Algebra Problem An inﬁnite sequence a0, a1, a2, . . . of real numbers satisﬁes the condition an = |an+1 −an+2| for every n \geq0 with a0 and a1 positive and distinct. Can this sequence be bounded? Solution We claim that the sequence {an} must be unbounded. The condition of the sequence is equivalent to an > 0 and an+1 = an+an−1...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A5

IMO 1996 SL A5 Origin: ROM | Category: Algebra Problem Let $P(x)$ be the real polynomial function $P(x) = ax^3 + bx^2 + cx + d.$ Prove that if $|P(x)| \leq 1 \quad \text{for all } x \text{ such that } |x| \leq 1,$ then $|a| + |b| + |c| + |d| \leq 7.$ Solution Considering the polynomials $\pm P(\pm x)$ we may assume w.l.o.g. that $a, b \geq 0$....

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A3

IMO 2002 SL A3 Origin: POL | Category: Algebra Problem Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+ d, where a, b, c, d are integers and a ̸= 0. Suppose that xP(x) = yP(y) for inﬁnitely many pairs x, y of integers with x ̸= y. Prove that the equation P(x) = 0 has an integer root. Solution Let x, y be distinct integers satisfying xP(x)...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 10

IMO 1974 SL 10 Problem II 4 (FIN 3)IMO2 Let \triangleABC be a triangle. Prove that there exists a point D on the side AB such that CD is the geometric mean of AD and BD if and only if \sqrt sin A sin B \leqsin C 2 . Solution If we set \angleACD = \gamma1 and \angleBCD = \gamma2 for a point D on the segment AB, then by...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 8

IMO 1978 SL 8 Origin: GBR Problem Let S be the set of all the odd positive integers that are not multiples of 5 and that are less than 30m, m being an arbitrary positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two diﬀerent integers, one of which divides the other? Solution Take the subset {ai} =...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G5

IMO 2000 SL G5 Origin: GBR | Category: Geometry Problem The tangents at B and A to the circumcircle of an acute- angled triangle ABC meet the tangent at C at T and U respectively. AT meets BC at P, and Q is the midpoint of AP; BU meets CA at R, and S is the midpoint of BR. Prove that \angleABQ = \angleBAS. Determine, in terms of ratios of...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 29

IMO 1991 SL 29 Origin: FIN Problem We call a set S on the real line R superinvariant if for any stretching A of the set by the transformation taking x to A(x) = x0 + a(x −x0) there exists a translation B, B(x) = x + b, such that the images of S under A and B agree; i.e., for any x \inS there is a y \inS such...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G3

IMO 1994 SL G3 Origin: RUS | Category: Geometry Problem A circle \omega is tangent to two parallel lines l1 and l2. A second circle \omega1 is tangent to l1 at A and to \omega externally at C. A third circle \omega2 is tangent to l2 at B, to \omega externally at D, and to \omega1 externally at E. AD intersects BC at Q. Prove that Q is the circumcenter...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 14

IMO 1991 SL 14 Origin: POL Problem Let a, b, c be integers and p an odd prime number. Prove that if f(x) = ax2 + bx + c is a perfect square for 2p −1 consecutive integer values of x, then p divides b2 −4ac. Solution Suppose that f(x0), f(x0 + 1), . . . , f(x0 + 2p −2) are squares. If p | a and p ∤b,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 24

IMO 1993 SL 24 Origin: USA Problem Prove that a b + 2c + 3d + b c + 2d + 3a + c d + 2a + 3b + d a + 2b + 3c \geq2 for all positive real numbers a, b, c, d. Solution By the Cauchy–Schwarz inequality, if x1, x2, . . . , xn and y1, y2, . . . , yn are positive numbers,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 18

IMO 1979 SL 18 Origin: POL Problem Let m positive integers a1, . . . , am be given. Prove that there exist fewer than 2m positive integers b1, . . . , bn such that all sums of dis- tinct bk’s are distinct and all ai (i \leqm) occur among them. Solution Let us write all ai in binary representation. For S \subseteq{1, 2, . . ., m} let...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 16

IMO 1987 SL 16 Origin: FRG Problem Let S be a set of n elements. We denote the number of all permutations of S that have exactly k ﬁxed points by pn(k). Prove: (a) n k=0 kpn(k) = n!; (b) n k=0(k −1)2pn(k) = n!. Solution We assume that S = {1, 2, . . ., n}, and use the obvious fact n k=0 pn(k) = n! (0) (a)...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 17

IMO 1982 SL 17 Problem C5 (USS 5) The right triangles ABC and AB1C1 are similar and have opposite orientation. The right angles are at C and C1, and we also have ∡CAB = ∡C1AB1. Let M be the point of intersection of the lines BC1 and B1C. Prove that if the lines AM and CC1 exist, they are perpendic- ular. Solution Let A be the origin of the Cartesian...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N5

IMO 1999 SL N5 Origin: ARM | Category: Number Theory Problem Let n, k be positive integers such that n is not divisible by 3 and k \geqn. Prove that there exists a positive integer m that is divisible by n and the sum of whose digits in decimal representation is k. Solution Since one can arbitrarily add zeros at the end of m, which increases divis- ibility by 2...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 3

IMO 1970 SL 3 Origin: BUL Problem In the tetrahedron SABC the angle BSC is a right angle, and the projection of the vertex S to the plane ABC is the intersection of the altitudes of the triangle ABC. Let z be the radius of the inscribed circle of the triangle ABC. Prove that SA2 + SB2 + SC2 \geq18z2. Solution We shall use the following lemma Lemma. If an...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 20

IMO 1984 SL 20 Origin: USA Problem Determine all pairs (a, b) of positive real numbers with a ̸= 1 such that loga b < loga+1(b + 1). Solution Deﬁne the set S = R+ ∖{1}. The given inequality is equivalent to ln b/ln a < ln (b + 1)/ln (a + 1). If b = 1, it is obvious that each a \inS satisﬁes this inequality. Suppose now that...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 16

IMO 1983 SL 16 Origin: GDR Problem Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn, with a0, a1, . . . , an \inR and 0 \leqa0 = an \leqa1 = an−1 \leq\cdot \cdot \cdot \leqa[n/2] = a[(n+1)/2]. Prove that if f \inF(m) and g \inF(n), then fg \inF(m + n). Solution Set hn,i(x) = xi + \cdot \cdot \cdot + xn−i, 2i \leqn. The set...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 13

IMO 1975 SL 13 Origin: ROM Problem Let A0, A1, . . . , An be points in a plane such that (i) A0A1 \leq1 2A1A2 \leq\cdot \cdot \cdot \leq 2n−1 An−1An and (ii) 0 < ∡A0A1A2 < ∡A1A2A3 < \cdot \cdot \cdot < ∡An−2An−1An < 180◦, where all these angles have the same orientation. Prove that the segments AkAk+1, AmAm+1 do not intersect for each k and n such...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N4

IMO 2000 SL N4 Origin: BRA | Category: Number Theory Problem Determine all triples of positive integers (a, m, n) such that am + 1 divides (a + 1)n. Solution Trivially all triples (a, 1, n) and (1, m, n) are solutions. Assume now that a > 1 and m > 1. If m is even, then am + 1 \equiv(−1)m + 1 \equiv2 (mod a + 1), which implies...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 1

IMO 1973 SL 1 Origin: BUL Problem Let a tetrahedron ABCD be inscribed in a sphere S. Find the locus of points P inside the sphere S for which the equality AP PA1 BP PB1 CP PC1 DP PD1 = 4 holds, where A1, B1, C1, and D1 are the intersection points of S with the lines AP, BP, CP, and DP, respectively. Solution The condition of the point P...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 6

IMO 1976 SL 6 Origin: NET Problem A rectangular box can be ﬁlled completely with unit cubes. If one places cubes with volume 2 in the box such that their edges are parallel to the edges of the box, one can ﬁll exactly 40% of the box. Determine all possible (interior) sizes of the box. Solution Suppose a1 \leqa2 \leqa3 are the dimensions of the box. If we set bi...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 18

IMO 1997 SL 18 Origin: GBR Problem The altitudes through the vertices A, B, C of an acute-angled triangle ABC meet the opposite sides at D, E, F, respectively. The line through D parallel to EF meets the lines AC and AB at Q and R, re- spectively. The line EF meets BC at P. Prove that the circumcircle of the triangle PQR passes through the midpoint of BC. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 25

IMO 1983 SL 25 Origin: USS Problem Prove that every partition of 3-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every a \inR+, there are points M and N inside that subset such that distance between M and N is exactly a. Solution Suppose the contrary, that R3 = P1 \cupP2 \cupP3 is a partition such that a1...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 2

IMO 1997 SL 2 Origin: CAN Problem Let R1, R2, . . . be the family of ﬁnite sequences of positive inte- gers deﬁned by the following rules: R1 = (1), and if Rn−1 = (x1, . . . , xs), then Rn = (1, 2, . . . , x1, 1, 2, . . ., x2, . . . , 1, 2, . . . , xs, n). For...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 10

IMO 1978 SL 10 Origin: NET Problem An international society has its members in 6 diﬀerent countries. The list of members contains 1978 names, numbered 1, 2, . . . , Solution Assume the opposite. One of the countries, say A, contains at least 330 members a1, a2, . . . , a330 of the society ( 6 \cdot 329 = 1974). Consider the diﬀerences a330 −ai, = 1, 2,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 19

IMO 1993 SL 19 Origin: ROM Problem Let a, b, n be positive integers, b > 1 and bn −1 | a. Show that the representation of the number a in the base b contains at least n digits diﬀerent from zero. Solution Let s be the minimum number of nonzero digits that can appear in the b- adic representation of any number divisible by bn −1. Among all numbers...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 13

IMO 1979 SL 13 Origin: GRE Problem Show that 20 60 < sin 20◦< 21 60. Solution From elementary trigonometry we have sin 3t = 3 sin t −4 sin3 t. Hence, if we denote y = sin 20◦, we have \sqrt 3/2 = sin 60◦= 3y −4y3. Obviously 0 < y < 1/2 = sin 30◦. The function f(x) = 3x −4x3 is strictly increasing on [0, 1/2) because...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 3

IMO 1977 SL 3 Origin: FRG Problem Let a and b be natural numbers and let q and r be the quotient and remainder respectively when a2 + b2 is divided by a + b. Determine the numbers a and b if q2 + r = 1977.

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 4

IMO 1998 SL 4 Origin: ARM Problem Let M and N be points inside triangle ABC such that \angleMAB = \angleNAC and \angleMBA = \angleNBC. Prove that AM \cdot AN AB \cdot AC + BM \cdot BN BA \cdot BC + CM \cdot CN CA \cdot CB = 1. Solution Let K be the point on the ray BN with \angleBCK = \angleBMA. Since \angleKBC = \angleABM, we get \triangleBCK...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 8

IMO 1968 SL 8 Origin: ROM Problem Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E, F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B, C, D$ of trapezoids that satisfy the following: (i) $|AB| \leq a$ ($a$ fixed); (ii) $|EF| = l$...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 15

IMO 1989 SL 15 Origin: IRE Problem Let a, b, c, d, m, n be positive integers such that a2+b2+c2+d2 = 1989, a+b+c+d = m2, and the largest of a, b, c, d is n2. Determine, with proof, the values of m and n. Solution By Cauchy’s inequality, 44 < \sqrt 1989 < a + b + c + d \leq \sqrt 2 \cdot 1989 < 90. Since m2 =...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 11

IMO 1978 SL 11 Origin: SWE Problem A function f : I \toR, deﬁned on an interval I, is called concave if f(\thetax + (1 −\theta)y) \geq\thetaf(x) + (1 −\theta)f(y) for all x, y \inI and 0 \leq\theta \leq1. Assume that the functions f1, . . . , fn, having all nonnegative values, are concave. Prove that the function (f1f2 . . . fn)1/n is concave. Solution Set F(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G6

IMO 1996 SL G6 Origin: ARM | Category: Geometry Problem Let the sides of two rectangles be ${a, b}$ and ${c, d}$ with $a < c \leq d < b$ and $ab < cd$. Prove that the first rectangle can be placed within the second one if and only if $(b^2 - a^2)^2 \leq (bd - ac)^2 + (bc - ad)^2.$ Solution Denote by $ABCD$ and $EFGH$ the two rectangles,...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 22

IMO 1983 SL 22 Origin: SWE Problem Let n be a positive integer having at least two diﬀerent prime factors. Show that there exists a permutation a1, a2, . . . , an of the integers 1, 2, . . . , n such that n k=1 k \cdot cos 2\piak n = 0. Solution Decompose n into n = st, where the greatest common divisor of s and...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 5

IMO 1986 SL 5 Origin: FRG Problem The set S = {2, 5, 13} has the property that for every a, b \inS, a ̸= b, the number ab −1 is a perfect square. Show that for every positive integer d not in S, the set S \cup{d} does not have the above property. Solution Suppose that for every a, b \in{2, 5, 13, d}, a ̸= b, the number...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 12

IMO 1985 SL 12 Problem 3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in x, y, and z is deﬁned by P0(x, y, z) = 1 and by Pm(x, y, z) = (x + z)(y + z)Pm−1(x, y, z + 1) −z2Pm−1(x, y, z) for m > 0. Prove that each Pm(x, y, z) is symmetric, in other words, is unaltered...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A3

IMO 2000 SL A3 Origin: BLR | Category: Algebra Problem Find all pairs of functions f : R \toR, g : R \toR such that f(x + g(y)) = xf(y) −yf(x) + g(x) for all x, y \inR. Solution Let us ﬁrst solve the problem under the assumption that g(\alpha) = 0 for some \alpha. Setting y = \alpha in the given equation yields g(x) = (\alpha+1)f(x)−xf(\alpha). Then the given...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 23

IMO 1993 SL 23 Origin: GBR Problem A ﬁnite set of (distinct) positive integers is called a “DS-set” if each of the integers divides the sum of them all. Prove that every ﬁnite set of positive integers is a subset of some DS-set. Solution Let the given numbers be a1, . . . , an. Put s = a1 + \cdot \cdot \cdot + an and m = lcm(a1, ....

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 22

IMO 1990 SL 22 Origin: ROM Problem Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules oﬀer round-trip service between the cities they serve. Prove that at least one of the airlines can oﬀer two disjoint round trips each containing an odd number of landings. Solution We can assume without loss of...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 13

IMO 1993 SL 13 Origin: IRE Problem Let S be the set of all pairs (m, n) of relatively prime positive integers m, n with n even and m < n. For s = (m, n) \inS write n = 2kn0, where k, n0 are positive integers with n0 odd and deﬁne f(s) = (n0, m + n −n0). Prove that f is a function from S to S and...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 15

IMO 1982 SL 15 Problem C3 (CAN 5) Show that 1 −sa 1 −s \leq(1 + s)a−1 holds for every 1 ̸= s > 0 real and 0 < a \leq1 rational. Solution Let a = k/n, where n, k \inN, n \geqk. Putting tn = s, the given inequality becomes 1−tk 1−tn \leq(1 + tn)k/n−1, or equivalently (1 + t + \cdot \cdot \cdot + tk−1)n(1 + tn)n−k \leq(1...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 5

IMO 1997 SL 5 Origin: ROM Problem Let ABCD be a regular tetrahedron and M, N distinct points in the planes ABC and ADC respectively. Show that the segments MN, BN, MD are the sides of a triangle. Solution We shall prove ﬁrst the 2-dimensional analogue: Lemma. Given an equilateral triangle ABC and two points M, N on the sides AB and AC respectively, there exists a triangle with sides...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 14

IMO 1978 SL 14 Origin: VIE Problem Prove that it is possible to place 2n(2n + 1) parallelepipedic (rectangular) pieces of soap of dimensions 1 \times 2 \times (n + 1) in a cubic box with edge 2n + 1 if and only if n is even or n = 1. Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N6

IMO 2002 SL N6 Origin: ROM | Category: Number Theory Problem Find all pairs of positive integers m, n \geq3 for which there exist inﬁnitely many positive integers a such that am + a −1 an + a2 −1 is itself an integer. Solution Suppose that (m, n) is such a pair. Assume that division of the polynomial F(x) = xm + x −1 by G(x) = xn + x2...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 16

IMO 1989 SL 16 Origin: ISR Problem The set {a0, a1, . . . , an} of real numbers satisﬁes the following conditions: (i) a0 = an = 0; (ii) for 1 \leqk \leqn −1, ak = c + n−1 i=k ai−k(ai + ai+1). Prove that c \leq 4n. Solution Deﬁne Sk = k i=0 ai (k = 0, 1, . . . , n) and S−1 = 0....

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 3

IMO 1990 SL 3 Origin: CZS Problem On a circle, 2n −1 (n \geq3) diﬀerent points are given. Find the minimal natural number N with the property that whenever N of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly n of the given 2n −1 points. Solution A segment connecting two points which divides the...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 12

IMO 1976 SL 12 Origin: VIE Problem The polynomial 1976(x+x2+\cdot \cdot \cdot+xn) is decomposed into a sum of polynomials of the form a1x + a2x2 + . . . + anxn, where a1, a2, \cdot \cdot \cdot , an are distinct positive integers not greater than n. Find all values of n for which such a decomposition is possible. Solution Suppose the decomposition into k polynomials is possible. The sum...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 18

IMO 1992 SL 18 Origin: USA Problem Let [x] denote the greatest integer less than or equal to x. Pick any x1 in [0, 1) and deﬁne the sequence x1, x2, x3, . . . by xn+1 = 0 if xn = 0 and xn+1 = 1/xn −[1/xn] otherwise. Prove that x1 + x2 + \cdot \cdot \cdot + xn < F1 F2 F2 F3 \cdot \cdot \cdot + Fn...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 7

IMO 1986 SL 7 Origin: FRA Problem Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot < xn < 1 and set x0 = 0, xn+1 = 1. Suppose that these numbers satisfy the following system of equations: n+1 j=0, j̸=i xi −xj = 0 where i = 1, 2, . . ., n. (1) Prove that xn+1−i...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 3

IMO 1971 SL 3 Origin: GDR Problem Knowing that the system x + y + z = 3, x3 + y3 + z3 = 15, x4 + y4 + z4 = 35, has a real solution x, y, z for which x2 + y2 + z2 < 10, ﬁnd the value of x5 + y5 + z5 for that solution. Solution Let x, y, z be a solution of the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N3

IMO 1995 SL N3 Origin: CZE | Category: Number Theory Problem Determine all integers n > 3 such that there are n points A1, A2, . . . , An in the plane that satisfy the following two conditions si- multaneously: (a) No three lie on the same line. (b) There exist real numbers p1, p2, . . . , pn such that the area of \triangleAiAjAk is equal to...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N5

IMO 1995 SL N5 Origin: IRE | Category: Number Theory Problem At a meeting of 12k people, each person exchanges greetings with exactly 3k + 6 others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? Solution For each two people let n be the number of people exchanging greetings with both of them. To determine n in...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 5

IMO 1971 SL 5 Origin: HUN Problem Let a, b, c, d, e be real numbers. Prove that the expression (a−b)(a−c)(a−d)(a−e)+(b−a)(b−c)(b−d)(b−e)+(c−a)(c−b)(c−d)(c−e) (d −a)(d −b)(d −c)(d −e) + (e −a)(e −b)(e −c)(e −d) is nonnegative. Solution Without loss of generality, we may assume that a \geqb \geqc \geqd \geqe. Then a −b = −(b −a) \geq0, a −c \geqb −c \geq0, a −d \geqb −d \geq0 and a −e \geqb −e...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 12

IMO 1997 SL 12 Origin: ITA Problem Let p be a prime number and let f(x) be a polynomial of degree d with integer coeﬃcients such that: (i) f(0) = 0, f(1) = 1; (ii) for every positive integer n, the remainder of the division of f(n) by p is either 0 or 1. Prove that d \geqp −1. Solution Lemma. For any polynomial P of degree at most n,...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 7

IMO 1990 SL 7 Origin: GRE Problem Let f(0) = f(1) = 0 and f(n + 2) = 4n+2f(n + 1) −16n+1f(n) + n \cdot 2n2, n = 0, 1, 2, 3, . . .. Show that the numbers f(1989), f(1990), f(1991) are divisible by 13. Solution Let f(n) = g(n)2n2 for all n. The recursion then transforms into g(n + 2) −2g(n + 1) + g(n) = n \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 17

IMO 1985 SL 17 Problem 6a.(SWE 3)IMO6 The sequence f1, f2, . . . , fn, . . . of functions is deﬁned for x > 0 recursively by f1(x) = x, fn+1(x) = fn(x) fn(x) + 1 n . Prove that there exists one and only one positive number a such that 0 < fn(a) < fn+1(a) < 1 for all integers n \geq1. Solution The statement...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 7

IMO 1992 SL 7 Origin: IND Problem Circles G, G1, G2 are three circles related to each other as follows: Circles G1 and G2 are externally tangent to one another at a point W and both these circles are internally tangent to the circle G. Points A, B, C are located on the circle G as follows: Line BC is a direct common tangent to the pair of circles G1...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 16

IMO 1973 SL 16 Origin: CUB Problem Given a, \theta \inR, m \inN, and P(x) = x2m −2|a|mxm cos \theta+a2m, factorize P(x) as a product of m real quadratic polynomials. Solution First, we have P(x) = Q(x)R(x) for Q(x) = xm −|a|mei\theta and R(x) = xm −|a|me−i\theta, where eiϕ means of course cos ϕ + i sin ϕ. It remains to factor both Q and R. Suppose that Q(x) =...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G6

IMO 2001 SL G6 Origin: IND | Category: Geometry Problem Let ABC be a triangle and P an exterior point in the plane of the triangle. Suppose AP, BP, CP meet the sides BC, CA, AB (or ex- tensions thereof) in D, E, F, respectively. Suppose further that the areas of triangles PBD, PCE, PAF are all equal. Prove that each of these areas is equal to the area of...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 11

IMO 1968 SL 11 Origin: ROM Problem Find all solutions $(x_1, x_2, \ldots, x_n)$ of the equation $$1 + \frac{1}{x_1} + \frac{x_1 + 1}{x_1x_2} + \frac{(x_1 + 1)(x_2 + 1)}{x_1x_2x_3} + \cdots + \frac{(x_1 + 1) \cdots (x_{n-1} + 1)}{x_1x_2 \cdots x_n} = 0.$$ Solution Introducing $y_i = \frac{1}{x_i}$, we transform our equation to $$\begin{aligned} 0 &= 1 + y_1 + (1 + y_1)y_2 + \cdots + (1 + y_1)...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 26

IMO 1998 SL 26 Origin: IND Problem In a contest, there are m candidates and n judges, where n \geq3 is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most k candidates. Prove that k m \geqn−1 2n . Solution Put n = 2r + 1. Since each of the n pairs of...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 25

IMO 1997 SL 25 Origin: POL Problem The bisectors of angles A, B, C of a triangle ABC meet its cir- cumcircle again at the points K, L, M, respectively. Let R be an internal point on the side AB. The points P and Q are deﬁned by the following conditions: RP is parallel to AK, and BP is perpendicular to BL; RQ is parallel to BL, and AQ is...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G2

IMO 2004 SL G2 Origin: KAZ | Category: Geometry Problem The circle \Gamma and the line ℓdo not intersect. Let AB be the diameter of \Gamma perpendicular to ℓ, with B closer to ℓthan A. An arbitrary point C ̸= A, B is chosen on \Gamma. The line AC intersects ℓat D. The line DE is tangent to \Gamma at E, with B and E on the same side of...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 11

IMO 1970 SL 11 Origin: SWE Problem Let P, Q, R be polynomials and let S(x) = P(x3) + xQ(x3) + x2R(x3) be a polynomial of degree n whose roots x1, . . . , xn are distinct. Construct with the aid of the polynomials P, Q, R a polynomial T of degree n that has the roots x3 1, x3 2, . . . , x3 n. Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 13

IMO 1983 SL 13 Origin: LUX Problem Let E be the set of 19833 points of the space R3 all three of whose coordinates are integers between 0 and 1982 (including 0 and 1982). A coloring of E is a map from E to the set {red, blue}. How many colorings of E are there satisfying the following property: The number of red vertices among the 8 vertices of any...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 1

IMO 1983 SL 1 Origin: AUS Problem The localities P1, P2, . . . , P1983 are served by ten international airlines A1, A2, . . . , A10. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules oﬀer round-trip ﬂights. Prove that at least one of the airlines can oﬀer a round trip with an odd number...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 28

IMO 1989 SL 28 Origin: ROM Problem Consider in a plane \Pi the points O, A1, A2, A3, A4 such that \sigma(OAiAj) \geq1 for all i, j = 1, 2, 3, 4, i ̸= j. Prove that there is at least one pair i0, j0 \in{1, 2, 3, 4} such that \sigma(OAi0Aj0) \geq \sqrt 2. (We have denoted by \sigma(OAiAj) the area of triangle OAiAj.) Solution Assume w.l.o.g. that the...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 4

IMO 1978 SL 4 Origin: CZS Problem Let T1 be a triangle having a, b, c as lengths of its sides and let T2 be another triangle having u, v, w as lengths of its sides. If P, Q are the areas of the two triangles, prove that 16PQ \leqa2(−u2 + v2 + w2) + b2(u2 −v2 + w2) + c2(u2 + v2 −w2). When does equality hold? Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G6

IMO 2000 SL G6 Origin: ARG | Category: Geometry Problem Let ABCD be a convex quadrilateral with AB not parallel to CD, let X be a point inside ABCD such that ∡ADX = ∡BCX < 90◦ and ∡DAX = ∡CBX < 90◦. If Y is the point of intersection of the perpendicular bisectors of AB and CD, prove that ∡AY B = 2∡ADX. Solution By the condition of the problem,...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 1

IMO 1975 SL 1 Origin: FRA Problem There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions: (i) Every route includes exactly three ports; (ii) No two routes contain the same three ports; (iii) The series oﬀers exactly two routes to each tourist who desires to visit two diﬀerent arbitrary ports? Solution First, we observe that there cannot exist three...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 1

IMO 1997 SL 1 Origin: BLR Problem An inﬁnite square grid is colored in the chessboard pattern. For any pair of positive integers m, n consider a right-angled triangle whose vertices are grid points and whose legs, of lengths m and n, run along the lines of the grid. Let Sb be the total area of the black part of the triangle and Sw the total area of its white...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 22

IMO 1997 SL 22 Origin: UKR Problem (a) Do there exist functions f : R \toR and g : R \toR such that f(g(x)) = x2 and g(f(x)) = x3 for all x \inR? (b) Do there exist functions f : R \toR and g : R \toR such that f(g(x)) = x2 and g(f(x)) = x4 for all x \inR? Solution (a) Suppose that f and g are such...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 19

IMO 1990 SL 19 Origin: POL Problem Let P be a point inside a regular tetrahedron T of unit volume. The four planes passing through P and parallel to the faces of T partition T into 14 pieces. Let f(P) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 1

IMO 1987 SL 1 Origin: AUS Problem Let f be a function that satisﬁes the following conditions: (i) If x > y and f(y) −y \geqv \geqf(x) −x, then f(z) = v + z, for some number z between x and y. (ii) The equation f(x) = 0 has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A1

IMO 2003 SL A1 Origin: USA | Category: Algebra Problem Let aij, i = 1, 2, 3, j = 1, 2, 3, be real numbers such that aij is positive for i = j and negative for i ̸= j. Prove that there exist positive real numbers c1, c2, c3 such that the num- bers a11c1 + a12c2 + a13c3, a21c1 + a22c2 + a23c3, a31c1 + a32c2 + a33c3...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G6

IMO 2004 SL G6 Origin: GBR | Category: Geometry Problem Let P be a convex polygon. Prove that there is a convex hexagon that is contained in P and that occupies at least 75 percent of the area of P. Solution Let ABC be the triangle of maximum area S contained in P (it exists because of compactness of P). Draw parallels to BC, CA, AB through A, B, C,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C1

IMO 1994 SL C1 Origin: UKR | Category: Combinatorics Problem On a 5 \times 5 board, two players alternately mark numbers on empty cells. The ﬁrst player always marks 1’s, the second 0’s. One number is marked per turn, until the board is ﬁlled. For each of the nine 3 \times 3 squares the sum of the nine numbers on its cells is computed. Denote by A the maximum of...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 17

IMO 1997 SL 17 Origin: CZE Problem Find all pairs of integers x, y \geq1 satisfying the equation xy2 = yx. Solution We note ﬁrst that x and y must be powers of the same positive integer. Indeed, if x = p\alpha1 1 \cdot \cdot \cdot p\alphak k and y = p\beta1 1 \cdot \cdot \cdot p\betak k (some of \alphai and \betai may be 0, but not both for...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 18

IMO 1986 SL 18 Origin: TUR Problem Let AX, BY, CZ be three cevians concurrent at an inte- rior point D of a triangle ABC. Prove that if two of the quadrangles DY AZ, DZBX, DXCY are circumscribable, so is the third. Solution We shall use the following criterion for a quadrangle to be circumscribable. Lemma. The quadrangle AY DZ is circumscribable if and only if DB − DC =...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 10

IMO 1988 SL 10 Origin: GDR Problem Let N = {1, 2, . . ., n}, n \geq2. A collection F = {A1, . . . , At} of subsets Ai \subseteqN, i = 1, . . . , t, is said to be separating if for every pair {x, y} \subseteqN, there is a set Ai \inF such that Ai \cap{x, y} contains just one element. A collection F...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 12

IMO 1992 SL 12 Origin: NET Problem Let f, g, and a be polynomials with real coeﬃcients, f and g in one variable and a in two variables. Suppose f(x) −f(y) = a(x, y)(g(x) −g(y)) for all x, y \inR. Prove that there exists a polynomial h with f(x) = h(g(x)) for all x \inR. Solution Let us set deg f = n and deg g = m. We shall...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C8

IMO 2004 SL C8 Origin: POL | Category: Combinatorics Problem For a ﬁnite graph G, let f(G) be the number of triangles and g(G) the number of tetrahedra formed by edges of G. Find the least constant c such that g(G)3 \leqc \cdot f(G)4 for every graph G. Solution Denote by V1, . . . , Vn the vertices of a graph G and by E the set of its...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 21

IMO 1992 SL 21 Origin: GBR Problem For each positive integer n, denote by s(n) the greatest integer such that for all positive integers k \leqs(n), n2 can be expressed as a sum of squares of k positive integers. (a) Prove that s(n) \leqn2 −14 for all n \geq4. (b) Find a number n such that s(n) = n2 −14. (c) Prove that there exist inﬁnitely many positive integers n...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 2

IMO 1984 SL 2 Origin: CAN Problem Prove: (a) There are inﬁnitely many triples of positive integers m, n, p such that 4mn −m −n = p2 −1. (b) There are no positive integers m, n, p such that 4mn −m −n = p2. Solution (a) For m = t(t −1)/2 and n = t(t + 1)/2 we have 4mn −m −n = (t2 −1)2 −1. (b) Suppose that 4mn...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A2

IMO 1996 SL A2 Origin: IRE | Category: Algebra Problem Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be real numbers such that $a_1^k + a_2^k + \cdots + a_n^k \geq 0$ for all integers $k > 0$. Let $p = \max{|a_1|, \ldots, |a_n|}$. Prove that $p = a_1$ and that $(x - a_1)(x - a_2) \cdots (x - a_n) \leq x^n - a_1 x^{n-1} + \cdots + (-1)^n a_n$...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 17

IMO 1991 SL 17 Origin: HKG Problem Find all positive integer solutions x, y, z of the equation 3x + 4y = 5z. Solution Taking the equation 3x + 4y = 5z (x, y, z > 0) modulo 3, we get that 5z \equiv1 (mod 3), hence z is even, say z = 2z1. The equation then becomes 3x = 52z1 −4y = (5z1 −2y)(5z1 + 2y). Each factor 5z1...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 10

IMO 1984 SL 10 Origin: GBR Problem Prove that the product of ﬁve consecutive positive integers cannot be the square of an integer. Solution Suppose that the product of some ﬁve consecutive numbers is a square. It is easily seen that among them at least one, say n, is divisible neither by 2 nor 3. Since n is coprime to the remaining four numbers, it is itself a square of...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 17

IMO 1973 SL 17 Origin: POL Problem Let F be a nonempty set of functions f : R \toR of the form f(x) = ax + b, where a and b are real numbers and a ̸= 0. Suppose that F satisﬁes the following conditions: (1) If f, g \inF, then g ◦f \inF, where (g ◦f)(x) = g[f(x)]. (2) If f \inF and f(x) = ax + b, then...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 20

IMO 1990 SL 20 Origin: POL Problem Prove that every integer k greater than 1 has a multiple that is less than k4 and can be written in the decimal system with at most four diﬀerent digits. Solution Let n be the unique integer such that 2n−1 \leqk < 2n. Let S(n) be the set of numbers less than 10n that are written with only the digits {0, 1} in...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 20

IMO 1985 SL 20 Origin: GBR Problem A circle whose center is on the side ED of the cyclic quadrilateral BCDE touches the other three sides. Prove that EB+CD = ED. Solution Let O be the center of the circle touching the three sides of BCDE and let F \in(ED) be the point such that EF = EB. Then \angleEFB = 90◦− \angleE/2 = \angleC/2 = \angleOCB, which implies that...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 4

IMO 1976 SL 4 Problem (GBR 1a)IMO6 For all positive integral n, un+1 = un(u2 n−1 −2) −u1, u0 = 2, and u1 = 5/2. Prove that 3 log2[un] = 2n −(−1)n, where [x] is the integral part of x.

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C7

IMO 1994 SL C7 Origin: BRA | Category: Combinatorics Problem Prove that for any integer n \geq2, there exists a set of 2n−1 points in the plane such that no 3 lie on a line and no 2n are the vertices of a convex 2n-gon. Solution Deﬁne Sn recursively as follows: Let S2 = {(0, 0), (1, 1)} and Sn+1 = Sn \cupTn, where Tn = {(x + 2n−1, y...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 27

IMO 1990 SL 27 Origin: USS Problem Find all natural numbers n for which every natural number whose decimal representation has n −1 digits 1 and one digit 7 is prime. Solution Let us denote by An(k) the n-digit number which consists of n −1 ones and one digit seven in the k + 1-th rightmost position (0 \leqk < n). Then An(k) = (10n + 54 \cdot 10k −1)/9....

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A5

IMO 2004 SL A5 Origin: THA | Category: Algebra Problem Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality a + 6b + b + 6c + c + 6a \leq abc. Solution By the general mean inequality (M1 \leqM3), the LHS of the inequality to be proved does not exceed E = 3\sqrt a + 1 b...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 9

IMO 1997 SL 9 Origin: USA Problem Let A1A2A3 be a nonisosceles triangle with incenter I. Let Ci, i = 1, 2, 3, be the smaller circle through I tangent to AiAi+1 and AiAi+2 (the addition of indices being mod 3). Let Bi, i = 1, 2, 3, be the second point of intersection of Ci+1 and Ci+2. Prove that the circumcenters of the triangles A1B1I, A2B2I, A3B3I are collinear....

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A3

IMO 1994 SL A3 Origin: GBR | Category: Algebra Problem Let S be the set of real numbers greater than −1. Find all functions f : S \toS such that f(x + f(y) + xf(y)) = y + f(x) + yf(x) for all x and y in S, and f(x)/x is strictly increasing for −1 < x < 0 and for 0 < x. Solution The last condition implies that...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C6

IMO 2003 SL C6 Origin: SAF | Category: Combinatorics Problem Let f(k) be the number of integers n that satisfy the following conditions: (i) 0 \leqn < 10k, so n has exactly k digits (in decimal notation), with leading zeros allowed; (ii) the digits of n can be permuted in such a way that they yield an integer divisible by 11. Prove that f(2m) = 10f(2m −1) for every positive...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 13

IMO 1971 SL 13 Origin: SWE Problem Consider the n \times n array of nonnegative integers ⎛ ⎜ ⎜ ⎜ ⎝ a11 a12 . . . a1n a21 a22 \cdot \cdot \cdot a2n ... ... ... an1 an2 . . . ann ⎞ ⎟ ⎟ ⎟ ⎠, with the following property: If an element aij is zero, then the sum of the elements of the ith row and the jth...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 9

IMO 1987 SL 9 Origin: HUN Problem Does there exist a set M in usual Euclidean space such that for every plane \lambda the intersection M \cap\lambda is ﬁnite and nonempty? Solution The answer is yes. Consider the curve C = {(x, y, z) | x = t, y = t3, z = t5, t \inR}. Any plane deﬁned by an equation of the form ax+by+cz+d = 0 intersects the...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A3

IMO 2004 SL A3 Origin: CAN | Category: Algebra Problem Does there exist a function s: Q \to{−1, 1} such that if x and y are distinct rational numbers satisfying xy = 1 or x + y \in{0, 1}, then s(x)s(y) = −1? Justify your answer. Solution The answer is yes. Every rational number x > 0 can be uniquely expressed as a continued fraction of the form a0 +...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 15

IMO 1993 SL 15 Origin: MCD Problem For three points A, B, C in the plane we deﬁne m(ABC) to be the smallest length of the three altitudes of the triangle ABC, where in the case of A, B, C collinear, m(ABC) = 0. Let A, B, C be given points in the plane. Prove that for any point X in the plane, m(ABC) \leqm(ABX) + m(AXC) + m(XBC). Solution...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G7

IMO 1996 SL G7 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $AO$ meet the circle $BOC$ again in $A'$, let $BO$ meet the circle $COA$ again in $B'$, and let $CO$ meet the circle $AOB$ again in $C'$. Prove that $$OA' \cdot OB' \cdot OC' \geq 8R^3.$$ When does equality hold? Solution Let $A_1$ be the point...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 9

IMO 1972 SL 9 Origin: NET Problem Find all solutions in positive real numbers xi (i = 1, 2, 3, 4, 5) of the following system of inequalities: (x2 1 −x3x5)(x2 2 −x3x5) \leq0, (i) (x2 2 −x4x1)(x2 3 −x4x1) \leq0, (ii) (x2 3 −x5x2)(x2 4 −x5x2) \leq0, (iii) (x2 4 −x1x3)(x2 5 −x1x3) \leq0, (iv) (x2 5 −x2x4)(x2 1 −x2x4) \leq0. (v) Solution Clearly x1 = x2 = x3...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 2

IMO 1988 SL 2 Origin: BUL Problem Let n be a positive integer. Find the number of odd coeﬃcients of the polynomial un(x) = (x2 + x + 1)n. Solution For polynomials f(x), g(x) with integer coeﬃcients, we use the notation f(x) ∼g(x) if all the coeﬃcients of f −g are even. Let n = 2s. It is immediately shown by induction that (x2 + x + 1)2s ∼x2s+1 +...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A5

IMO 1994 SL A5 Origin: POL | Category: Algebra Problem Let f(x) = x2+1 2x for x ̸= 0. Deﬁne f (0)(x) = x and f (n)(x) = f(f (n−1)(x)) for all positive integers n and x ̸= 0. Prove that for all nonnegative integers n and x ̸= −1, 0, or 1, f (n)(x) f (n+1)(x) = 1 + f x+1 x−1 2n. Solution If f (n)(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 28

IMO 1998 SL 28 Origin: IRN Problem A solitaire game is played on an m \times n rectangular board, using mn markers that are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one can take away one marker...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 9

IMO 1981 SL 9 Origin: FRG Problem A sequence (an) is deﬁned by means of the recursion a1 = 1, an+1 = 1 + 4an + \sqrt1 + 24an . Find an explicit formula for an. Solution If we put 1 + 24an = b2 n, the given recurrent relation becomes 3b2 n+1 = 3 2 + b2 n 6 + bn = 2 3 2 + bn 2 ,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G4

IMO 2003 SL G4 Origin: ARM | Category: Geometry Problem Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1, \Gamma3 are externally tangent at P, and \Gamma2, \Gamma4 are externally tangent at the same point P. Suppose that \Gamma1 and \Gamma2; \Gamma2 and \Gamma3; \Gamma3 and \Gamma4; \Gamma4 and \Gamma1 meet at A, B, C, D, respectively, and that all these points are diﬀerent from P. Prove that...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G8

IMO 2002 SL G8 Origin: ARM | Category: Geometry Problem Let S1 and S2 be circles meeting at the points A and B. A line through A meets S1 at C and S2 at D. Points M, N, K lie on the line segments CD, BC, BD respectively, with MN parallel to BD and MK parallel to BC. Let E and F be points on those arcs BC of S1...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A6

IMO 2002 SL A6 Origin: IRN | Category: Algebra Problem Let A be a nonempty set of positive integers. Suppose that there are positive integers b1, . . . , bn and c1, . . . , cn such that (i) for each i the set biA + ci = {bia + ci | a \inA} is a subset of A, and (ii) the sets biA + ci and bjA...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 2

IMO 1971 SL 2 Origin: BUL Problem Prove that for every natural number m \geq1 there exists a ﬁnite set Sm of points in the plane satisfying the following condition: If A is any point in Sm, then there are exactly m points in Sm whose distance to A equals 1. Solution We will construct such a set Sm of 2m points. Take vectors u1, . . . , um...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 14

IMO 1988 SL 14 Origin: HUN Problem For what values of n does there exist an n \times n array of entries −1, 0, or 1 such that the 2n sums obtained by summing the elements of the rows and the columns are all diﬀerent? Solution Consider an array [aij] of the given property and denote the sums of the rows and the columns by ri and cj respectively. Among...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G3

IMO 2000 SL G3 Origin: IND | Category: Geometry Problem Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Show that there exist points D, E, and F on sides BC, CA, and AB respectively such that OD + DH = OE + EH = OF + FH and the lines AD, BE, and CF are concurrent. Solution Let L be the point symmetric to...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 12

IMO 1973 SL 12 Origin: SWE Problem Consider the two square matrices A = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ and B = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ with entries 1 and −1. The following operations will be called elementary: (1) Changing signs of all...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 25

IMO 1991 SL 25 Origin: USA Problem Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between 0 and 1 (inclusive). Prove that for some index i between 1 and n −1 the inequality xi(1 −xi+1) \geq1 4x1(1 −xn) holds. Solution Since replacing x1 by 1 can only reduce the set of indices i for which the desired inequality holds, we may assume x1...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 12

IMO 1971 SL 12 Origin: POL Problem Two congruent equilateral triangles ABC and A′B′C′ in the plane are given. Show that the midpoints of the segments AA′, BB′, CC′ either are collinear or form an equilateral triangle. Solution Let us start with the case A = A′. If the triangles ABC and A′B′C′ are oppositely oriented, then they are symmetric with respect to some axis, and the statement is true....

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 6

IMO 1988 SL 6 Origin: CZS Problem In a given tetrahedron ABCD let K and L be the centers of edges AB and CD respectively. Prove that every plane that contains the line KL divides the tetrahedron into two parts of equal volume. Solution Let \omega be the plane through AB, parallel to CD. Deﬁne the point trans- formation f : X "\toX′ in space as follows. If X \inKL,...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 2

IMO 1968 SL 2 Origin: ROM Problem Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others. Solution The sides a, b, c of a triangle ABC with \angleABC = 2\angleBAC satisfy b2 = a(a + c) (this statement is the lemma in (SL98-7)). Taking into account the remaining condition that a,...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 24

IMO 1991 SL 24 Origin: IND Problem An odd integer n \geq3 is said to be “nice” if there is at least one permutation a1, a2, . . . , an of 1, 2, . . ., n such that the n sums a1 −a2 + a3 −\cdot \cdot \cdot −an−1 + an, a2 −a3 + a4 −\cdot \cdot \cdot −an + a1, a3 −a4 + a5 −\cdot \cdot \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N4

IMO 1996 SL N4 Origin: BUL | Category: Number Theory Problem Find all positive integers $a$ and $b$ for which $$\left[\frac{a^2}{b}\right] + \left[\frac{b^2}{a}\right] = \left[\frac{a^2 + b^2}{ab}\right] + ab$$ where as usual, $[t]$ refers to greatest integer that is less than or equal to $t$. Solution Let $a, b \in \mathbb{N}$ satisfy the given equation. It is not possible that $a = b$ (since it leads to $a^2 + 2...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 8

IMO 1972 SL 8 Origin: GBR Problem Let m and n be nonnegative integers. Prove that m!n!(m+ n)! divides (2m)!(2n)!. Solution Let f(m, n) = (2m)!(2n)! m!n!(m+n)!. Then it is directly shown that f(m, n) = 4f(m, n −1) −f(m + 1, n −1), and thus n may be successively reduced until one obtains f(m, n) = r crf(r, 0). Now f(r, 0) is a simple binomial coeﬃcient, and...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 16

IMO 1981 SL 16 Origin: GBR Problem A sequence of real numbers u1, u2, u3, . . . is determined by u1 and the following recurrence relation for n \geq1: 4un+1 = 3\sqrt64un + 15. Describe, with proof, the behavior of un as n \to\infty. Solution The sequence {un} is bounded, whatever u1 is. Indeed, assume the opposite, and let um be the ﬁrst member of the sequence such that...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 18

IMO 1993 SL 18 Origin: POL Problem Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in {0, 1}, in which no six consecutive blocks are equal. Prove that Sn \to\infty as n \to\infty. Solution Let Bn be the set of sequences with the stated property (Sn = |Bn|). We shall prove by induction on n that Sn \geq3 2Sn−1 for every n....

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N4

IMO 2002 SL N4 Origin: GER | Category: Number Theory Problem Is there a positive integer m such that the equation a + 1 b + 1 c + 1 abc = m a + b + c has inﬁnitely many solutions in positive integers a, b, c? Solution For a = b = c = 1 we obtain m = 12. We claim that the given equation has inﬁnitely...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C4

IMO 2004 SL C4 Origin: POL | Category: Combinatorics Problem Consider a matrix of size n\timesn whose entries are real numbers of absolute value not exceeding 1, and the sum of all entries is 0. Let n be an even positive integer. Determine the least number C such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding C in absolute...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A1

IMO 2000 SL A1 Origin: USA | Category: Algebra Problem Let a, b, c be positive real numbers with product 1. Prove that a −1 + 1 b b −1 + 1 c c −1 + 1 a \leq1. Solution Elementary computation gives a −1 + 1 b b −1 + 1 c = ab−a+ a c −b+ 1 −1...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N3

IMO 1999 SL N3 Origin: RUS | Category: Number Theory Problem Prove that there exist two strictly increasing sequences (an) and (bn) such that an(an + 1) divides b2 n + 1 for every natural number n. Solution We ﬁrst prove the following lemma. Lemma. For d, c \inN and d2 | c2 + 1 there exists b \inN such that d2(d2 + 1) | b2 + 1. Proof. It...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 13

IMO 1987 SL 13 Origin: GDR Problem Is it possible to put 1987 points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area? Solution We claim that the points Pi(i, i2), i = 1, 2, . . . , 1987, satisfy the conditions. In fact: (i) PiPj = (i −j)2 + (i2 −j2)2...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 19

IMO 1992 SL 19 Origin: IRE Problem Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime and suppose there exists an integer z such that p divides f(z). Prove that there exist integers z1, z2, . . . , z8 such that if g(x) = (x −z1)(x −z2) \cdot \cdot \cdot (x −z8), then all coeﬃcients of f(x) −g(x) are...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 19

IMO 1982 SL 19 Problem C7 (CZS 3) Let M be the set of real numbers of the form m+n \sqrt m2+n2 , where m and n are positive integers. Prove that for every pair x \inM, y \inM with x < y, there exists an element z \inM such that x < z < y. Solution Let us set x = m/n. Since f(x) = (m + n)/ \sqrt...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 9

IMO 1975 SL 9 Origin: NET Problem Let f(x) be a continuous function deﬁned on the closed interval 0 \leqx \leq1. Let G(f) denote the graph of f(x): G(f) = {(x, y) \inR2 | 0 \leq x \leq1, y = f(x)}. Let Ga(f) denote the graph of the translated function f(x −a) (translated over a distance a), deﬁned by Ga(f) = {(x, y) \in R2 | a \leqx \leqa +...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G7

IMO 1995 SL G7 Origin: LAT | Category: Geometry Problem O is a point inside a convex quadrilateral ABCD of area S. K, L, M, and N are interior points of the sides AB, BC, CD, and DA respectively. If OKBL and OMDN are parallelograms, prove that \sqrt S \geq\sqrtS1 + \sqrtS2, where S1 and S2 are the areas of ONAK and OLCM respectively. Solution If O lies on AC,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G2

IMO 1994 SL G2 Origin: UKR | Category: Geometry Problem ABCD is a quadrilateral with BC parallel to AD. M is the midpoint of CD, P that of MA and Q that of MB. The lines DP and CQ meet at N. Prove that N is not outside triangle ABM.8 Solution Although it does not seem to have been noticed at the jury, the state- ment of the problem is...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N1

IMO 2003 SL N1 Origin: POL | Category: Number Theory Problem Let m be a ﬁxed integer greater than 1. The sequence x0, x1, x2, . . . is deﬁned as follows: xi = . 2i, if 0 \leqi \leqm −1; m j=1 xi−j, if i \geqm. Find the greatest k for which the sequence contains k consecutive terms divisible by m. Solution Let ri be the remainder when xi...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 15

IMO 1968 SL 15 Origin: GBR Problem Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum $$\frac{n+1}{2} + \frac{n+2}{2^2} + \cdots + \frac{n+2^i}{2^{i+1}} + \cdots .$$ Solution Set $$f(n)= \frac{n+1}{2} + \frac{n+2}{2^2} + \cdots + \frac{n+2^i}{2^{i+1}} + \cdots .$$ We prove by induction that $f(n)=n$. This obviously holds for...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G6

IMO 1999 SL G6 Origin: RUS | Category: Geometry Problem Two circles Ω1 and Ω2 touch internally the circle Ωin M and N, and the center of Ω2 is on Ω1. The common chord of the circles Ω1 and Ω2 intersects Ωin A and B. MA and MB intersect Ω1 in C and D. Prove that Ω2 is tangent to CD. Solution We ﬁrst introduce the following lemmas. Lemma 1....

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 13

IMO 1986 SL 13 Origin: FRG Problem A particle moves from (0, 0) to (n, n) directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At (n, y), y < n, it stays there if a head comes up and at (x, n), x < n, it stays there if a tail comes up. Let k be...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G1

IMO 1994 SL G1 Origin: FRA | Category: Geometry Problem A semicircle \Gamma is drawn on one side of a straight line l. C and D are points on \Gamma. The tangents to \Gamma at C and D meet l at B and A respectively, with the center of the semicircle between them. Let E be the point of intersection of AC and BD, and F the point on l...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 27

IMO 1998 SL 27 Origin: BLR Problem Ten points such that no three of them lie on a line are marked in the plane. Each pair of points is connected with a segment. Each of these segments is painted with one of k colors in such a way that for any k of the ten points, there are k segments each joining two of them with no two being painted...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 18

IMO 1968 SL 18 Origin: ITA Problem If an acute-angled triangle ABC is given, construct an equilat- eral triangle A′B′C′ in space such that lines AA′, BB′, CC′ pass through a given point. Solution The required construction is not feasible. In fact, let us consider the special case \angleBOC = 135◦, \angleAOC = 120◦, \angleAOB = 90◦, where AA′ \capBB′ \cap CC′ = {O}. Denoting OA′, OB′, OC′ by a,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 2

IMO 1979 SL 2 Origin: BEL Problem From a bag containing 5 pairs of socks, each pair a diﬀerent color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of diﬀerent colors held outside the bag. What is the...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 6

IMO 1984 SL 6 Origin: CAN Problem Let c be a positive integer. The sequence {fn} is deﬁned as follows: f1 = 1, f2 = c, fn+1 = 2fn −fn−1 + 2 (n \geq2). Show that for each k \inN there exists r \inN such that fkfk+1 = fr. Solution From the given recurrence we infer fn+1−fn = fn−fn−1+2. Consequently, fn+1 −fn = (f2 −f1) + 2(n −1) = c...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 10

IMO 1983 SL 10 Origin: FIN Problem Let p and q be integers. Show that there exists an interval I of length 1/q and a polynomial P with integral coeﬃcients such that P(x) −p q < 1 q2 for all x \inI. Solution Choose P(x) = p q (qx −1)2n+1 + 1 , I = [1/2q, 3/2q]. Then all the coef- ﬁcients of P are integers, and P(x)...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 23

IMO 1983 SL 23 Origin: USS Problem Let K be one of the two intersection points of the circles W1 and W2. Let O1 and O2 be the centers of W1 and W2. The two common tangents to the circles meet W1 and W2 respectively in P1 and P2, the ﬁrst tangent, and Q1 and Q2, the second tangent. Let M1 and M2 be the midpoints of P1Q1 and P2Q2,...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 6

IMO 1973 SL 6 Origin: POL Problem Does there exist a ﬁnite set M of points in space, not all in the same plane, such that for each two points A, B \inM there exist two other points C, D \inM such that lines AB and CD are parallel? Solution Yes. Take for M the set of vertices of a cube ABCDEFGH and two points I, J symmetric to the...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 1

IMO 1991 SL 1 Origin: PHI Problem Let ABC be any triangle and P any point in its interior. Let P1, P2 be the feet of the perpendiculars from P to the two sides AC and BC. Draw AP and BP, and from C drop perpendiculars to AP and BP. Let Q1 and Q2 be the feet of these perpendiculars. Prove that the lines Q1P2, Q2P1, and AB are concurrent....

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N1

IMO 2004 SL N1 Origin: BLR | Category: Number Theory Problem Let \tau(n) denote the number of positive divisors of the positive integer n. Prove that there exist inﬁnitely many positive integers a such that the equation \tau(an) = n does not have a positive integer solution n. Solution Setting m = an we reduce the given equation to m/\tau(m) = a. Let us show that for a = pp−1...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 2

IMO 1977 SL 2 Origin: CZS Problem A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let k be a circle with radius r \geq2, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle k that has a neighboring point lying...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 23

IMO 1989 SL 23 Origin: POL Problem We consider permutations (x1, . . . , x2n) of the set {1, . . . , 2n} such that |xi −xi+1| = n for at least one i \in{1, . . ., 2n−1}. For every natural number n, ﬁnd out whether permutations with this property are more or less numerous than the remaining permutations of {1, . . . , 2n}. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 4

IMO 1985 SL 4 Origin: AUS Problem Each of the numbers in the set N = {1, 2, 3, . . ., n −1}, where n \geq3, is colored with one of two colors, say red or black, so that: (i) i and n −i always receive the same color, and (ii) for some j \inN, relatively prime to n, i and |j −i| receive the same color for all...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C5

IMO 2001 SL C5 Origin: FIN | Category: Combinatorics Problem Find all ﬁnite sequences (x0, x1, . . . , xn) such that for every j, 0 \leqj \leqn, xj equals the number of times j appears in the sequence. Solution Let (x0, x1, . . . , xn) be any such sequence: its terms are clearly nonnegative integers. Also, x0 = 0 yields a contradiction, so x0 > 0....

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 21

IMO 1968 SL 21 Origin: CZS Problem Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive integers y such that a0 | y; (a0 + a1) | (y + a1); . . . ; (a0 + an) | (y + an). Solution The given conditions are equivalent to y −a0 being divisible by a0, a0 + a1, a0 +a2, . . . ,...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 8

IMO 1984 SL 8 Origin: ROM Problem In a plane two diﬀerent points O and A are given. For each point X ̸= O of the plane denote by \alpha(X) the angle AOX measured in radians (0 \leq\alpha(X) < 2\pi) and by C(X) the circle with center O and radius OX + \alpha(X) OX . Suppose each point of the plane is colored by one of a ﬁnite number of...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 4

IMO 1988 SL 4 Origin: CZS Problem An n \times n chessboard (n \geq2) is numbered by the numbers 1, 2, . . . , n2 (every number occurs once). Prove that there exist two neigh- boring (which share a common edge) squares such that their numbers diﬀer by at least n. Solution Suppose that the numbers of any two neighboring squares diﬀer by at most n −1. For k...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A3

IMO 2003 SL A3 Origin: GEO | Category: Algebra Problem Consider pairs of sequences of positive real numbers a1 \geq a2 \geqa3 \geq\cdot \cdot \cdot , b1 \geqb2 \geqb3 \geq\cdot \cdot \cdot and the sums An = a1 + \cdot \cdot \cdot + an, Bn = b1 + \cdot \cdot \cdot + bn, n = 1, 2, . . . . For any pair deﬁne ci = min{ai, bi} and...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C7

IMO 2002 SL C7 Origin: NZL | Category: Combinatorics Problem Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets? Solution For a graph G on 120 vertices (i.e., people at the party), write q(G) for the number of weak quartets in G. Our solution will...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 15

IMO 1987 SL 15 Origin: FRG Problem Suppose x1, x2, . . . , xn are real numbers with x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = 1. Prove that for any integer k > 1 there are integers ei not all 0 and with |ei| < k such that |e1x1 + e2x2 + \cdot \cdot \cdot + enxn| \leq(k −1)\sqrtn kn −1 . Solution...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 11

IMO 1973 SL 11 Origin: SWE Problem Determine the minimum of a2 + b2 if a and b are real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. Solution Putting x + 1 x = t we also get x2 + x2 = t2 −2, and the given equation reduces to t2 + at + b...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A1

IMO 2001 SL A1 Origin: IND | Category: Algebra Problem Let T denote the set of all ordered triples (p, q, r) of nonneg- ative integers. Find all functions f : T \toR such that f(p, q, r) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ if pqr = 0, 1 + 1 6 (f(p + 1, q −1, r) + f(p −1, q + 1, r) +f(p...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 13

IMO 1998 SL 13 Origin: BUL Problem Determine the least possible value of f(1998), where f is a function from the set N of positive integers into itself such that for all m, n \inN, f(n2f(m)) = m[f(n)]2. Solution Denote by F the set of functions considered. Let f \inF, and let f(1) = a. Putting n = 1 and m = 1 we obtain f(f(z)) = a2z and f(az2)...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 17

IMO 1988 SL 17 Origin: ISR Problem In the convex pentagon ABCDE, the sides BC, CD, DE have the same length. Moreover, each diagonal of the pentagon is parallel to a side (AC is parallel to DE, BD is parallel to AE, etc.). Prove that ABCDE is a regular pentagon. Solution Let AC and AD meet BE in R, S, respectively. Then by the conditions of the problem, \angleAEB =...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 13

IMO 1981 SL 13 Origin: ROM Problem Let P be a polynomial of degree n satisfying P(k) = n + 1 k −1 for k = 0, 1, . . ., n. Determine P(n + 1). Solution Lemma. For any polynomial P of degree at most n, n+1 i=0 (−1)i n + 1 i P(i) = 0. (1) Proof. We shall use induction on n. For n =...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 26

IMO 1993 SL 26 Origin: VIE Problem Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d = Solution Set f(a, b, c, d) = abc + bcd + cda + dab −176 27 abcd = ab(c + d) + cd a + b −176 27 ab . If a + b −176 a b \leq0, by the arithmetic-geometric inequality we have f(a, b, c, d) \leqab(c...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C7

IMO 2004 SL C7 Origin: EST | Category: Combinatorics Problem Determine all m \times n rectangles that can be covered with hooks made up of 6 unit squares, as in the ﬁgure: Rotations and reﬂections of hooks are allowed. The rectangle must be covered without gaps and overlaps. No part of a hook may cover area outside the rectangle. Solution Suppose that an m\timesn rectangle can be covered by “hooks”....

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 8

IMO 1974 SL 8 Problem II 2 (NET 3)IMO5 If a, b, c, d are arbitrary positive real numbers, ﬁnd all possible values of S = a a + b + d + b a + b + c + c b + c + d + d a + c + d. Solution It is clear that a a + b + c + d + b a +...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 3

IMO 1974 SL 3 Problem I 3 (SWE 3)IMO6 Let P(x) be a polynomial with integer coeﬃcients. If n(P) is the number of (distinct) integers k such that P 2(k) = 1, prove that n(P) −deg(P) \leq2, where deg(P) denotes the degree of the polynomial P. Solution For deg(P) \leq2 the statement is obvious, since n(P) \leqdeg(P 2) = 2 deg(P) \leqdeg(P) + 2. Suppose now that deg(P) \geq3 and...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A1

IMO 1995 SL A1 Origin: RUS | Category: Algebra Problem Let a, b, and c be positive real numbers such that abc = 1. Prove that a3(b + c) + b3(a + c) + c3(a + b) \geq3 2. Solution Let x = 1 a, y = 1 b, z = 1 c. Then xyz = 1 and S = a3(b + c) + b3(c + a) + c3(a +...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G7

IMO 2004 SL G7 Origin: RUS | Category: Geometry Problem For a given triangle ABC, let X be a variable point on the line BC such that C lies between B and X and the incircles of the triangles ABX and ACX intersect at two distinct points P and Q. Prove that the line PQ passes through a point independent of X. Solution The proof uses the following observation: Lemma....

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 22

IMO 1985 SL 22 Origin: USS Problem A circle with center O passes through points A and C and intersects the sides AB and BC of the triangle ABC at points K and N, respectively. The circumscribed circles of the triangles ABC and KBN intersect at two distinct points B and M. Prove that \angleOMB = 90◦. Solution Assume that \triangleABC is acute (the case of an obtuse \triangleABC is...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 11

IMO 1976 SL 11 Origin: VIE Problem Prove that there exist inﬁnitely many positive integers n such that the decimal representation of 5n contains a block of 1976 consecutive zeros. Solution We shall show by induction that 52k −1 = 2k+2qk for each k = 0, 1, . . . , where qk \inN. Indeed, the statement is true for k = 0, and if it holds for some k...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N8

IMO 1995 SL N8 Origin: GER | Category: Number Theory Problem Let p be an odd prime. Determine positive integers x and y for which x \leqy and \sqrt2p −\sqrtx −\sqrty is nonnegative and as small as possible. Solution Let u = \sqrt2p −\sqrtx −\sqrty and v = u(2\sqrt2p −u) = 2p −(\sqrt2p −u)2 = 2p −x −y −\sqrt4xy for x, y \inN, x \leqy. Obviously u \geq0 if and...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A2

IMO 2002 SL A2 Origin: YUG | Category: Algebra Problem Let a1, a2, . . . be an inﬁnite sequence of real numbers for which there exists a real number c with 0 \leqai \leqc for all i such that |ai −aj| \geq i + j for all i, j with i ̸= j. Prove that c \geq1. Solution For n \geq2, let (k1, k2, . . . , kn)...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 15

IMO 1992 SL 15 Origin: PRK Problem Does there exist a set M with the following properties? (i) The set M consists of 1992 natural numbers. (ii) Every element in M and the sum of any number of elements have the form mk (m, k \inN, k \geq2). Solution The result follows from the following lemma by taking n = 1992\cdot1993 and M = {d, 2d, . . . ,...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G3

IMO 1996 SL G3 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ be an acute-angled triangle with $BC > CA$. Let $O$ be the circumcenter, $H$ its orthocenter, and $F$ the foot of its altitude $CH$. Let the perpendicular to $OF$ at $F$ meet the side $CA$ at $P$. Prove that $\angle FHP = \angle BAC.$ Possible second part: What happens if $|BC| \leq |CA|$ (the triangle still being...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 20

IMO 1989 SL 20 Origin: NET Problem Given a set S in the plane containing n points and satis- fying the conditions: (i) no three points of S are collinear, (ii) for every point P of S there exist at least k points in S that have the same distance to P, prove that the following inequality holds: k < 1 2 + \sqrt 2n. Solution Suppose k \geq1/2 +...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C2

IMO 2003 SL C2 Origin: GEO | Category: Combinatorics Problem Let D1, . . . , Dn be closed disks in the plane. (A closed disk is a region bounded by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 2003 disks Di. Prove that there exists disk Dk that intersects at most 7 \cdot 2003 −1 other disks Di....

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 17

IMO 1981 SL 17 Origin: USS Problem Three equal circles touch the sides of a triangle and have one common point O. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle ABC and the point O are collinear. Solution Let us denote by SA, SB, SC the centers of the given circles, where SA lies on the bisector of \angleA, etc. Then...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 25

IMO 1989 SL 25 Origin: KOR Problem Let a, b be integers that are not perfect squares. Prove that if x2 −ay2 −bz2 + abw2 = 0 has a nontrivial solution in integers, then so does x2 −ay2 −bz2 = 0. Solution We may assume w.l.o.g. that a > 0 (because a, b < 0 is impossible, and a, b ̸= 0 from the condition of the problem). Let (x0,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 19

IMO 1979 SL 19 Origin: ROM Problem Consider the sequences (an), (bn) deﬁned by a1 = 3, b1 = 100, an+1 = 3an, bn+1 = 100bn. Find the smallest integer m for which bm > a100. Solution Let us deﬁne ij for two positive integers i and j in the following way: i1 = i and ij+1 = iij for all positive integers j. Thus we must ﬁnd the smallest...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 5

IMO 1972 SL 5 Origin: GDR Problem Prove the following assertion: The four altitudes of a tetrahe- dron ABCD intersect in a point if and only if AB2 + CD2 = BC2 + AD2 = CA2 + BD2. Solution Lemma. If X, Y, Z, T are points in space, then the lines XZ and Y T are perpendicular if and only if XY 2 + ZT 2 = Y Z2...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 1

IMO 1982 SL 1 Problem A1 (GBR 3)IMO1 The function f(n) is deﬁned for all positive integers n and takes on nonnegative integer values. Also, for all m, n, f(m + n) −f(m) −f(n) = 0 or 1; f(2) = 0, f(3) > 0, and f(9999) = 3333. Determine f(1982). Solution From f(1) + f(1) \leqf(2) = 0 we obtain f(1) = 0. Since 0 < f(3) \leq f(1) +...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 16

IMO 1979 SL 16 Origin: ISR Problem Let K denote the set {a, b, c, d, e}. F is a collection of 16 diﬀerent subsets of K, and it is known that any three members of F have at least one element in common. Show that all 16 members of F have exactly one element in common. Solution Obviously, no two elements of F can be complements of each other....

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 23

IMO 1988 SL 23 Origin: SIN Problem Let Q be the center of the inscribed circle of a triangle ABC. Prove that for any point P, a(PA)2+b(PB)2+c(PC)2 = a(QA)2+b(QB)2+c(QC)2+(a+b+c)(QP)2, where a = BC, b = CA, and c = AB. Solution Denote by R the intersection point of lines AQ and BC. We know that BR : RC = c : b and AQ : QR = (b + c)...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 4

IMO 1971 SL 4 Origin: GBR Problem We are given two mutually tangent circles in the plane, with radii r1, r2. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of r1 and r2 and the condition for the solvability of the problem. Solution In the coordinate system in which the x-axis passes through the centers of the circles...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 21

IMO 1979 SL 21 Origin: USS Problem Let N be the number of integral solutions of the equation x2 −y2 = z3 −t3 satisfying the condition 0 \leqx, y, z, t \leq106, and let M be the number of integral solutions of the equation x2 −y2 = z3 −t3 + 1 satisfying the condition 0 \leqx, y, z, t \leq106. Prove that N > M. Solution Let f(n) be the...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 14

IMO 1992 SL 14 Origin: POL Problem For any positive integer x deﬁne g(x) = greatest odd divisor of x, f(x) = . x/2 + x/g(x), if x is even; 2(x+1)/2, if x is odd. Construct the sequence x1 = 1, xn+1 = f(xn). Show that the number 1992 appears in this sequence, determine the least n such that xn = 1992, and determine whether n is unique. Solution We...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A5

IMO 1999 SL A5 Origin: JAP | Category: Algebra Problem Find all the functions f : R \toR that satisfy f(x −f(y)) = f(f(y)) + xf(y) + f(x) −1 for all x, y \inR. Solution Let A = {f(x) | x \inR} and f(0) = c. Plugging in x = y = 0 we get f(−c) = f(c) + c −1, hence c ̸= 0. If x \inA, then taking...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 4

IMO 1979 SL 4 Origin: BUL Problem A pentagonal prism A1A2 . . . A5B1B2 . . . B5 is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism has its three edges of the same color. Prove that all edges of...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 15

IMO 1973 SL 15 Origin: CUB Problem Prove that for all n \inN the following is true: 2n n k=1 sin k\pi 2n + 1 = \sqrt 2n + 1. Solution If z = cos \theta + i sin \theta, then z −z−1 = 2i sin \theta. Now put z = cos \pi 2n+1 + i sin \pi 2n+1. Using de Moivre’s formula we transform the required equality into A...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 17

IMO 1968 SL 17 Origin: GBR Problem Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x$, $OB = y$, $OC = z$, if and only if $x+y \geq z$, $y+z \geq x$, $z+x \geq y$ (the points $O, A, B, C$ are coplanar).

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G3

IMO 1995 SL G3 Origin: TUR | Category: Geometry Problem The incircle of ABC touches BC, CA, and AB at D, E, and F respectively. X is a point inside ABC such that the incircle of XBC touches BC at D also, and touches CX and XB at Y and Z, respectively. Prove that EFZY is a cyclic quadrilateral. Solution If EF is parallel to BC, \triangleABC must be isosceles...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 3

IMO 1984 SL 3 Origin: USS Problem Find all positive integers n such that n = d2 6 + d2 7 −1, where 1 = d1 < d2 < \cdot \cdot \cdot < dk = n are all positive divisors of the number n. Solution From the equality n = d2 6 + d2 7 −1 we see that d6 and d7 are relatively prime and d7 | d2 6...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 16

IMO 1988 SL 16 Origin: IRE Problem Show that the solution set of the inequality k=1 k x −k \geq5 is a union of disjoint intervals the sum of whose lengths is 1988. Solution Let f(x) = 70 k=1 k x−k. For all integers i = 1, . . . , 70 we have that f(x) tends to plus inﬁnity as x tends downward to i, and f(x) tends...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 11

IMO 1982 SL 11 Problem B5 (CAN 3) (a) Find the rearrangement {a1, . . . , an} of {1, 2, . . ., n} that maximizes a1a2 + a2a3 + \cdot \cdot \cdot + ana1 = Q. (b) Find the rearrangement that minimizes Q. Solution (a) Suppose {a1, a2, . . . , an} is the arrangement that yields the maxi- mal value Qmax of Q. Note that the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 6

IMO 1998 SL 6 Origin: POL Problem Let ABCDEF be a convex hexagon such that \angleB +\angleD +\angleF = 360◦and AB BC \cdot CD DE \cdot EF FA = 1. Prove that BC CA \cdot AE EF \cdot FD DB = 1. Solution Let P be the point such that \triangleCDP and \triangleCBA are similar and equally oriented. Since then \angleDCP = \angleBCA and BC CA = DC CP ,...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N4

IMO 1999 SL N4 Origin: FRA | Category: Number Theory Problem Denote by S the set of all primes p such that the decimal representation of 1/p has its fundamental period divisible by 3. For every p \inS such that 1/p has its fundamental period 3r one may write 1/p = 0.a1a2 . . . a3ra1a2 . . . a3r . . . , where r = r(p); for every...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 9

IMO 1990 SL 9 Origin: HUN Problem The incenter of the triangle ABC is K. The midpoint of AB is C1 and that of AC is B1. The lines C1K and AC meet at B2, the lines B1K and AB at C2. If the areas of the triangles AB2C2 and ABC are equal, what is the measure of angle \angleCAB? Solution Let a, b, c be the lengths of the...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 16

IMO 1985 SL 16 Problem 5b.(BEL 2) If possible, construct an equilateral triangle whose three vertices are on three given circles. Solution Let the three circles be \alpha(A, a), \beta(B, b), and \gamma(C, c), and assume c \leqa, b. We denote by RX,ϕ the rotation around X through an angle ϕ. Let PQR be an equilateral triangle, say of positive orientation (the case of negatively oriented \trianglePQR is analogous), with...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 15

IMO 1983 SL 15 Origin: POL Problem Decide whether there exists a set M of natural numbers satis- fying the following conditions: (i) For any natural number m > 1 there are a, b \inM such that a+b = m. (ii) If a, b, c, d \inM, a, b, c, d > 10 and a + b = c + d, then a = c or a = d. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 28

IMO 1991 SL 28 Origin: NET Problem Given a real number a > 1, construct an inﬁnite and bounded sequence x0, x1, x2, . . . such that for all natural numbers i and j, i ̸= j, the following inequality holds: |xi −xj||i −j|a \geq1. Solution Let xn = c(n \sqrt 2 −[n \sqrt 2]) for some constant c > 0. For i > j, putting p = [i...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A5

IMO 1995 SL A5 Origin: UKR | Category: Algebra Problem Let R be the set of real numbers. Does there exist a function f : R \toR that simultaneously satisﬁes the following three conditions? (a) There is a positive number M such that −M \leqf(x) \leqM for all x. (b) f(1) = 1. (c) If x ̸= 0, then f x + 1 x2 = f(x) + f...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 3

IMO 1983 SL 3 Origin: BEL Problem We say that a set E of points of the Euclidian plane is “Pythagorean” if for any partition of E into two sets A and B, at least one of the sets contains the vertices of a right-angled triangle. Decide whether the following sets are Pythagorean: (a) a circle; (b) an equilateral triangle (that is, the set of three vertices and the points...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 6

IMO 1977 SL 6 Origin: GDR Problem Let n be a positive integer. How many integer solutions (i, j, k, l), 1 \leqi, j, k, l \leqn, does the following system of inequalities have: 1 \leq−j + k + l \leqn 1 \leq i −k + l \leqn 1 \leq i −j + l \leqn 1 \leq i + j −k \leqn ?

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C1

IMO 2000 SL C1 Origin: HUN | Category: Combinatorics Problem A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one, and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two diﬀerent boxes and announces the sum of numbers on those cards. Given this information, the...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 8

IMO 1977 SL 8 Origin: GBR Problem Let S be a convex quadrilateral ABCD and O a point inside it. The feet of the perpendiculars from O to AB, BC, CD, DA are A1, B1, C1, D1 respectively. The feet of the perpendiculars from O to the sides of Si, the quadrilateral AiBiCiDi, are Ai+1Bi+1Ci+1Di+1, where i = 1, 2, 3. Prove that S4 is similar to S.

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 1

IMO 1976 SL 1 Origin: BUL Problem Let ABC be a triangle with bisectors AA1, BB1, CC1 (A1 \in BC, etc.) and M their common point. Consider the triangles MB1A, MC1A, MC1B, MA1B, MA1C, MB1C, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then AB = BC = CA. Solution Let r denote the common inradius. Some two of the four triangles...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 18

IMO 1989 SL 18 Origin: MON Problem Given a convex polygon A1A2 . . . An with area S, and a point M in the same plane, determine the area of polygon M1M2 . . . Mn, where Mi is the image of M under rotation R\alpha Ai around Ai by \alpha, i = 1, 2, . . . , n. Solution Consider the triangle MAiMi. Obviously, the point Mi...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 3

IMO 1993 SL 3 Origin: SPA Problem Consider the triangle ABC, its circumcircle k with center O and radius R, and its incircle with center I and radius r. Another circle kc is tangent to the sides CA, CB at D, E, respectively, and it is internally tangent to k. Show that the incenter I is the midpoint of DE.

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 7

IMO 1988 SL 7 Origin: FRA Problem Let a be the greatest positive root of the equation x3−3x2+1 = 0. Show that [a1788] and [a1988] are both divisible by 17. ([x] denotes the integer part of x.) Solution The algebraic equation x3 −3x2 + 1 = 0 admits three real roots \beta, \gamma, a, with −0.6 < \beta < −0.5, 0.6 < \gamma < 0.7, \sqrt 8 < a <...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 13

IMO 1990 SL 13 Origin: IRE Problem An eccentric mathematician has a ladder with n rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers a rungs of the ladder, and when he descends, each step he takes covers b rungs of the ladder, where a and b are ﬁxed positive integers. By a sequence of ascending and descending steps he...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N6

IMO 2004 SL N6 Origin: IRE | Category: Number Theory Problem Given an integer n > 1, denote by Pn the product of all positive integers x less than n and such that n divides x2 −1. For each n > 1, ﬁnd the remainder of Pn on division by n. Solution Let Sn = {x \inN | x \leqn, n | x2 −1}. It is easy to check that...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A6

IMO 2004 SL A6 Origin: RUS | Category: Algebra Problem Find all functions f : R \toR satisfying the equation f x2 + y2 + 2f(xy) = (f(x + y))2 for all x, y \inR. Solution Let us make the substitution z = x + y, t = xy. Given z, t \inR, x, y are real if and only if 4t \leqz2. Deﬁne g(x) = 2(f(x) −x)....

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 5

IMO 1987 SL 5 Origin: GBR Problem Find, with proof, the point P in the interior of an acute-angled triangle ABC for which BL2 +CM2 +AN2 is a minimum, where L, M, N are the feet of the perpendiculars from P to BC, CA, AB respectively. Solution Assuming the notation a = BC, b = AC, c = AB; x = BL, y = CM, z = AN, from the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C1

IMO 1999 SL C1 Origin: IND | Category: Combinatorics Problem Let n \geq1 be an integer. A path from (0, 0) to (n, n) in the xy plane is a chain of consecutive unit moves either to the right (move denoted by E) or upwards (move denoted by N), all the moves being made inside the half-plane x \geqy. A step in a path is the occurrence of two consecutive...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N1

IMO 2001 SL N1 Origin: AUS | Category: Number Theory Problem Prove that there is no positive integer n such that for k = 1, 2, . . . , 9, the leftmost digit (in decimal notation) of (n + k)! equals k. Solution For each positive integer x, deﬁne \alpha(x) = x/10r if r is the positive integer satisfying 10r \leqx < 10r+1. Observe that if \alpha(x)\alpha(y) < 10...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 18

IMO 1987 SL 18 Origin: ROM Problem For any integer r \geq1, determine the smallest integer h(r) \geq1 such that for any partition of the set {1, 2, . . ., h(r)} into r classes, there are integers a \geq0, 1 \leqx \leqy, such that a + x, a + y, a + x + y belong to the same class. Solution Note ﬁrst that the statement that some a...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C5

IMO 1996 SL C5 Origin: FRA | Category: Combinatorics Problem Let $p, q, n$ be three positive integers with $p + q < n$. Let $(x_0, x_1, \dots, x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions: $x_0 = x_n = 0$. For each $i$ with $1 \leq i \leq n$, either $x_i - x_{i-1} = p$ or $x_i - x_{i-1} = -q$. Show that there exists a pair...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N3

IMO 2003 SL N3 Origin: BUL | Category: Number Theory Problem Determine all pairs (a, b) of positive integers such that a2 2ab2 −b3 + 1 is a positive integer. Solution Let a and b be positive integers for which a2 2ab2−b3+1 = k is a positive integer. Since k > 0, it follows that 2ab2 \geqb3, so 2a \geqb. If 2a > b, then from 2ab2 −b3 + 1...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 3

IMO 1986 SL 3 Origin: USA Problem Let A, B, and C be three points on the edge of a circular chord such that B is due west of C and ABC is an equilateral triangle whose side is 86 meters long. A boy swam from A directly toward B. After covering a distance of x meters, he turned and swam westward, reaching the shore after covering a distance of...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 3

IMO 1997 SL 3 Origin: GER Problem For each ﬁnite set U of nonzero vectors in the plane we deﬁne l(U) to be the length of the vector that is the sum of all vectors in U. Given a ﬁnite set V of nonzero vectors in the plane, a subset B of V is said to be maximal if l(B) is greater than or equal to l(A) for each nonempty...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 5

IMO 1974 SL 5 Problem I 5 (GBR 3) Let Ar, Br, Cr be points on the circumference of a given circle S. From the triangle ArBrCr, called \triangler, the triangle \triangler+1 is ob- tained by constructing the points Ar+1, Br+1, Cr+1 on S such that Ar+1Ar is parallel to BrCr, Br+1Br is parallel to CrAr, and Cr+1Cr is parallel to ArBr. Each angle of \triangle1 is an integer number...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 22

IMO 1989 SL 22 Origin: PHI Problem Prove that the set {1, 2, . . ., 1989} can be expressed as the disjoint union of 17 subsets A1, A2, . . . , A17 such that: (i) each Ai contains the same number of elements; (ii) the sum of all elements of each Ai is the same for i = 1, 2, . . . , 17. Solution The statement...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 20

IMO 1986 SL 20 Origin: CAN Problem Prove that the sum of the face angles at each vertex of a tetra- hedron is a straight angle if and only if the faces are congruent triangles. Solution If the faces of the tetrahedron ABCD are congruent triangles, we must have AB = CD, AC = BD, and AD = BC. Then the sum of angles at A is \angleBAC + \angleCAD...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C5

IMO 2002 SL C5 Origin: BRA | Category: Combinatorics Problem Let r \geq2 be a ﬁxed positive integer, and let F be an inﬁnite family of sets, each of size r, no two of which are disjoint. Prove that there exists a set of size r −1 that meets each set in F. Solution Assume to the contrary that no set of size less than r meets all sets in...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 9

IMO 1971 SL 9 Origin: POL Problem Let Tk = k −1 for k = 1, 2, 3, 4 and T2k−1 = T2k−2 + 2k−2, T2k = T2k−5 + 2k (k \geq3). Show that for all k, 1 + T2n−1 = 7 2n−1 and 1 + T2n = 7 2n−1 , where [x] denotes the greatest integer not exceeding x. Solution We use induction. Since T1 = 0, T2 =...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 13

IMO 1978 SL 13 Origin: USA Problem Given any point P in the interior of a sphere with ra- dius R, three mutually perpendicular segments PA, PB, PC are drawn terminating on the sphere and having one common vertex in P. Con- sider the rectangular parallelepiped of which PA, PB, PC are coterminal edges. Find the locus of the point Q that is diagonally opposite P in the parallelepiped when...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 21

IMO 1991 SL 21 Origin: HKG Problem Let f(x) be a monic polynomial of degree 1991 with integer coeﬃcients. Deﬁne g(x) = f 2(x) −9. Show that the number of distinct integer solutions of g(x) = 0 cannot exceed 1995. Solution The polynomial g(x) factorizes as g(x) = f(x)2 −9 = (f(x)−3)(f(x)+3). If one of the equations f(x) + 3 = 0 and f(x) −3 = 0 has no integer...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 3

IMO 1978 SL 3 Origin: CUB Problem Let n > m \geq1 be natural numbers such that the groups of the last three digits in the decimal representation of 1978m, 1978n coincide. Find the ordered pair (m, n) of such m, n for which m + n is minimal. Solution What we need are m, n for which 1978m(1978n−m −1) is divisible by 1000 = 8 \cdot125. Since 1978n−m −1...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 20

IMO 1988 SL 20 Origin: MON Problem Find the least natural number n such that if the set {1, 2, . . ., n} is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third. Solution Suppose that An = {1, 2, . . ., n} is partitioned into Bn and Cn, and that...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 4

IMO 1984 SL 4 Origin: MON Problem Let d be the sum of the lengths of all diagonals of a convex polygon of n (n > 3) vertices and let p be its perimeter. Prove that n −3 < d p < 1 n n + 1 −2 . Solution Consider the convex n-gon A1A2 . . . An (the indices are considered modulo n). For any diagonal...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A2

IMO 2001 SL A2 Origin: POL | Category: Algebra Problem Let a0, a1, a2, . . . be an arbitrary inﬁnite sequence of positive numbers. Show that the inequality 1 + an > an−1 n\sqrt 2 holds for inﬁnitely many positive integers n. Solution It follows from Bernoulli’s inequality that for each n \inN, 1 + 1 n n \geq2, or n\sqrt 2 \leq1 + 1 n. Consequently, it...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A5

IMO 2002 SL A5 Origin: IND | Category: Algebra Problem Let n be a positive integer that is not a perfect cube. Deﬁne real numbers a, b, c by a = 3\sqrtn, b = a −[a], c = b −[b], where [x] denotes the integer part of x. Prove that there are inﬁnitely many such integers n with the property that there exist integers r, s, t, not all zero,...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 11

IMO 1975 SL 11 Origin: GBR Problem Let a1, a2, a3, . . . be any inﬁnite increasing sequence of pos- itive integers. (For every integer i > 0, ai+1 > ai.) Prove that there are inﬁnitely many m for which positive integers x, y, h, k can be found such that 0 < h < k < m and am = xah + yak. Solution Let (aki) be the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 8

IMO 1998 SL 8 Origin: IND Problem Let ABC be a triangle such that \angleA = 90◦and \angleB < \angleC. The tangent at A to its circumcircle \omega meets the line BC at D. Let E be the reﬂection of A across BC, X the foot of the perpendicular from A to BE, and Y the midpoint of AX. Let the line BY meet \omega again at Z. Prove that...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C6

IMO 2001 SL C6 Origin: CAN | Category: Combinatorics Problem For a positive integer n deﬁne a sequence of zeros and ones to be balanced if it contains n zeros and n ones. Two balanced sequences a and b are neighbors if you can move one of the 2n symbols of a to another position to form b. For instance, when n = 4, the balanced sequences 01101001 and 00110101...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 4

IMO 1997 SL 4 Origin: IRN Problem An n \times n matrix with entries from {1, 2, . . ., 2n −1} is called a coveralls matrix if for each i the union of the ith row and the ith column contains 2n −1 distinct entries. Show that: (a) There exist no coveralls matrices for n = 1997. (b) Coveralls matrices exist for inﬁnitely many values of n. Solution (a)...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 20

IMO 1987 SL 20 Origin: USS Problem Let f(x) = x2 + x + p, p \inN. Prove that if the numbers f(0), f(1), . . . , f([ p/3 ]) are primes, then all the numbers f(0), f(1), . . . , f(p −2) are primes. Solution Let y be the smallest nonnegative integer with y \leqp −2 for which f(y) is a composite number. Denote by q the...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 5

IMO 1990 SL 5 Origin: FRA Problem Given \triangleABC with no side equal to another side, let G, K, and H be its centroid, incenter, and orthocenter, respectively. Prove that \angleGKH > 90◦. Solution Let O be the circumcenter of ABC, E the midpoint of OH, and R and r the radii of the circumcircle and incircle respectively. We use the following facts from elementary geometry: −−\to OH = 3−−\to...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 1

IMO 1989 SL 1 Origin: AUS Problem Let ABC be a triangle. The bisector of angle A meets the circumcircle of triangle ABC in A1. Points B1 and C1 are deﬁned similarly. Let AA1 meet the lines that bisect the two external angles at B and C in point A0. Deﬁne B0 and C0 similarly. If SX1X2...Xn denotes the area of the polygon X1X2 . . . Xn, prove that...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 10

IMO 1986 SL 10 Origin: HUN Problem Three persons A, B, C, are playing the following game: A k- element subset of the set {1, . . . , 1986} is randomly chosen, with an equal probability of each choice, where k is a ﬁxed positive integer less than or equal to 1986. The winner is A, B or C, respectively, if the sum of the chosen numbers leaves a...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C1

IMO 2001 SL C1 Origin: COL | Category: Combinatorics Problem Let A = (a1, a2, . . . , a2001) be a sequence of positive integers. Let m be the number of 3-element subsequences (ai, aj, ak) with 1 \leqi < j < k \leq2001 such that aj = ai + 1 and ak = aj + 1. Considering all such sequences A, ﬁnd the greatest value of m. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N6

IMO 1995 SL N6 Origin: POL | Category: Number Theory Problem Let p be an odd prime. Find the number of p-element subsets A of {1, 2, . . ., 2p} such that the sum of all elements of A is divisible by p. Solution We shall consider the set M = {0, 1, . . ., 2p −1} instead. Let M1 = {0, 1, . . ., p −1}...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C3

IMO 2003 SL C3 Origin: LIT | Category: Combinatorics Problem Let n \geq5 be a given integer. Determine the largest integer k for which there exists a polygon with n vertices (convex or not, with non-self-intersecting boundary) having k internal right angles. Solution Suppose that k of the angles of an n-gon are right. Since the other n −k angles are less than 360◦and the sum of the angles is...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 8

IMO 1970 SL 8 Origin: POL Problem Given a point M on the side AB of the triangle ABC, let r1 and r2 be the radii of the inscribed circles of the triangles ACM and BCM respectively and let \rho1 and \rho2 be the radii of the excircles of the triangles ACM and BCM at the sides AM and BM respectively. Let r and \rho denote the radii of the...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N7

IMO 2004 SL N7 Origin: BUL | Category: Number Theory Problem Let p be an odd prime and n a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length pn. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by pn+1. Solution We...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 5

IMO 1983 SL 5 Origin: BRA Problem Consider the set of all strictly decreasing sequences of n natural numbers having the property that in each sequence no term divides any other term of the sequence. Let A = (aj) and B = (bj) be any two such sequences. We say that A precedes B if for some k, ak < bk and ai = bi for i < k. Find...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 6

IMO 1992 SL 6 Origin: IND Problem Find all functions f : R \toR such that f(x2 + f(y)) = y + f(x)2 for all x, y in R. Solution It is easy to see that f is injective and surjective. From f(x2 + f(y)) = f((−x)2 + f(y)) it follows that f(x)2 = (f(−x))2, which implies f(−x) = −f(x) because f is injective. Furthermore, there exists z \inR such...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 13

IMO 1997 SL 13 Origin: IND Problem In town A, there are n girls and n boys, and each girl knows each boy. In town B, there are n girls g1, g2, . . . , gn and 2n −1 boys b1, b2, . . ., b2n−1. The girl gi, i = 1, 2, . . ., n, knows the boys b1, b2, . . . , b2i−1, and no...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 22

IMO 1998 SL 22 Origin: UKR Problem A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number x in the array can be changed into either \lceilx\rceilor \lfloorx\rfloorso that the row sums and column sums remain unchanged. (Note that \lceilx\rceilis the least integer greater than or equal to x, while \lfloorx\rflooris the greatest integer...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 7

IMO 1998 SL 7 Origin: GBR Problem Let ABC be a triangle such that \angleACB = 2\angleABC. Let D be the point on the side BC such that CD = 2BD. The segment AD is extended to E so that AD = DE. Prove that \angleECB + 180◦= 2\angleEBC. Solution We shall use the following result. Lemma. In a triangle ABC with BC = a, CA = b, and AB...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 28

IMO 1990 SL 28 Origin: USS Problem Prove that on the coordinate plane it is impossible to draw a closed broken line such that (i) the coordinates of each vertex are rational; (ii) the length each of its edges is 1; (iii) the line has an odd number of vertices. Solution Let us ﬁrst prove the following lemma. Lemma. Let (b′/a′, d′/c′) and (b′′/a′′, d′′/c′′) be two points with rational...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 9

IMO 1984 SL 9 Origin: POL Problem Let a, b, c be positive numbers with \sqrta+ \sqrt b+\sqrtc = \sqrt 2 . Prove that the system of equations \sqrty −a + \sqrtz −a = 1, \sqrt z −b + \sqrt x −b = 1, \sqrtx −c + \sqrty −c = 1, has exactly one solution (x, y, z) in real numbers. Solution Let us show ﬁrst that the system has...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 3

IMO 1982 SL 3 Problem A3 (USS 4)IMO3 Consider the inﬁnite sequences {xn} of positive real numbers with the following properties: x0 = 1 and for all i \geq0, xi+1 \leqxi. (a) Prove that for every such sequence there is an n \geq1 such that x2 x1 + x2 x2 + \cdot \cdot \cdot + x2 n−1 xn \geq3.999. (b) Find such a sequence for which x2 x1 + x2...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 24

IMO 1990 SL 24 Origin: THA Problem Let a, b, c, d be nonnegative real numbers such that ab + bc + cd + da = 1. Show that a3 b + c + d + b3 a + c + d + c3 a + b + d + d3 a + b + c \geq1 3. Solution Let us denote A = b + c + d, B...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 6

IMO 1989 SL 6 Origin: CZS Problem For a triangle ABC, let k be its circumcircle with radius r. The bisectors of the inner angles A, B, and C of the triangle intersect respec- tively the circle k again at points A′, B′, and C′. Prove the inequality 16Q3 \geq27r4P, where Q and P are the areas of the triangles A′B′C′ and ABC respec- tively. Solution Let us denote the...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A5

IMO 2000 SL A5 Origin: BLR | Category: Algebra Problem Let n \geq2 be a positive integer and \lambda a positive real number. Initially there are n ﬂeas on a horizontal line, not all at the same point. We deﬁne a move of choosing two ﬂeas at some points A and B, with A to the left of B, and letting the ﬂea from A jump over the ﬂea from...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 5

IMO 1981 SL 5 Origin: COL Problem A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer. Solution There are four types of small cubes upon disassembling: (1) 8 cubes with three...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 8

IMO 1987 SL 8 Origin: HUN Problem (a) Let (m, k) = 1. Prove that there exist integers a1, a2, . . . , am and b1, b2, . . . , bk such that each product aibj (i = 1, 2, . . ., m; j = 1, 2, . . . , k) gives a diﬀerent residue when divided by mk. (b) Let (m, k) > 1. Prove...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 5

IMO 1984 SL 5 Origin: FRG Problem Let x, y, z be nonnegative real numbers with x+y +z = 1. Show that 0 \leqxy + yz + zx −2xyz \leq7 27. Solution Let f(x, y, z) = xy + yz + zx −2xyz. The ﬁrst inequality follows imme- diately by adding xy \geqxyz, yz \geqxyz, and zx \geqxyz (in fact, a stronger inequality xy + yz + zx −9xyz \geq0...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 9

IMO 1982 SL 9 Problem B3 (GBR 1) Let ABC be a triangle, and let P be a point inside it such that ∡PAC = ∡PBC. The perpendiculars from P to BC and CA meet these lines at L and M, respectively, and D is the midpoint of AB. Prove that DL = DM. Solution Let X and Y be the midpoints of the segments AP and BP. Then DY...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 6

IMO 1990 SL 6 Origin: FRG Problem Two players A and B play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number n0 > 1 is given. Knowing n2k, player A may choose any n2k+1 \inN such that n2k \leqn2k+1 \leqn2 2k. Then player B chooses a number n2k+2 \inN such that n2k+1 n2k+2 = pr, where p is...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 1

IMO 1998 SL 1 Origin: LUX Problem A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD. Prove that ABCD is cyclic if and only if triangles ABP and CDP have equal areas. Solution We begin with the following observation: Suppose that P lies in \triangleAEB, where E is the intersection of AC and BD (the other cases...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 13

IMO 1989 SL 13 Origin: ICE Problem The quadrilateral ABCD has the following properties: (i) AB = AD + BC; (ii) there is a point P inside it at a distance x from the side CD such that AP = x + AD and BP = x + BC. Show that \sqrtx \geq \sqrt AD + \sqrt BC . Solution Let us construct the circles \sigma1 with center A and...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 15

IMO 1991 SL 15 Origin: USS Problem Let an be the last nonzero digit in the decimal representation of the number n!. Does the sequence a1, a2, . . . , an, . . . become periodic after a ﬁnite number of terms? Solution Assume that the sequence has the period T . We can ﬁnd integers k > m > 0, as large as we like, such that 10k...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 5

IMO 1978 SL 5 Origin: GDR Problem For every integer d \geq1, let Md be the set of all positive integers that cannot be written as a sum of an arithmetic progression with diﬀerence d, having at least two terms and consisting of positive integers. Let A = M1, B = M2 ∖{2}, C = M3. Prove that every c \inC may be written in a unique way as c...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 11

IMO 1985 SL 11 Problem 3a.(USS 3) Find a method by which one can compute the coeﬃcients of P(x) = x6 + a1x5 + \cdot \cdot \cdot + a6 from the roots of P(x) = 0 by performing not more than 15 additions and 15 multiplications. Solution Let −x1, . . . , −x6 be the roots of the polynomial. Let sk,i (k \leqi \leq6) denote the sum of all...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 2

IMO 1972 SL 2 Origin: CZS Problem We are given 3n points A1, A2, . . . , A3n in the plane, no three of them collinear. Prove that one can construct n disjoint triangles with vertices at the points Ai. Solution We use induction. For n = 1 the assertion is obvious. Assume that it is true for a positive integer n. Let A1, A2, . . . ,...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 30

IMO 1988 SL 30 Origin: USS Problem A point M is chosen on the side AC of the triangle ABC in such a way that the radii of the circles inscribed in the triangles ABM and BMC are equal. Prove that BM 2 = ∆cot B 2 , where ∆is the area of the triangle ABC. Solution Let ∆1, s1, r′ denote the area, semiperimeter, and inradius of triangle ABM,...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 9

IMO 1978 SL 9 Origin: GBR Problem Let {f(n)} be a strictly increasing sequence of positive integers: 0 < f(1) < f(2) < f(3) < \cdot \cdot \cdot . Of the positive integers not belonging to the sequence, the nth in order of magnitude is f(f(n)) + 1. Determine f(240). Solution Since the nth missing number (gap) is f(f(n))+1 and f(f(n)) is a member of the sequence, there are exactly...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 10

IMO 1975 SL 10 Origin: GBR Problem The function f(x, y) is a homogeneous polynomial of the nth degree in x and y. If f(1, 0) = 1 and for all a, b, c, f(a + b, c) + f(b + c, a) + f(c + a, b) = 0, prove that f(x, y) = (x −2y)(x + y)n−1. Solution We shall prove that for all x, y with x+y...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 24

IMO 1983 SL 24 Origin: USS Problem Let dn be the last nonzero digit of the decimal representation of n!. Prove that dn is aperiodic; that is, there do not exist T and n0 such that for all n \geqn0, dn+T = dn. Solution See the solution of (SL91-15).

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 9

IMO 1985 SL 9 Problem 2a.(USA 3) Determine the radius of a sphere S that passes through the centroids of each face of a given tetrahedron T inscribed in a unit sphere with center O. Also, determine the distance from O to the center of S as a function of the edges of T . Solution Let −\toa , −\tob , −\toc , −\tod denote the vectors −\to OA, −−\to...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G2

IMO 2000 SL G2 Origin: RUS | Category: Geometry Problem Two circles G1 and G2 intersect at M and N. Let AB be the line tangent to these circles at A and B, respectively, such that M lies closer to AB than N. Let CD be the line parallel to AB and passing through M, with C on G1 and D on G2. Lines AC and BD meet at E;...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 8

IMO 1976 SL 8 Origin: SWE Problem Let P be a polynomial with real coeﬃcients such that P(x) > 0 if x > 0. Prove that there exist polynomials Q and R with nonnegative coeﬃcients such that P(x) = Q(x) R(x) if x > 0. Solution Every polynomial with real coeﬃcients can be factored as a product of linear and quadratic polynomials with real coeﬃcients. Thus it suﬃces to prove...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 2

IMO 1970 SL 2 Origin: ROM Problem Let a and b be the bases of two number systems and let An = x1x2 . . . xn (a), An+1 = x0x1x2 . . . xn (a), Bn = x1x2 . . . xn(b), Bn+1 = x0x1x2 . . . xn(b), be numbers in the number systems with respective bases a and b, so that x0, x1, x2, . . ....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 32

IMO 1989 SL 32 Origin: USA Problem The vertex A of the acute triangle ABC is equidistant from the circumcenter O and the orthocenter H. Determine all possible values for the measure of angle A. Solution Let CC′ be an altitude, and R the circumradius. Then, since AH = R, we have AC′ = |R sin B| and hence (1) CC′ = |R sin B tan A|. On the other...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 2

IMO 1973 SL 2 Origin: CZS Problem Given a circle K, ﬁnd the locus of vertices A of parallelograms ABCD with diagonals AC \leqBD, such that BD is inside K. Solution Let D′ be the reﬂection of D across A. Since BCAD′ is then a parallel- ogram, the condition BD \geqAC is equivalent to BD \geqBD′, which is in turn equivalent to \angleBAD \geq\angleBAD′, i.e. to \angleBAD \geq90◦. Thus the...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A4

IMO 2004 SL A4 Origin: KOR | Category: Algebra Problem Find all polynomials P(x) with real coeﬃcients that satisfy the equality P(a −b) + P(b −c) + P(c −a) = 2P(a + b + c) for all triples a, b, c of real numbers such that ab + bc + ca = 0. Solution Let P(x) = a0 + a1x + \cdot \cdot \cdot + anxn. For every x \inR...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G7

IMO 2001 SL G7 Origin: BUL | Category: Geometry Problem Let O be an interior point of acute triangle ABC. Let A1 lie on BC with OA1 perpendicular to BC. Deﬁne B1 on CA and C1 on AB similarly. Prove that O is the circumcenter of ABC if and only if the perimeter of A1B1C1 is not less than any one of the perimeters of AB1C1, BC1A1, and CA1B1. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 19

IMO 1987 SL 19 Origin: USS Problem Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta + \gamma < \pi, \alpha + \beta > \gamma, \beta + \gamma > \alpha, \gamma + \alpha > \beta. Prove that with the segments of lengths sin \alpha, sin \beta, sin \gamma we can construct a triangle and that its area is not greater than 8(sin 2\alpha + sin 2\beta...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 5

IMO 1982 SL 5 Problem A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its diagonals Ai−1Ai+1 is divided into the same ratio \lambda 1−\lambda, where 0 < \lambda < 1, by a point Bi in such a way that Ai, Bi, and Bi+2 are collinear (i \equiv 1, . . . , 6 (mod 6)). Compute \lambda. Solution We ﬁrst observe that \triangleA5B4A4 ∼= \triangleA3B2A2. Since \angleA5A3A2...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 12

IMO 1974 SL 12 Problem II 6 (USS 1) In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of diﬀerent lengths. Prove that one can form a word of arbitrary length that does not contain any nonallowed word. Solution We say that a word is good if it doesn’t...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 2

IMO 1991 SL 2 Origin: JAP Problem For an acute triangle ABC, M is the midpoint of the segment BC, P is a point on the segment AM such that PM = BM, H is the foot of the perpendicular line from P to BC, Q is the point of intersection of segment AB and the line passing through H that is perpendicular to PB, and ﬁnally, R is the...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 10

IMO 1987 SL 10 Origin: ICE Problem Let S1 and S2 be two spheres with distinct radii that touch externally. The spheres lie inside a cone C, and each sphere touches the cone in a full circle. Inside the cone there are n additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone C, both of the spheres S1 and S2 externally,...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G5

IMO 2002 SL G5 Origin: AUS | Category: Geometry Problem For any set S of ﬁve points in the plane, no three of which are collinear, let M(S) and m(S) denote the greatest and smallest areas, respectively, of triangles determined by three points from S. What is the minimum possible value of M(S)/m(S)? Solution When S is the set of vertices of a regular pentagon, then it is easily veriﬁed...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 10

IMO 1976 SL 10 Origin: USA Problem Find the largest number obtainable as the product of pos- itive integers whose sum is 1976. Solution Let a1 < a2 < \cdot \cdot \cdot < an be positive integers whose sum is 1976. Let M denote the maximal value of a1a2 \cdot \cdot \cdot an. We make the following observa- tions: (1) a1 = 1 does not yield the maximum, since replacing...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 9

IMO 1993 SL 9 Origin: IND Problem (a) Show that the set Q+ of all positive rational numbers can be par- titioned into three disjoint subsets A, B, C satisfying the following conditions: BA = B, B2 = C, BC = A, where HK stands for the set {hk | h \inH, k \inK} for any two subsets H, K of Q+ and H2 stands for HH. (b) Show that...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 27

IMO 1991 SL 27 Origin: POL Problem Determine the maximum value of the sum i<j xixj(xi + xj) over all n-tuples (x1, . . . , xn), satisfying xi \geq0 and n i=1 xi = 1. Solution Write F(x1, . . . , xn) = i<j xixj(xi+xj). Choose an n-tuple (x1, . . . , xn), n i=1 xi = 1, xi \geq0 with at least three nonzero...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 13

IMO 1984 SL 13 Origin: BUL Problem Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume 1 does not exceed 3\pi. Solution Let Z be the given cylinder of radius r, altitude h, and volume \pir2h = 1, k1 and k2 the circles surrounding its bases, and V the volume of an inscribed tetrahedron ABCD. We claim that there is no loss of generality...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 4

IMO 1990 SL 4 Origin: CZS Problem Assume that the set of all positive integers is decomposed into r (disjoint) subsets A1 \cupA2 \cup\cdot \cdot \cdot Ar = N. Prove that one of them, say Ai, has the following property: There exists a positive m such that for any k one can ﬁnd numbers a1, a2, . . . , ak in Ai with 0 < aj+1 −aj \leqm (1...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 14

IMO 1998 SL 14 Origin: GBR Problem Determine all pairs (x, y) of positive integers such that x2y+ x + y is divisible by xy2 + y + 7. Solution If x2y + x + y is divisible by xy2 + y + 7, then so is the number y(x2y + x + y) −x(xy2 + y + 7) = y2 −7x. If y2 −7x \geq0, then since y2 −7x...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 19

IMO 1991 SL 19 Origin: IRE Problem Let a be a rational number with 0 < a < 1 and suppose that cos 3\pia + 2 cos2\pia = 0. (Angle measurements are in radians.) Prove that a = 2/3. Solution Set x = cos(\pia). The given equation is equivalent to 4x3+4x2−3x−2 = 0, which factorizes as (2x + 1)(2x2 + x −2) = 0. The case 2x + 1 =...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 16

IMO 1992 SL 16 Origin: KOR Problem Prove that N = 5125−1 525−1 is a composite number. Solution Observe that x4 + x3 + x2 + x + 1 = (x2 + 3x + 1)2 −5x(x + 1)2. Thus for x = 525 we have N = x4 + x3 + x2 + x + 1 = (x2 + 3x + 1 −513(x + 1))(x2 + 3x + 1 +...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 2

IMO 1986 SL 2 Origin: SWE Problem Let f(x) = xn where n is a ﬁxed positive integer and x = 1, 2, . . . . Is the decimal expansion a = 0.f(1)f(2)f(3) . . . rational for any value of n? The decimal expansion of a is deﬁned as follows: If f(x) = d1(x)d2(x) . . . . . . dr(x)(x) is the decimal expansion of f(x), then...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 7

IMO 1983 SL 7 Origin: CAN Problem Let a be a positive integer and let {an} be deﬁned by a0 = 0 and an+1 = (an + 1)a + (a + 1)an + 2 a(a + 1)an(an + 1) (n = 1, 2 . . .). Show that for each positive integer n, an is a positive integer. Solution Clearly, each an is positive and \sqrtan+1 = \sqrtan \sqrta +...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 2

IMO 1987 SL 2 Origin: USA Problem At a party attended by n married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques C1, C2, . . . , Ck with the following prop- erty: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A4

IMO 2003 SL A4 Origin: IRE | Category: Algebra Problem Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be real numbers. (a) Prove that ⎛ ⎝ n i,j=1 |xi −xj| ⎞ ⎠ \leq2(n2 −1) n i,j=1 (xi −xj)2. (b) Show that equality holds if and only if x1, . . . , xn is an arithmetic progession. Solution By the Cauchy–Schwarz inequality...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 12

IMO 1977 SL 12 Origin: NET Problem On the sides of a square ABCD one constructs inwardly equilateral triangles ABK, BCL, CDM, DAN. Prove that the midpoints of the four segments KL, LM, MN, NK, together with the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN, are the 12 vertices of a regular dodecagon.

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 24

IMO 1989 SL 24 Origin: POL Problem For points A1, . . . , A5 on the sphere of radius 1, what is the maximum value that min1\leqi,j\leq5 AiAj can take? Determine all conﬁgura- tions for which this maximum is attained. (Or: determine the diameter of any set {A1, . . . , A5} for which this maximum is attained.) Solution Instead of Euclidean distance, we will use the angles...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 30

IMO 1991 SL 30 Origin: BUL Problem Two students A and B are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student A: “Can you tell the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A6

IMO 1995 SL A6 Origin: JAP | Category: Algebra Problem Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers such that xi < xi+1 for 1 \leqi \leqn −1. Prove that n(n −1) i n−1 i=1 (n −i)xi ⎛ ⎝ n j=2 (j −1)xj ⎞ ⎠. Solution Let yi = xi+1 + \cdot \cdot \cdot + xn, Y...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 12

IMO 1983 SL 12 Origin: GBR Problem Find all functions f deﬁned on the positive real numbers and taking positive real values that satisfy the following conditions: (i) f(xf(y)) = yf(x) for all positive real x, y. (ii) f(x) \to0 as x \to+\infty. Solution Putting y = x in (1) we see that there exist positive real numbers z such that f(z) = z (this is true for every z...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 7

IMO 1987 SL 7 Origin: NET Problem Given ﬁve real numbers u0, u1, u2, u3, u4, prove that it is always possible to ﬁnd ﬁve real numbers v0, v1, v2, v3, v4 that satisfy the following conditions: (i) ui −vi \inN. (ii) 0\leqi<j\leq4(vi −vj)2 < 4. Solution For all real numbers v the following inequality holds: 0\leqi<j\leq4 (vi −vj)2 \leq5 i=0 (vi −v)2. (1) Indeed, 0\leqi<j\leq4...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 7

IMO 1973 SL 7 Origin: POL Problem Given a tetrahedron ABCD, let x = AB \cdot CD, y = AC \cdot BD, and z = AD \cdot BC. Prove that there exists a triangle with edges x, y, z. Solution The result follows immediately from Ptolemy’s inequality.

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 1

IMO 1985 SL 1 Origin: MON Problem Given a set M of 1985 positive integers, none of which has a prime divisor larger than 26, prove that the set has four distinct elements whose geometric mean is an integer. Solution Since there are 9 primes (p1 = 2 < p2 = 3 < \cdot \cdot \cdot < p9 = 23) less than 26, each number xj \inM is of the...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 1

IMO 1979 SL 1 Origin: BEL Problem Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length). Solution We prove more generally, by induction on n, that any 2n-gon with equal edges and opposite edges parallel to each other can be dissected. For n = 2 the...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N7

IMO 1994 SL N7 Origin: GBR | Category: Number Theory Problem A wobbly number is a positive integer whose digits in base 10 are alternately nonzero and zero, the units digit being nonzero. Determine all positive integers that do not divide any wobbly number. Solution A multiple of 10 does not divide any wobbly number. Also, if 25 | n, then every multiple of n ends with 25, 50, 75,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G5

IMO 1994 SL G5 Origin: CYP | Category: Geometry Problem A line l does not meet a circle \omega with center O. E is the point on l such that OE is perpendicular to l. M is any point on l other than E. The tangents from M to \omega touch it at A and B. C is the point on MA such that EC is perpendicular to MA. D...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 17

IMO 1984 SL 17 Origin: FRG Problem In a permutation (x1, x2, . . . , xn) of the set 1, 2, . . . , n we call a pair (xi, xj) discordant if i < j and xi > xj. Let d(n, k) be the number of such permutations with exactly k discordant pairs. Find d(n, 2) and d(n, 3). Solution For any m = 0, 1, ....

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 14

IMO 1984 SL 14 Origin: ROM Problem Let ABCD be a convex quadrilateral for which the circle with diameter AB is tangent to the line CD. Show that the circle with diameter CD is tangent to the line AB if and only if the lines BC and AD are parallel. Solution Let M and N be the midpoints of AB and CD, and let M ′, N ′ be their...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 13

IMO 1968 SL 13 Origin: POL Problem Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ $(A_iA_k = B_iB_k)$, prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented. Solution Translating one of the triangles if necessary, we may assume w.l.o.g. that $B_1 \equiv A_1$. We also assume that $B_2 \not\equiv A_2$ and $B_3 \not\equiv A_3$, since the result is obvious...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C5

IMO 2004 SL C5 Origin: NZL | Category: Combinatorics Problem Let N be a positive integer. Two players A and B, taking turns, write numbers from the set {1, . . . , N} on a blackboard. A begins the game by writing 1 on his ﬁrst move. Then, if a player has written n on a certain move, his adversary is allowed to write n+1 or 2n (provided the...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C3

IMO 2000 SL C3 Origin: COL | Category: Combinatorics Problem Let n \geq4 be a ﬁxed positive integer. Given a set S = {P1, P2, . . . , Pn} of points in the plane such that no three are collinear and no four concyclic, let at, 1 \leqt \leqn, be the number of circles PiPjPk that contain Pt in their interior, and let m(S) = a1 + a2 +...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 10

IMO 1997 SL 10 Origin: CZE Problem Find all positive integers k for which the following statement is true: If F(x) is a polynomial with integer coeﬃcients satisfying the condition 0 \leqF(c) \leqk for each c \in{0, 1, . . ., k + 1}, then F(0) = F(1) = \cdot \cdot \cdot = F(k + 1). Solution Suppose that k \geq4. Consider any polynomial F(x) with integer coeﬃ- cients such...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A3

IMO 1995 SL A3 Origin: UKR | Category: Algebra Problem Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers such that 2 \leqai \leq3 for i = 1, 2, . . ., n. If s = a1 + a2 + \cdot \cdot \cdot + an, prove that a2 1 + a2 2 −a2 a1 + a2 −a3 a2 2 + a2...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 1

IMO 1993 SL 1 Origin: BRA Problem Show that there exists a ﬁnite set A \subsetR2 such that for every X \inA there are points Y1, Y2, . . . , Y1993 in A such that the distance between X and Yi is equal to 1, for every i. Solution First we notice that for a rational point O (i.e., with rational coordinates), there exist 1993 rational points in each...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 10

IMO 1970 SL 10 Origin: SWE Problem Let 1 = a0 \leqa1 \leqa2 \leq\cdot \cdot \cdot \leqan \leq\cdot \cdot \cdot be a sequence of real numbers. Consider the sequence b1, b2, . . . deﬁned by: bn = n k=1 1 −ak−1 ak \sqrtak . Prove that: (a) For all natural numbers n, 0 \leqbn < 2. (b) Given an arbitrary 0 \leqb < 2, there is...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 11

IMO 1981 SL 11 Origin: NET Problem On a semicircle with unit radius four consecutive chords AB, BC, CD, DE with lengths a, b, c, d, respectively, are given. Prove that a2 + b2 + c2 + d2 + abc + bcd < 4. Solution Let us denote the center of the semicircle by O, and \angleAOB = 2\alpha, \angleBOC = 2\beta, AC = m, CE = n. We claim...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 26

IMO 1979 SL 26 Origin: YUG Problem Prove that the functional equations f(x + y) = f(x) + f(y), and f(x + y + xy) = f(x) + f(y) + f(xy) (x, y \inR) are equivalent. Solution Let us assume that f(x + y) = f(x) + f(y) for all reals. In this case we trivially apply the equation to get f(x + y + xy) = f(x + y)...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C3

IMO 2002 SL C3 Origin: COL | Category: Combinatorics Problem Let n be a positive integer. A sequence of n positive integers (not necessarily distinct) is called full if it satisﬁes the following condition: For each positive integer k \geq2, if the number k appears in the sequence, then so does the number k −1, and moreover, the ﬁrst occurrence of k −1 comes before the last occurrence of k....

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 18

IMO 1981 SL 18 Origin: USS Problem Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet. Solution Let C be the convex hull of the set of...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G7

IMO 2000 SL G7 Origin: IRN | Category: Geometry Problem Ten gangsters are standing on a ﬂat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed? Solution The problem can be reformulated in the following way: Given...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 10

IMO 1991 SL 10 Origin: USA Problem Suppose G is a connected graph with n edges. Prove that it is possible to label the edges of G from 1 to n in such a way that in every vertex v of G with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1. Solution We start at some vertex v0 and...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 5

IMO 1992 SL 5 Origin: COL Problem Let ABCD be a convex quadrilateral such that AC = BD. Equilateral triangles are constructed on the sides of the quadrilat- eral. Let O1, O2, O3, O4 be the centers of the triangles constructed on AB, BC, CD, DA respectively. Show that O1O3 is perpendicular to O2O4. Solution Denote by K, L, M, and N the midpoints of the sides AB, BC, CD,...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G4

IMO 1995 SL G4 Origin: UKR | Category: Geometry Problem An acute triangle ABC is given. Points A1 and A2 are taken on the side BC (with A2 between A1 and C), B1 and B2 on the side AC (with B2 between B1 and A), and C1 and C2 on the side AB (with C2 between C1 and B) such that \angleAA1A2 = \angleAA2A1 = \angleBB1B2 = \angleBB2B1 = \angleCC1C2...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N2

IMO 2001 SL N2 Origin: COL | Category: Number Theory Problem Consider the system x + y = z + u, 2xy = zu. Find the greatest value of the real constant m such that m \leqx/y for every positive integer solution x, y, z, u of the system with x \geqy. Solution We shall ﬁnd the general solution to the system. Squaring both sides of the ﬁrst equation and...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A1

IMO 2004 SL A1 Origin: KOR | Category: Algebra Problem Let n \geq3 be an integer and t1, t2, . . . , tn positive real numbers such that n2 + 1 > (t1 + t2 + \cdot \cdot \cdot + tn) 1 t1 1 t2 \cdot \cdot \cdot + 1 tn . Show that ti, tj, tk are the side lengths of a triangle for all i,...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 6

IMO 1971 SL 6 Origin: HUN Problem Let n \geq2 be a natural number. Find a way to assign nat- ural numbers to the vertices of a regular 2n-gon such that the following conditions are satisﬁed: (1) only digits 1 and 2 are used; (2) each number consists of exactly n digits; (3) diﬀerent numbers are assigned to diﬀerent vertices; (4) the numbers assigned to two neighboring vertices diﬀer at...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G1

IMO 1996 SL G1 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A$, $B$, $C$. Let $E$ be the foot of the altitude from $B$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet $HR$ in $X$. Prove that $EX\parallel AP$. Solution We first show that $H$ is the common orthocenter of the triangles...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C2

IMO 2002 SL C2 Origin: ARM | Category: Combinatorics Problem For n an odd positive integer, the unit squares of an n \times n chessboard are colored alternately black and white, with the four corners colored black. A tromino is an L-shape formed by three connected unit squares. For which values of n is it possible to cover all the black squares with nonoverlapping trominos? When it is possible, what...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 4

IMO 1981 SL 4 Origin: CAN Problem Let {fn} be the Fibonacci sequence {1, 1, 2, 3, 5, . . .}. (a) Find all pairs (a, b) of real numbers such that for each n, afn + bfn+1 is a member of the sequence. (b) Find all pairs (u, v) of positive real numbers such that for each n, uf 2 n + vf 2 n+1 is a member of...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 15

IMO 1971 SL 15 Origin: USS Problem Natural numbers from 1 to 99 (not necessarily distinct) are written on 99 cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by 100. Show that all the cards contain the same number. Solution Assume the opposite. Then one can numerate the cards 1 to 99, with a...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 6

IMO 1981 SL 6 Origin: CUB Problem Let P(z) and Q(z) be complex-variable polynomials, with degree not less than 1. Let Pk = {z \inC | P(z) = k}, Qk = {z \inC | Q(z) = k}. Let also P0 = Q0 and P1 = Q1. Prove that P(z) \equivQ(z). Solution Assume w.l.o.g. that n = degP \geqdegQ, and let P0 = {z1, z2, . . . , zk}, P1...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A9

IMO 1996 SL A9 Origin: POL | Category: Algebra Problem Let the sequence $a(n)$, $n = 1, 2, 3, \dots$, be generated as follows: $a(1) = 0$, and for $n > 1$, $$a(n) = a\bigl(\lfloor n/2 \rfloor\bigr) + (-1)^n \frac{n(n+1)}{2}.$$ (Here $\lfloor t \rfloor$ denotes the greatest integer $\leq t$.) (a) Determine the maximum and minimum value of $a(n)$ over $n \leq 1996$ and find all $n \leq 1996$ for...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 1

IMO 1974 SL 1 Problem I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam- inations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are diﬀerent positive integers. The ﬁnal test scores were Alice Betty Carol If Betty placed ﬁrst in the arithmetic examination, who placed second in the spelling examination? Solution...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 19

IMO 1968 SL 19 Origin: ITA Problem We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0$, $1$, $2$, $\ldots$ from it we obtain points with abscissas $n = 0, 1, 2, \ldots$ respectively. How many points among them should we take to ensure that some two of them are less than...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N2

IMO 1999 SL N2 Origin: ARM | Category: Number Theory Problem Prove that every positive rational number can be repre- sented in the form a3 + b3 c3 + d3 , where a, b, c, d are positive integers. Solution We ﬁrst prove that every rational number in the interval (1, 2) can be represented in the form a3+b3 a3+d3 . Taking b, d such that b ̸= d and...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 8

IMO 1990 SL 8 Origin: HUN Problem For a given positive integer k denote the square of the sum of its digits by f1(k) and let fn+1(k) = f1(fn(k)). Determine the value of f1991(21990). Solution Since 21990 < 8700 < 10700, we have f1(21990) < (9\cdot700)2 < 4\cdot107. We then have f2(21990) < (3+9\cdot7)2 < 4900 and ﬁnally f3(21990) < (3+9\cdot3)2 = 302. It is easily shown that fk(n) \equivfk−1(n)2...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G6

IMO 2002 SL G6 Origin: UKR | Category: Geometry Problem Let n \geq3 be a positive integer. Let C1, C2, C3, . . . , Cn be unit circles in the plane, with centers O1, O2, O3, . . . , On respectively. If no line meets more than two of the circles, prove that 1\leqi<j\leqn OiOj \leq(n −1)\pi . Solution Let l(MN) denote the length of the shorter...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N4

IMO 2003 SL N4 Origin: ROM | Category: Number Theory Problem Let b be an integer greater than 5. For each positive integer n, consider the number xn = 11 . . .1 n−1 22 . . .2 n 5, written in base b. Prove that the following condition holds if and only if b = 10: There exists a positive integer M such...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 3

IMO 1991 SL 3 Origin: PRK Problem Let S be any point on the circumscribed circle of \trianglePQR. Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by l(S, PQR). Suppose that the hexagon ABCDEF is inscribed in a circle. Show that the four lines l(A, BDF), l(B, ACE), l(D, ABF), and l(E, ABC) intersect...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N5

IMO 1994 SL N5 Origin: ROM | Category: Number Theory Problem For any positive integer k, Ak is the subset of {k+1, k+ 2, . . . , 2k} consisting of all elements whose digits in base 2 contain exactly three 1’s. Let f(k) denote the number of elements in Ak. (a) Prove that for any positive integer m, f(k) = m has at least one Solution (a) Denote by...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 10

IMO 1982 SL 10 Problem B4 (BRA 1) A box contains p white balls and q black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have diﬀerent colors, the white ball is put back into the box. This procedure is...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G3

IMO 2002 SL G3 Origin: KOR | Category: Geometry Problem The circle S has center O, and BC is a diameter of S. Let A be a point of S such that ∡AOB < 120◦. Let D be the midpoint of the arc AB that does not contain C. The line through O parallel to DA meets the line AC at I. The perpendicular bisector of OA meets S at...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G9

IMO 1996 SL G9 Origin: UKR | Category: Geometry Problem In the plane are given a point $O$ and a polygon $F$ (not necessarily convex). Let $P$ denote the perimeter of $F$, $D$ the sum of the distances from $O$ to the vertices of $F$, and $H$ the sum of the distances from $O$ to the lines containing the sides of $F$. Prove that $$D^2 - H^2 \geq \frac{P^2}{4}.$$ Solution...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G8

IMO 1996 SL G8 Origin: RUS | Category: Geometry Problem Let $\triangle ABCD$ be a convex quadrilateral, and let $R_A$, $R_B$, $R_C$, and $R_D$ denote the circumradii of the triangles $\triangle DAB$, $\triangle ABC$, $\triangle BCD$, and $\triangle CDA$ respectively. Prove that $$R_A+R_C>R_B+R_D$$ if and only if $$\angle A+\angle C>\angle B+\angle D.$$ Solution Let the diagonals $AC$ and $BD$ meet at $X$. Either $\angle AXB$ or $\angle AXD$ is greater...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 5

IMO 1993 SL 5 Origin: FIN Problem On an inﬁnite chessboard, a solitaire game is played as follows: At the start, we have n2 pieces occupying n2 squares that form a square of side n. The only allowed move is a jump horizontally or vertically over an occupied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers n can the...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 13

IMO 1992 SL 13 Origin: NZL Problem Find all integer triples (p, q, r) such that 1 < p < q < r and (p −1)(q −1)(r −1) is a divisor of (pqr −1). Solution Let us deﬁne F(p, q, r) = (pqr −1) (p −1)(q −1)(r −1) = 1 + p −1 + q −1 + r −1 + (p −1)(q −1) + (q −1)(r −1) + (r −1)(p...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 24

IMO 1968 SL 24 Origin: MON Problem Find the number of all $n$-digit numbers for which some fixed digit stands only in the $i$th $(1 < i < n)$ place and the last $j$ digits are distinct.$^3$ Solution If the $i$th digit is $0$, then the result is $\frac{9^{k-j}9!}{(10-j)!}$ if $i > k-j$ and $\frac{9^{k-j-1}9!}{(9-j)!}$ otherwise. If the $i$th digit is not $0$, then the above results are multiplied by...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 8

IMO 1991 SL 8 Origin: NET Problem Let S be a set of n points in the plane. No three points of S are collinear. Prove that there exists a set P containing 2n −5 points satisfying the following condition: In the interior of every triangle whose three vertices are elements of S lies a point that is an element of P. Solution Let P1(x1, y1), P2(x2, y2), . ....

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 7

IMO 1974 SL 7 Problem II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k, and m the least common multiple of b1, . . . , bk. Prove that the greatest common divisor of a1 m b1 , . . . , ak m bk equals the greatest common divisor of a1, . . . , ak. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 14

IMO 1968 SL 14 Origin: BUL Problem A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$. Find the locus of the center of the circumcircle of triangle $XAY$. Solution Let $O$, $D$, $E$ be the circumcenter of $\triangle ABC$ and the midpoints of $AB$ and $AC$, and given arbitrary $X \in AB$ and...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 3

IMO 1987 SL 3 Origin: FIN Problem Does there exist a second-degree polynomial p(x, y) in two variables such that every nonnegative integer n equals p(k, m) for one and only one ordered pair (k, m) of nonnegative integers? Solution The answer: yes. Set p(k, m) = k + [1 + 2 + \cdot \cdot \cdot + (k + m)] = (k + m)2 + 3k + m . It...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 10

IMO 1977 SL 10 Origin: NET Problem Let n be an integer greater than 2. Deﬁne V = {1 + kn | k = 1, 2, . . . }. A number p \inV is called indecomposable in V if it is not possible to ﬁnd numbers q1, q2 \inV such that q1q2 = p. Prove that there exists a number N \inV that can be factorized into indecomposable factors...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N1

IMO 2000 SL N1 Origin: JAP | Category: Number Theory Problem Determine all positive integers n \geq2 that satisfy the following condition: For all integers a, b relatively prime to n, a \equivb (mod n) if and only if ab \equiv1 (mod n). Solution The given condition is obviously equivalent to a2 \equiv1 (mod n) for all inte- gers a coprime to n. Let n = p\alpha1 1 p\alpha2 2...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 12

IMO 1968 SL 12 Origin: POL Problem If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that $$\left(1 + \frac{a}{b}\right)^m + \left(1 + \frac{b}{a}\right)^m \geq 2^{m+1}.$$ Solution The given inequality is equivalent to $$\frac{(a + b)^m}{b^m} + \frac{(a + b)^m}{a^m} \geq 2^{m+1},$$ which can be rewritten as $$\left(\frac{1}{a^m} + \frac{1}{b^m}\right) \geq \frac{2^{m+1}}{(a + b)^m}.$$ Since $f(x) = \frac{1}{x^m}$ is a convex function for every $m...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 3

IMO 1968 SL 3 Origin: POL Problem Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle. Solution A triangle cannot be formed out of three lengths if and only if one of them is larger than the sum of the other two. Let us assume this is the case for all triplets of edges out...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 5

IMO 1998 SL 5 Origin: FRA Problem Let ABC be a triangle, H its orthocenter, O its circumcenter, and R its circumradius. Let D be the reﬂection of A across BC, E that of B across CA, and F that of C across AB. Prove that D, E, and F are collinear if and only if OH = 2R. Solution Let G be the centroid of \triangleABC and H the...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 20

IMO 1983 SL 20 Origin: ROM Problem Solve the system of equations x1|x1| = x2|x2| + (x1 −a)|x1 −a|, x2|x2| = x3|x3| + (x2 −a)|x2 −a|, \cdot \cdot \cdot xn|xn| = x1|x1| + (xn −a)|xn −a|, in the set of real numbers, where a > 0. Solution If (x1, x2, . . . , xn) satisﬁes the system with parameter a, then (−x1, −x2, . . . , −xn) satisﬁes...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 9

IMO 1968 SL 9 Origin: ROM Problem Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC, CA, AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$ Prove that the left-hand side attains its maximum when $M$...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 14

IMO 1989 SL 14 Origin: IND Problem A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals. Solution Lemma 1. In a quadrilateral ABCD circumscribed about a circle, with points of tangency P, Q, R, S on DA, AB, BC, CD respectively,...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 12

IMO 1972 SL 12 Origin: USS Problem A set of 10 positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements. Solution First we observe that it is not essential to require the subsets to be disjoint (if they aren’t, one simply excludes their intersection). There are 210−1...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 12

IMO 1991 SL 12 Origin: CHN Problem Let S = {1, 2, 3, . . ., 280}. Find the minimal natural num- ber n such that in any n-element subset of S there are ﬁve numbers that are pairwise relatively prime. Solution Let Am be the set of those elements of S which are divisible by m. By the inclusion-exclusion principle, the number of elements divisible by 2, 3, 5...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 21

IMO 1990 SL 21 Problem (ROM 1′) Let n be a composite natural number and p a proper divisor of n. Find the binary representation of the smallest natural number N such that (1+2p+2n−p)N−1 2n is an integer. Solution We must solve the congruence (1 + 2p + 2n−p)N \equiv1 (mod 2n). Since (1 + 2p + 2n−p) and 2n are coprime, there clearly exists a unique N satisfying this...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 8

IMO 1983 SL 8 Origin: SPA Problem In a test, 3n students participate, who are located in three rows of n students in each. The students leave the test room one by one. If N1(t), N2(t), N3(t) denote the numbers of students in the ﬁrst, second, and third row respectively at time t, ﬁnd the probability that for each t during the test, |Ni(t) −Nj(t)| < 2, i ̸= j,...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 7

IMO 1975 SL 7 Origin: GDR Problem Prove that from x + y = 1 (x, y \inR) it follows that xm+1 n j=0 m + j j yj + yn+1 m i=0 n + i i xi = 1 (m, n = 0, 1, 2, . . .). Solution We use induction on m. Denote by Sm the left-hand side of the equality to be...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 4

IMO 1968 SL 4 Origin: BUL Problem Let $a$, $b$, $c$ be real numbers. Prove that the system of equations $$\begin{cases} ax_1^2 + bx_1 + c = x_2, \ ax_2^2 + bx_2 + c = x_3, \ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \ ax_{n-1}^2 + bx_{n-1} + c = x_n, \ ax_n^2 + bx_n + c = x_1, \end{cases}$$ has a unique real...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 25

IMO 1968 SL 25 Origin: MON Problem Given k parallel lines and a few points on each of them, ﬁnd the number of all possible triangles with vertices at these given points.4 Solution The answer is 1\leqp<q<r\leqk npnqnr + 1\leqp<q\leqk np nq nq np .

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 22

IMO 1987 SL 22 Origin: VIE Problem Does there exist a function f : N \toN, such that f(f(n)) = n + 1987 for every natural number n? Solution Suppose that there exists such function f. Then we obtain f(n + 1987) = f(f(f(n))) = f(n) + 1987 for all n \inN, and from here, by induction, f(n + 1987t) = f(n) + 1987t for all n, t \inN. Further,...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A7

IMO 2004 SL A7 Origin: IRE | Category: Algebra Problem Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by gn their geometric mean, and by A1, A2, . . . , An the sequence of arithmetic means deﬁned by Ak = a1+a2+\cdot\cdot\cdot+ak k , k = 1, 2, . . ., n. Let Gn be the geometric mean of A1, A2, ....

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C2

IMO 2001 SL C2 Origin: CAN | Category: Combinatorics Problem Let n be an odd integer greater than 1 and let c1, c2, . . . , cn be integers. For each permutation a = (a1, a2, . . . , an) of {1, 2, . . ., n}, deﬁne S(a) = n i=1 ciai. Prove that there exist permutations a ̸= b of {1, 2, . . ., n}...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 19

IMO 1989 SL 19 Origin: MON Problem A positive integer is written in each square of an m\timesn board. The allowed move is to add an integer k to each of two adjacent numbers in such a way that no negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and suﬃcient condition for it to be possible for all the numbers to...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N3

IMO 2000 SL N3 Origin: RUS | Category: Number Theory Problem Does there exist a positive integer n such that n has exactly 2000 prime divisors and 2n + 1 is divisible by n? Solution More generally, we will prove by induction on k that for each k \inN there exists nk \inN that has exactly k distinct prime divisors such that nk | 2nk + 1 and 3 |...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 11

IMO 1998 SL 11 Origin: RUS Problem Let x, y, and z be positive real numbers such that xyz = 1. Prove that x3 (1 + y)(1 + z) + y3 (1 + z)(1 + x) + z3 (1 + x)(1 + y) \geq3 4. Solution The given inequality is equivalent to x3(x + 1) + y3(y + 1) + z3(z + 1) \geq 4(x + 1)(y + 1)(z +...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C4

IMO 1994 SL C4 Origin: EST | Category: Combinatorics Problem There are n + 1 ﬁxed positions in a row, labeled 0 to n in increasing order from right to left. Cards numbered 0 to n are shuﬄed and dealt, one in each position. The object of the game is to have card i in the ith position for 0 \leqi \leqn. If this has not been achieved, the following...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N6

IMO 1999 SL N6 Origin: BLR | Category: Number Theory Problem Prove that for every real number M there exists an inﬁnite arithmetic progression such that: (i) each term is a positive integer and the common diﬀerence is not di- visible by 10; (ii) the sum of the digits of each term (in decimal representation) exceeds M. Solution Let N be the smallest integer greater than M. We take the...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A6

IMO 2003 SL A6 Origin: USA | Category: Algebra Problem Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn) be two sequences of positive real numbers. Suppose (z2, z3, . . . , z2n) is a sequence of positive real numbers such that z2 i+j \geqxiyj for all 1 \leqi, j \leqn. Let M = max{z2, . . ....

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N6

IMO 2000 SL N6 Origin: ROM | Category: Number Theory Problem Show that the set of positive integers that cannot be repre- sented as a sum of distinct perfect squares is ﬁnite. Solution Suppose that a natural number N satisﬁes N = a2 1 + \cdot \cdot \cdot + a2 k, 2N = b2 1 +\cdot \cdot \cdot+b2 l , where ai, bj are natural numbers such that none of...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 1

IMO 1984 SL 1 Origin: FRA Problem Find all solutions of the following system of n equations in n variables: x1|x1| −(x1 −a)|x1 −a| = x2|x2|, x2|x2| −(x2 −a)|x2 −a| = x3|x3|, \cdot \cdot \cdot xn|xn| −(xn −a)|xn −a| = x1|x1|, where a is a given number. Solution This is the same problem as (SL83-20).

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 10

IMO 1968 SL 10 Origin: ROM Problem Consider two segments of length $a, b$ ($a > b$) and a segment of length $c = \sqrt{ab}$. (a) For what values of $a/b$ can these segments be sides of a triangle? (b) For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled?

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 16

IMO 1986 SL 16 Origin: ISR Problem Let A, B be adjacent vertices of a regular n-gon in the plane and let O be its center. Now let the triangle ABO glide around the polygon in such a way that the points A and B move along the whole circumference of the polygon. Describe the ﬁgure traced by the vertex O. Solution Let Z be the center of the polygon....

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G2

IMO 2003 SL G2 Origin: GRE | Category: Geometry Problem Three distinct points A, B, C are ﬁxed on a line in this order. Let \Gamma be a circle passing through A and C whose center does not lie on the line AC. Denote by P the intersection of the tangents to \Gamma at A and C. Suppose \Gamma meets the segment PB at Q. Prove that the intersection of...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 19

IMO 1983 SL 19 Origin: ROM Problem Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 = Fn+1 + Fn (n \geq1), and P(x) the polynomial of degree 990 satisfying P(k) = Fk, for k = 992, . . ., 1982. Prove that P(1983) = F1983 −1. Solution For all n, there exists a unique polynomial Pn of degree n such that Pn(k) = Fk for n...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 16

IMO 1998 SL 16 Origin: UKR Problem Determine the smallest integer n \geq4 for which one can choose four diﬀerent numbers a, b, c, and d from any n distinct integers such that a + b −c −d is divisible by 20. Solution Let S be a set of integers such that for no four distinct elements a, b, c, d \in S, it holds that 20 | a +...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 8

IMO 1992 SL 8 Origin: IND Problem Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: (i) its side lengths are 1, 2, 3, . . ., 1992 in some order; (ii) the polygon is circumscribable about a circle. Alternative formulation. Does there exist a 1992-gon with side lengths 1, 2, 3, . . ., 1992 circumscribed about a circle? Answer the...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 7

IMO 1993 SL 7 Origin: GEO Problem Let a, b, c be given integers a > 0, ac −b2 = P = P1 \cdot \cdot \cdot Pm where P1, . . . , Pm are (distinct) prime numbers. Let M(n) denote the num- ber of pairs of integers (x, y) for which ax2 + 2bxy + cy2 = n. Prove that M(n) is ﬁnite and M(n) = M(P k \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 8

IMO 1975 SL 8 Origin: NET Problem On the sides of an arbitrary triangle ABC, triangles BPC, CQA, and ARB are externally erected such that ∡PBC = ∡CAQ = 45◦, ∡BCP = ∡QCA = 30◦, ∡ABR = ∡BAR = 15◦. Prove that ∡QRP = 90◦and QR = RP. Solution Let K and L be the feet of perpendiculars from P and Q to BC and AC respectively. Let M, N...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 11

IMO 1979 SL 11 Origin: GDR Problem Given real numbers x1, x2, . . . , xn (n \geq2), with xi \geq1/n (i = 1, 2, . . ., n) and with x2 1 +x2 2 +\cdot \cdot \cdot+x2 n = 1, ﬁnd whether the product P = x1x2x3 \cdot \cdot \cdot xn has a greatest and/or least value and if so, give these values. Solution Let us deﬁne yi...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G1

IMO 2002 SL G1 Origin: FRA | Category: Geometry Problem Let B be a point on a circle S1, and let A be a point distinct from B on the tangent at B to S1. Let C be a point not on S1 such that the line segment AC meets S1 at two distinct points. Let S2 be the circle touching AC at C and touching S1 at a point...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 9

IMO 1988 SL 9 Origin: FRG Problem Let a and b be two positive integers such that ab+1 divides a2 + b2. Show that a2+b2 ab+1 is a perfect square. Solution Let us assume a2+b2 ab+1 = k \inN. We then have a2 −kab + b2 = k. Let us assume that k is not an integer square, which implies k \geq2. Now we observe the minimal pair (a, b)...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 14

IMO 1983 SL 14 Origin: POL Problem Prove or disprove: From the interval [1, . . . , 30000] one can select a set of 1000 integers containing no arithmetic triple (three consecutive numbers of an arithmetic progression). Solution Let Tn be the set of all nonnegative integers whose ternary representations consist of at most n digits and do not contain a digit 2. The cardinality of Tn is 2n,...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 20

IMO 1991 SL 20 Origin: IRE Problem Let \alpha be the positive root of the equation x2 = 1991x + 1. For natural numbers m, n deﬁne m ∗n = mn + [\alpham][\alphan], where [x] is the greatest integer not exceeding x. Prove that for all natural numbers p, q, r, (p ∗q) ∗r = p ∗(q ∗r). Solution We prove the result with 1991 replaced by any positive integer...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 1

IMO 1977 SL 1 Origin: BUL Problem Let f : N \toN be a function that satisﬁes the inequality f(n + 1) > f(f(n)) for all n \inN. Prove that f(n) = n for all natural numbers n.

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 16

IMO 1968 SL 16 Origin: GBR Problem A polynomial $p(x) = a_0 x^k + a_1 x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by $m$ for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k! a_0$ is also divisible by $m$. Also prove that if $a_0, k, m$ are nonnegative integers for which...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G2

IMO 1995 SL G2 Origin: GER | Category: Geometry Problem Let A, B, and C be noncollinear points. Prove that there is a unique point X in the plane of ABC such that XA2 + XB2 + AB2 = XB2 + XC2 + BC2 = XC2 + XA2 + CA2. Solution Let A′, B′, C′ be the points symmetric to A, B, C with respect to the midpoints of BC,...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 5

IMO 1968 SL 5 Origin: BUL Problem Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality $(n + 1)h_{n+1} - n h_n > r.$ Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C1

IMO 2004 SL C1 Origin: PUR | Category: Combinatorics Problem There are 10001 students at a university. Some students join together to form several clubs (a student may belong to diﬀerent clubs). Some clubs join together to form several societies (a club may belong to diﬀerent societies). There are a total of k societies. Suppose that the following conditions hold: (i) Each pair of students are in exactly one club....

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 13

IMO 1982 SL 13 Problem C1 (NET 1)IMO2 A scalene triangle A1A2A3 is given with sides a1, a2, a3 (ai is the side opposite to Ai). For all i = 1, 2, 3, Mi is the midpoint of side ai, Ti is the point where the incircle touches side ai, and the reﬂection of Ti in the interior bisector of Ai yields the point Si. Prove that the lines M1S1,...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 23

IMO 1997 SL 23 Origin: GBR Problem Let ABCD be a convex quadrilateral and O the intersection of its diagonals AC and BD. If OA sin \angleA + OC sin \angleC = OB sin \angleB + OD sin \angleD, prove that ABCD is cyclic. Solution Let K, L, M, and N be the projections of O onto the lines AB, BC, CD, and DA, and let \alpha1, \alpha2, \alpha3, \alpha4,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G7

IMO 2003 SL G7 Origin: SAF | Category: Geometry Problem Let ABC be a triangle with semiperimeter s and inradius r. The semicircles with diameters BC, CA, AB are drawn outside of the triangle ABC. The circle tangent to all three semicircles has radius t. Prove that s 2 < t \leqs 2 + 1 − \sqrt r. Solution Let D, E, F be the midpoints of BC, CA, AB,...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 10

IMO 1992 SL 10 Origin: ITA Problem Let V be a ﬁnite subset of Euclidean space consisting of points (x, y, z) with integer coordinates. Let S1, S2, S3 be the projections of V onto the yz, xz, xy planes, respectively. Prove that |V |2 \leq|S1||S2||S3| (|X| denotes the number of elements of X). Solution Let us set S(x) = {(y, z) | (x, y, z) \inV }, Sy(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 3

IMO 1972 SL 3 Origin: CZS Problem Let x1, x2, . . . , xn be real numbers satisfying x1+x2+\cdot \cdot \cdot+xn = 0. Let m be the least and M the greatest among them. Prove that x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \leq−nmM. Solution We have for each k = 1, 2, . . . , n that m \leqxk \leqM, which gives...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 3

IMO 1989 SL 3 Origin: AUS Problem Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he ﬁnds that if he lays it ﬂat on the ﬂoor of either of his storerooms, then each corner of the carpet touches a diﬀerent wall of that room. He knows that the sides...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 1

IMO 1986 SL 1 Origin: GBR Problem Find, with proof, all functions f deﬁned on the nonnegative real numbers and taking nonnegative real values such that (i) f[xf(y)]f(y) = f(x + y), (ii) f(2) = 0 but f(x) ̸= 0 for 0 \leqx < 2. Solution If w > 2, then setting in (i) x = w −2, y = 2, we get f(w) = f((w − 2)f(w))f(2) = 0....

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G1

IMO 2000 SL G1 Origin: NET | Category: Geometry Problem In the plane we are given two circles intersecting at X and Y . Prove that there exist four points A, B, C, D with the following property: For every circle touching the two given circles at A and B, and meeting the line XY at C and D, each of the lines AC, AD, BC, BD passes through one...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 8

IMO 1982 SL 8 Problem B2 (POL 4) A convex, closed ﬁgure lies inside a given circle. The ﬁgure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ﬁgure are perpendicular). Prove that the center of the circle is a center of symmetry of the ﬁgure. Solution Let F be the given ﬁgure. Consider...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A3

IMO 2001 SL A3 Origin: ROM | Category: Algebra Problem Let x1, x2, . . . , xn be arbitrary real numbers. Prove the inequality x1 1 + x2 + x2 1 + x2 1 + x2 \cdot \cdot \cdot + xn 1 + x2 1 + \cdot \cdot \cdot + x2n < \sqrtn. Solution By the arithmetic–quadratic mean inequality, it suﬃces to prove that x2 (1 + x2 1)2...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A7

IMO 2000 SL A7 Origin: RUS | Category: Algebra Problem For a polynomial P of degree 2000 with distinct real co- eﬃcients let M(P) be the set of all polynomials that can be produced from P by permutation of its coeﬃcients. A polynomial P will be called n-independent if P(n) = 0 and we can get from any Q in M(P) a poly- nomial Q1 such that Q1(n) = 0...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 17

IMO 1989 SL 17 Origin: MON Problem Given seven points in the plane, some of them are connected by segments so that: (i) among any three of the given points, two are connected by a segment; (i) the number of segments is minimal. How many segments does a ﬁgure satisfying (i) and (ii) contain? Give an example of such a ﬁgure. Solution A ﬁgure consisting of 9 lines is shown...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 24

IMO 1995 SL 24 Problem S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 = x1995 and xi−1 + xi−1 = 2xi + 1 xi for i = 1, 2, . . ., 1995. Find the maximum value that x0 can have. Solution The given condition is equivalent to (2xi −xi−1)(xixi−1 −1) = 0, so either xi = 1 2xi−1 or xi = xi−1...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N6

IMO 2003 SL N6 Origin: FRA | Category: Number Theory Problem Let p be a prime number. Prove that there exists a prime number q such that for every integer n, the number np −p is not divisible by q. Solution Suppose that for every prime q, there exists an n for which np \equivp (mod q). Assume that q = kp + 1. By Fermat’s theorem we deduce that...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 4

IMO 1991 SL 4 Origin: FRA Problem Let ABC be a triangle and M an interior point in ABC. Show that at least one of the angles ∡MAB, ∡MBC, and ∡MCA is less than or equal to 30◦. Solution Assume the contrary, that \angleMAB, \angleMBC, \angleMCA are all greater than 30◦. By the sine Ceva theorem, it holds that sin \angleMAC sin \angleMBA sin \angleMCB = sin \angleMAB sin \angleMBC...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 7

IMO 1979 SL 7 Origin: FRG Problem Given that 1 −1 2 + 1 3 −1 4 + \cdot \cdot \cdot − 1318 + 1319 = p q , where p and q are natural numbers having no common factor, prove that p is divisible by 1979. Solution We denote the sum mentioned above by S. We have the following equali- ties: S = 1 −1 2 + 1 3...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N2

IMO 2000 SL N2 Origin: FRA | Category: Number Theory Problem For a positive integer n, let d(n) be the number of all positive divisors of n. Find all positive integers n such that d(n)3 = 4n. Solution Let n = p\alpha1 1 p\alpha2 2 \cdot \cdot \cdot p\alphak k be the factorization of n onto primes (p1 < p2 < \cdot \cdot \cdot < pk). Since 4n is a...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 7

IMO 1991 SL 7 Origin: CHN Problem Let O be the center of the circumsphere of a tetrahedron ABCD. Let L, M, N be the midpoints of BC, CA, AB respectively, and assume that AB+BC = AD+CD, BC+CA = BD+AD, and CA+AB = CD + BD. Prove that \angleLOM = \angleMON = \angleNOL. Solution The given equations imply AB = CD, AC = BD, AD = BC. Let L1, M1,...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 23

IMO 1968 SL 23 Origin: CZS Problem Find all complex numbers m such that polynomial x3 + y3 + z3 + mxyz can be represented as the product of three linear trinomials. Solution We may assume w.l.o.g. that in all the factors the coeﬃcient of x is 1. Suppose that x + ay + bz is one of the linear factors of p(x, y, z) = x3 + y3 +...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G2

IMO 1996 SL G2 Origin: CAN | Category: Geometry Problem Let $P$ be a point inside $\triangle ABC$ such that $\angle APB - \angle C = \angle APC - \angle B.$ Let $D, E$ be the incenters of $\triangle APB$ and $\triangle APC$, respectively. Show that $AP$, $BD$, and $CE$ meet in a point. Solution Let $X, Y, Z$ respectively be the feet of the perpendiculars from $P$ to $BC$,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 23

IMO 1979 SL 23 Origin: USA Problem Find all natural numbers n for which 28 + 211 + 2n is a perfect square. Solution It is easily veriﬁed that no solutions exist for n \leq8. Let us now assume that n > 8. We note that 28 + 211 + 2n = 28 \cdot (9 + 2n−8). Hence 9 + 2n−8 must also be a square, say 9 + 2n−8...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A5

IMO 2003 SL A5 Origin: KOR | Category: Algebra Problem Let R+ be the set of all positive real numbers. Find all functions f : R+ \toR+ that satisfy the following conditions: (i) f(xyz) + f(x) + f(y) + f(z) = f(\sqrtxy)f(\sqrtyz)f(\sqrtzx) for all x, y, z \in R+. (ii) f(x) < f(y) for all 1 \leqx < y. Solution Placing x = y = z = 1 in (i)...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N3

IMO 1994 SL N3 Origin: FIN | Category: Number Theory Problem Find a set A of positive integers such that for any inﬁnite set P of prime numbers, there exist positive integers m \inA and n ̸\inA, both the product of the same number of distinct elements of P. Solution Let A be the set of all numbers of the form p1p2 . . . pp1, where p1 < p2...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 26

IMO 1991 SL 26 Origin: CZS Problem Let n \geq2 be a natural number and let the real numbers p, a1, a2, . . . , an, b1, b2, . . . , bn satisfy 1/2 \leqp \leq1, 0 \leqai, 0 \leqbi \leqp, i = 1, . . . , n, and n i=1 ai = n i=1 bi = 1. Prove the inequality n i=1 bi n j=1...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G7

IMO 1999 SL G7 Origin: ARM | Category: Geometry Problem The point M inside the convex quadrilateral ABCD is such that MA = MC, \angleAMB = \angleMAD+\angleMCD, \angleCMD = \angleMCB+\angleMAB. Prove that AB \cdot CM = BC \cdot MD and BM \cdot AD = MA \cdot CD. Solution Let us construct a convex quadrilateral PQRS and an interior point T such that \trianglePTQ ∼= \triangleAMB, \triangleQTR ∼\triangleAMD, and \trianglePTS ∼\triangleCMD....

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 14

IMO 1982 SL 14 Problem C2 (AUS 4) Let ABCD be a convex plane quadrilateral and let A1 denote the circumcenter of \triangleBCD. Deﬁne B1, C1, D1 in a corresponding way. (a) Prove that either all of A1, B1, C1, D1 coincide in one point, or they are all distinct. Assuming the latter case, show that A1, C1 are on opposite sides of the line B1D1, and similarly, B1, D1...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C4

IMO 2001 SL C4 Origin: NZL | Category: Combinatorics Problem A set of three nonnegative integers {x, y, z} with x < y < z is called historic if {z −y, y −x} = {1776, 2001}. Show that the set of all nonnegative integers can be written as the union of disjoint historic sets. Solution For convenience let us write a = 1776, b = 2001, 0 < a <...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 17

IMO 1992 SL 17 Origin: SWE Problem Let \alpha(n) be the number of digits equal to one in the binary representation of a positive integer n. Prove that: (a) the inequality \alpha(n2) \leq1 2\alpha(n)(\alpha(n) + 1) holds; (b) the above inequality is an equality for inﬁnitely many positive integers; (c) there exists a sequence (ni)\infty 1 such that \alpha(n2 i )/\alpha(ni) \to0 as i \to\infty. Alternative parts: Prove that there...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 7

IMO 1982 SL 7 Problem B1 (CAN 2) Let p(x) be a cubic polynomial with integer coeﬃcients with leading coeﬃcient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1)+p(−1)−2(1+p(0)). Solution Let a, b, ab be the roots of the cubic polynomial P(x) = (x−a)(x−b)(x− ab). Observe that 2p(−1) = −2(1 + a)(1 + b)(1 + ab);...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 9

IMO 1977 SL 9 Origin: HUN Problem For which positive integers n do there exist two polynomials f and g with integer coeﬃcients of n variables x1, x2, . . . , xn such that the following equality is satisﬁed: n i=1 xi f(x1, x2, . . . , xn) = g(x2 1, x2 2, . . . , x2 n)?

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 2

IMO 1990 SL 2 Origin: CAN Problem Given n countries with three representatives each, m commit- tees A(1), A(2), . . . A(m) are called a cycle if (i) each committee has n members, one from each country; (ii) no two committees have the same membership; (iii) for i = 1, 2, . . . , m, committee A(i) and committee A(i + 1) have no member in common, where...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N1

IMO 1995 SL N1 Origin: ROM | Category: Number Theory Problem Let k be a positive integer. Prove that there are inﬁnitely many perfect squares of the form n2k −7, where n is a positive integer. Solution We show by induction on k that there exists a positive integer ak for which a2 k \equiv−7 (mod 2k). The statement of the problem follows, since every ak + r2k (r =...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A2

IMO 1994 SL A2 Origin: FRA | Category: Algebra Problem Let m and n be positive integers. The set A = {a1, a2, . . . , am} is a subset of {1, 2, . . . , n}. Whenever ai + aj \leqn, 1 \leqi \leqj \leqm, ai + aj also belongs to A. Prove that a1 + a2 + \cdot \cdot \cdot + am m \geqn + 1...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 4

IMO 1986 SL 4 Origin: CZS Problem Let n be a positive integer and let p be a prime number, p > 3. Find at least 3(n+1) [easier version: 2(n+1)] sequences of positive integers x, y, z satisfying xyz = pn(x + y + z) that do not diﬀer only by permutation. Solution Let x = p\alphax′, y = p\betay′, z = p\gammaz′ with p ∤x′y′z′ and \alpha \geq\beta \geq\gamma....

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A6

IMO 1999 SL A6 Origin: SWE | Category: Algebra Problem For n \geq3 and a1 \leqa2 \leq\cdot \cdot \cdot \leqan given real numbers we have the following instructions: (1) place the numbers in some order in a circle; (2) delete one of the numbers from the circle; (3) if just two numbers are remaining in the circle, let S be the sum of these two numbers. Otherwise, if there are...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C8

IMO 2001 SL C8 Origin: GER | Category: Combinatorics Problem Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that (i) each contestant solved at most six problems, and (ii) for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 12

IMO 1979 SL 12 Origin: GDR Problem Let R be a set of exactly 6 elements. A set F of subsets of R is called an S-family over R if and only if it satisﬁes the following three conditions: (i) For no two sets X, Y in F is X \subseteqY ; (ii) For any three sets X, Y, Z in F, X \cupY \cupZ ̸= R, (iii) % X\inF...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A3

IMO 1996 SL A3 Origin: GRE | Category: Algebra Problem Let $a>2$ be given, and define recursively $$a_0 = 1, \quad a_1 = a, \quad a_{n+1} = \left(\frac{a_n^2}{a_{n-1}^2}-2\right)a_n.$$ Show that for all $k \in \mathbb{N}$, we have $$\frac{1}{a_0+1} + \frac{1}{a_1+1} + \frac{1}{a_2+1} + \cdots + \frac{1}{a_k+1} < \frac{1}{2} + a - \sqrt{a^2-4}.$$ Solution Since $a_1>2$, it can be written as $a_1 = b + b^{-1}$ for some $b>0$. Furthermore, $$a_1^2...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 14

IMO 1981 SL 14 Origin: ROM Problem Prove that a convex pentagon (a ﬁve-sided polygon) ABCDE with equal sides and for which the interior angles satisfy the condition \angleA \geq\angleB \geq\angleC \geq\angleD \geq\angleE is a regular pentagon. Solution We need the following lemma. Lemma. If a convex quadrilateral PQRS satisﬁes PS = QR and \angleSPQ \geq \angleRQP, then \angleQRS \geq\anglePSR. Proof. If the lines PS and QR are parallel, then...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 11

IMO 1971 SL 11 Origin: POL Problem The matrix ⎛ ⎜ ⎝ a11 . . . a1n ... \cdot \cdot \cdot ... an1 . . . ann ⎞ ⎟ ⎠ satisﬁes the inequality n j=1 |aj1x1 + \cdot \cdot \cdot + ajnxn| \leqM for each choice of numbers xi equal to \pm1. Show that |a11 + a22 + \cdot \cdot \cdot + ann| \leqM. Solution We use induction. The statement...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 15

IMO 1978 SL 15 Origin: YUG Problem Let p be a prime and A = {a1, . . . , ap−1} an arbitrary subset of the set of natural numbers such that none of its elements is divisible by p. Let us deﬁne a mapping f from P(A) (the set of all subsets of A) to the set P = {0, 1, . . ., p −1} in the following...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 9

IMO 1986 SL 9 Origin: GDR Problem Prove or disprove: Given a ﬁnite set of points with integer coordinates in the plane, it is possible to color some of these points red and the remaining ones white in such a way that for any straight line L parallel to one of the coordinate axes, the number of red colored points and the number of white colored points on L diﬀer...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C1

IMO 2003 SL C1 Origin: BRA | Category: Combinatorics Problem Let A be a 101-element subset of the set S = {1, 2, . . . , 1000000}. Prove that there exist numbers t1, t2, . . . , t100 in S such that the sets Aj = {x + tj | x \inA}, j = 1, 2, . . . , 100, are pairwise disjoint. Solution Consider the set...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 23

IMO 1990 SL 23 Origin: ROM Problem Find all positive integers n having the property that 2n+1 n2 is an integer. Solution Let us assume n > 1. Obviously n is odd. Let p \geq3 be the smallest prime divisor of n. In this case (p −1, n) = 1. Since 2n + 1 | 22n −1, we have that p | 22n −1. Thus it follows from Fermat’s little...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 25

IMO 1979 SL 25 Origin: USA Problem Given a point P in a given plane \pi and also a given point Q not in \pi, show how to determine a point R in \pi such that QP+PR QR is a maximum. Solution Let us ﬁrst look for such a point R on a line l in \pi going through P. Let \angleQPR = 2\theta. Consider a point Q′ on l...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 16

IMO 1993 SL 16 Origin: MCD Problem Let n \inN, n \geq2, and A0 = (a01, a02, . . . , a0n) be any n-tuple of natural numbers such that 0 \leqa0i \leqi−1, for i = 1, . . . , n. The n-tuples A1 = (a11, a12, . . . , a1n), A2 = (a21, a22, . . . , a2n), . . . are deﬁned by ai+1,j =...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 5

IMO 1975 SL 5 Origin: SWE Problem Let M be the set of all positive integers that do not contain the digit 9 (base 10). If x1, . . . , xn are arbitrary but distinct elements in M, prove that n j=1 xj < 80. Solution There are exactly 8 \cdot 9k−1 k-digit numbers in M (the ﬁrst digit can be chosen in 8 ways, while any other...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C6

IMO 2004 SL C6 Origin: IRN | Category: Combinatorics Problem For an n \times n matrix A, let Xi be the set of entries in row i, and Yj the set of entries in column j, 1 \leqi, j \leqn. We say that A is golden if X1, . . . , Xn, Y1, . . . , Yn are distinct sets. Find the least integer n such that there...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N1

IMO 1996 SL N1 Origin: UKR | Category: Number Theory Problem Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this number and the next number on the circle, in a given direction (that is, the numbers $a,b,c,d$ are replaced by $a-b$, $b-c$, $c-d$, $d-a$). Is it possible after $1996$ such steps to have numbers $a,b,c,d$ such that the numbers...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 26

IMO 1997 SL 26 Origin: ITA Problem For every integer n \geq2 determine the minimum value that the sum a0 + a1 + \cdot \cdot \cdot + an can take for nonnegative numbers a0, a1, . . . , an satisfying the condition a0 = 1, ai \leqai+1 + ai+2 for i = 0, . . . , n −2. Solution Let us ﬁrst examine the case that all the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C5

IMO 1999 SL C5 Origin: BLR | Category: Combinatorics Problem Let n be an even positive integer. We say that two dif- ferent cells of an n \times n board are neighboring if they have a common side. Find the minimal number of cells on the n\timesn board that must be marked so that every cell (marked or not marked) has a marked neighboring cell. Solution Let n = 2k....

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 1

IMO 1968 SL 1 Origin: SWE Problem Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9{:}00$ the distance between them was $20$ miles; at $9{:}35$, $15$ miles; and at $9{:}55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance? Solution Since the ships are sailing with constant speeds and directions, the...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 4

IMO 1982 SL 4 Problem A4 (BUL 2) Determine all real values of the parameter a for which the equation 16x4 −ax3 + (2a + 17)x2 −ax + 16 = 0 has exactly four distinct real roots that form a geometric progression. Solution Suppose that a satisﬁes the requirements of the problem and that x, qx, q2x, q3x are the roots of the given equation. Then x ̸= 0 and...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 7

IMO 1984 SL 7 Origin: FRG Problem (a) Decide whether the ﬁelds of the 8 \times 8 chessboard can be numbered by the numbers 1, 2, . . ., 64 in such a way that the sum of the four numbers in each of its parts of one of the forms is divisible by four. (b) Solve the analogous problem for Solution It clearly suﬃces to solve the problem for...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G6

IMO 2003 SL G6 Origin: POL | Category: Geometry Problem Each pair of opposite sides of a convex hexagon has the following property: The distance between their midpoints is equal to \sqrt 3/2 times the sum of their lengths. Prove that all the angles of the hexagon are equal. Solution Let ABCDEF be the given hexagon. We shall use the following lemma. Lemma. If \angleXZY \geq60◦and if M is the...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 21

IMO 1985 SL 21 Origin: IRE Problem The tangents at B and C to the circumcircle of the acute-angled triangle ABC meet at X. Let M be the midpoint of BC. Prove that (a) \angleBAM = \angleCAX, and (b) AM AX = cos \angleBAC. Solution Let B1 and C1 be the points on the rays AC and AB respectively such that XB1 = XC = XB = XC1. Then \angleXB1C...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G4

IMO 2004 SL G4 Origin: POL | Category: Geometry Problem In a convex quadrilateral ABCD the diagonal BD does not bisect the angles ABC and CDA. The point P lies inside ABCD and satisﬁes \anglePBC = \angleDBA and \anglePDC = \angleBDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP. Solution Assume ﬁrst that the points A, B, C, D are concyclic. Let the lines...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A2

IMO 1999 SL A2 Origin: RUS | Category: Algebra Problem The numbers from 1 to n2 are randomly arranged in the cells of a n \times n square (n \geq2). For any pair of numbers situated in the same row or in the same column, the ratio of the greater number to the smaller one is calculated. Let us call the characteristic of the arrangement the smallest of these n2(n−1)...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 20

IMO 1979 SL 20 Origin: SWE Problem Given the integer n > 1 and the real number a > 0 determine the maximum of n−1 i=1 xixi+1 taken over all nonnegative numbers xi with sum a. Solution Let xk = max{x1, x2, . . . , xn}. Then xixi+1 \leqxixk for i = 1, 2, . . . , k −1 and xixi+1 \leqxkxi+1 for i = k, . ....

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G1

IMO 1995 SL G1 Origin: BUL | Category: Geometry Problem Let A, B, C, and D be distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . O is an arbitrary point on the line XY but not on AD. CO intersects the circle with diameter AC again at M, and BO intersects the other circle again at N....

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 17

IMO 1987 SL 17 Origin: ROM Problem Prove that there exists a four-coloring of the set M = {1, 2, . . ., 1987} such that any arithmetic progression with 10 terms in the set M is not monochromatic. Alternative formulation. Let M = {1, 2, . . ., 1987}. Prove that there is a function f : M \to{1, 2, 3, 4} that is not constant on every set...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 5

IMO 1976 SL 5 Origin: NET Problem Let a set of p equations be given, a11x1 + \cdot \cdot \cdot + a1qxq = 0, a21x1 + \cdot \cdot \cdot + a2qxq = 0, ... ap1x1 + \cdot \cdot \cdot + apqxq = 0, with coeﬃcients aij satisfying aij = −1, 0, or +1 for all i = 1, . . . , p and j = 1, . . ....

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 11

IMO 1984 SL 11 Origin: CAN Problem Let n be a natural number and a1, a2, . . . , a2n mutually distinct integers. Find all integers x satisfying (x −a1) \cdot (x −a2) \cdot \cdot \cdot (x −a2n) = (−1)n(n!)2. Solution Suppose that an integer x satisﬁes the equation. Then the numbers x − a1, x −a2, . . . , x −a2n are 2n distinct integers whose product...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N1

IMO 2002 SL N1 Origin: UZB | Category: Number Theory Problem What is the smallest positive integer t such that there exist integers x1, x2, . . . , xt with x3 1 + x3 2 + \cdot \cdot \cdot + x3 t = 20022002? Solution Consider the given equation modulo 9. Since each cube is congruent to either −1, 0 or 1, whereas 20022002 \equiv42002 = 4 \cdot 64667...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N5

IMO 2002 SL N5 Origin: IRN | Category: Number Theory Problem Let m, n \geq2 be positive integers, and let a1, a2, . . . , an be integers, none of which is a multiple of mn−1. Show that there exist integers e1, e2, . . . , en, not all zero, with |ei| < m for all i, such that e1a1 + e2a2 + \cdot \cdot \cdot + enan...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C4

IMO 2000 SL C4 Origin: CZE | Category: Combinatorics Problem Let n and k be positive integers such that n/2 < k \leq2n/3. Find the least number m for which it is possible to place m pawns on m squares of an n \times n chessboard so that no column or row contains a block of k adjacent unoccupied squares. Solution By a good placement of pawns we mean the...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 27

IMO 1989 SL 27 Origin: ROM Problem Let m be a positive odd integer, m \geq2. Find the smallest positive integer n such that 21989 divides mn −1. Solution Let n be the required exponent, and suppose n = 2kq, where q is an odd integer. Then we have mn −1 = (m2k −1)[(m2k(q−1) + \cdot \cdot \cdot + m2k + 1] = (m2k −1)A, where A is odd. Therefore...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N5

IMO 2000 SL N5 Origin: BUL | Category: Number Theory Problem Prove that there exist inﬁnitely many positive integers n such that p = nr, where p and r are respectively the semiperimeter and the inradius of a triangle with integer side lengths. Solution It is known that the area of the triangle is S = pr = p2/n and S = p(p −a)(p −b)(p −c). It follows that p3...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 14

IMO 1987 SL 14 Origin: FRG Problem How many words with n digits can be formed from the alphabet {0, 1, 2, 3, 4}, if neighboring digits must diﬀer by exactly one? Solution Let xn be the total number of counted words of length n, and yn, zn, un, zn, yn the numbers of counted words of length n starting with 0, 1, 2, 3, 4, respectively (indeed, by symmetry,...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A4

IMO 1996 SL A4 Origin: KOR | Category: Algebra Problem Let $a_1, a_2, \dots, a_n$ be nonnegative real numbers, not all zero. (a) Prove that $$x^n - a_1 x^{n-1} - a_2 x^{n-2} - \cdots - a_{n-1} x - a_n = 0$$ has precisely one positive real root. (b) Let $$A = \sum_{j=1}^{n} a_j, \quad B = \sum_{j=1}^{n} j a_j,$$ and let $R$ be the positive real root of the equation...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 11

IMO 1972 SL 11 Origin: NET Problem Consider a sequence of circles K1, K2, K3, K4, . . . of radii r1, r2, r3, r4, . . . , respectively, situated inside a triangle ABC. The circle K1 is tangent to AB and AC; K2 is tangent to K1, BA, and BC; K3 is tangent to K2, CA, and CB; K4 is tangent to K3, AB, and AC; etc. (a)...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G8

IMO 1999 SL G8 Origin: RUS | Category: Geometry Problem Points A, B, C divide the circumcircle Ωof the triangle ABC into three arcs. Let X be a variable point on the arc AB, and let O1, O2 be the incenters of the triangles CAX and CBX. Prove that the circumcircle of the triangle XO1O2 intersects Ωin a ﬁxed point. Solution We ﬁrst introduce the same lemma as in problem...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 25

IMO 1995 SL 25 Problem S3 (POL) For an integer x \geq1, let p(x) be the least prime that does not divide x, and deﬁne q(x) to be the product of all primes less than p(x). In particular, p(1) = 2. For x such that p(x) = 2, deﬁne q(x) = 1. Consider the sequence x0, x1, x2, . . . deﬁned by x0 = 1 and xn+1 = xnp(xn)...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 16

IMO 1984 SL 16 Origin: POL Problem Let a, b, c, d be odd positive integers such that a < b < c < d, ad = bc, and a + d = 2k, b + c = 2m for some integers k and m. Prove that a = 1. Solution From the ﬁrst two conditions we can easily conclude that a + d > b + c (indeed, (d...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 9

IMO 1998 SL 9 Origin: MON Problem Let a1, a2, . . . , an be positive real numbers such that a1 + a2 + \cdot \cdot \cdot + an < 1. Prove that a1a2 \cdot \cdot \cdot an[1 −(a1 + a2 + \cdot \cdot \cdot + an)] (a1 + a2 + \cdot \cdot \cdot + an)(1 −a1)(1 −a2) \cdot \cdot \cdot (1 −an) \leq nn+1 . Solution Set an+1...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 15

IMO 1977 SL 15 Origin: VIE Problem The length of a ﬁnite sequence is deﬁned as the number of terms of this sequence. Determine the maximal possible length of a ﬁnite sequence that satisﬁes the following condition: The sum of each seven successive terms is negative, and the sum of each eleven successive terms is positive.

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G6

IMO 1995 SL G6 Origin: USA | Category: Geometry Problem Let A1A2A3A4 be a tetrahedron, G its centroid, and A′ 1, A′ 2, A′ 3, and A′ 4 the points where the circumsphere of A1A2A3A4 in- tersects GA1, GA2, GA3, and GA4, respectively. Prove that GA1 \cdot GA2 \cdot GA3 \cdot GA4 \leqGA′ 1 \cdot GA′ 2 \cdot GA′ 3 \cdot GA′ and GA′ + GA′ + GA′ + GA′...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 2

IMO 1976 SL 2 Origin: BUL Problem Let a0, a1, . . . , an, an+1 be a sequence of real numbers satisfying the following conditions: a0 = an+1 = 0, |ak−1 −2ak + ak+1| \leq1 (k = 1, 2, . . ., n). Prove that |ak| \leqk(n+1−k) (k = 0, 1, . . ., n + 1). Solution Let us put bi = i(n + 1 −i)/2, and let...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 6

IMO 1986 SL 6 Origin: NET Problem Find four positive integers each not exceeding 70000 and each having more than 100 divisors. Solution There are ﬁve such numbers: 69300 = 22 \cdot 32 \cdot 52 \cdot 7 \cdot 11 : 3 \cdot 3 \cdot 3 \cdot 2 \cdot 2 = 108 divisors; 50400 = 25 \cdot 32 \cdot 52 \cdot 7 : 6 \cdot 3 \cdot 3 \cdot 2 =...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C5

IMO 1994 SL C5 Origin: SWE | Category: Combinatorics Problem At a round table are 1994 girls, playing a game with a deck of n cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G3

IMO 2003 SL G3 Origin: IND | Category: Geometry Problem Let ABC be a triangle and let P be a point in its interior. Denote by D, E, F the feet of the perpendiculars from P to the lines BC, CA, and AB, respectively. Suppose that AP 2 + PD2 = BP 2 + PE2 = CP 2 + PF 2. Denote by IA, IB, IC the excenters of the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G3

IMO 2001 SL G3 Origin: GBR | Category: Geometry Problem Let ABC be a triangle with centroid G. Determine, with proof, the position of the point P in the plane of ABC such that AP \cdot AG + BP \cdot BG + CP \cdot CG is a minimum, and express this minimum value in terms of the side lengths of ABC. Solution Let us investigate a more general problem, in...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 12

IMO 1998 SL 12 Origin: POL Problem Let n \geqk \geq0 be integers. The numbers c(n, k) are deﬁned as follows: c(n, 0) = c(n, n) = 1 for all n \geq0; c(n + 1, k) = 2kc(n, k) + c(n, k −1) for n \geqk \geq1. Prove that c(n, k) = c(n, n −k) for all n \geqk \geq0. Solution The assertion is clear for n = 0. We...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N2

IMO 2003 SL N2 Origin: USA | Category: Number Theory Problem Each positive integer a undergoes the following procedure in order to obtain the number d = d(a): (1) move the last digit of a to the ﬁrst position to obtain the number b; (2) square b to obtain the number c; (3) move the ﬁrst digit of c to the end to obtain the number d. (All the numbers...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 6

IMO 1975 SL 6 Origin: USS Problem Let A be the sum of the digits of the number 1616 and B the sum of the digits of the number A. Find the sum of the digits of the number B without calculating 1616. Solution Let us denote by C the sum of digits of B. We know that 1616 \equivA \equiv B \equivC (mod 9). Since 1616 = 264 =...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 2

IMO 1992 SL 2 Origin: CHN Problem Let R+ be the set of all nonnegative real numbers. Given two positive real numbers a and b, suppose that a mapping f : R+ \toR+ satisﬁes the functional equation f(f(x)) + af(x) = b(a + b)x. Prove that there exists a unique solution of this equation. Solution Let us deﬁne xn inductively as xn = f(xn−1), where x0 \geq0 is a ﬁxed...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 6

IMO 1985 SL 6 Origin: POL Problem Let xn = 2 + 3 + . . . + n\sqrtn. Prove that xn+1 −xn < 1 n!, n = 2, 3, . . . . Alternatives Solution Let us set xn,i = i i + i+1 i + 1 + \cdot \cdot \cdot + n\sqrtn, yn,i = xi−1 n+1,i + xi−2 n+1,ixn,i + \cdot \cdot \cdot + xi−1 n,i ....

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 22

IMO 1991 SL 22 Origin: USA Problem Real constants a, b, c are such that there is exactly one square all of whose vertices lie on the cubic curve y = x3 + ax2 + bx + c. Prove that the square has sides of length 4\sqrt 72. Solution Suppose w.l.o.g. that the center of the square is at the origin O(0, 0). We denote the curve y = f(x)...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C7

IMO 2001 SL C7 Origin: FRA | Category: Combinatorics Problem A pile of n pebbles is placed in a vertical column. This conﬁguration is modiﬁed according to the following rules. A pebble can be moved if it is at the top of a column that contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 10

IMO 1993 SL 10 Origin: IND Problem A natural number n is said to have the property P if whenever n divides an −1 for some integer a, n2 also necessarily divides an −1. (a) Show that every prime number has property P. (b) Show that there are inﬁnitely many composite numbers n that possess property P. Solution (a) Let n = p be a prime and let p |...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 6

IMO 1982 SL 6 Problem A6 (VIE 1)IMO6 Let S be a square with sides of length 100 and let L be a path within S that does not meet itself and that is composed of linear segments A0A1, A1A2, . . . , An−1An with A0 ̸= An. Suppose that for every point P of the boundary of S there is a point of L at a distance from...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 19

IMO 1986 SL 19 Origin: BUL Problem A tetrahedron ABCD is given such that AD = BC = a; AC = BD = b; AB \cdot CD = c2. Let f(P) = AP + BP + CP + DP, where P is an arbitrary point in space. Compute the least value of f(P). Solution Let M and N be the midpoints of segments AB and CD, respectively. The given conditions...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 1

IMO 1978 SL 1 Origin: BUL Problem The set M = {1, 2, . . ., 2n} is partitioned into k nonintersecting subsets M1, M2, . . . , Mk, where n \geqk3 + k. Prove that there exist even numbers 2j1, 2j2, . . . , 2jk+1 in M that are in one and the same subset Mi (1 \leqi \leqk) such that the numbers 2j1 −1, 2j2 −1,...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 20

IMO 1998 SL 20 Origin: ARG Problem Prove that for each positive integer n, there exists a positive integer with the following properties: (i) It has exactly n digits. (ii) None of the digits is 0. (iii) It is divisible by the sum of its digits. Solution We ﬁrst consider the special case n = 3r. Then the simplest choice 10n−1 11 . . . 1 (n digits) works. This...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C4

IMO 1996 SL C4 Origin: FIN | Category: Combinatorics Problem Determine whether or not there exist two disjoint infinite sets $A$ and $B$ of points in the plane satisfying the following conditions: No three points in $A \cup B$ are collinear, and the distance between any two points in $A \cup B$ is at least $1$. There is a point of $A$ in any triangle whose vertices are in $B$,...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 19

IMO 1985 SL 19 Origin: ISR Problem For which integers n \geq3 does there exist a regular n-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system? Solution Suppose that for some n > 6 there is a regular n-gon with vertices having integer coordinates, and that A1A2 . . . An is the smallest such n-gon, of side length a. If O...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 19

IMO 1997 SL 19 Origin: IRE Problem Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers. Prove that n k=1 ak \leq n k=1 \sqrt k(\sqrtak −\sqrtak+1). Solution Using that an+1 = 0 we can transform the desired inequality into a1 + a2 + \cdot \cdot \cdot + an+1 \leq \sqrt 1\sqrta1 + ( \sqrt 2 − \sqrt 1)\sqrta2 + \cdot \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 7

IMO 1985 SL 7 Problem 1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 < \cdot \cdot \cdot < xn < 2x1. Set P = x1x2 \cdot \cdot \cdot xn. Prove that if p is a prime number, k a positive integer, and P is divisible by pk, then P pk \geqn!. Solution Let ki \geq0 be the largest integer such that...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 4

IMO 1972 SL 4 Origin: GDR Problem Let n1, n2 be positive integers. Consider in a plane E two dis- joint sets of points M1 and M2 consisting of 2n1 and 2n2 points, respec- tively, and such that no three points of the union M1 \cupM2 are collinear. Prove that there exists a straightline g with the following property: Each of the two half-planes determined by g on E (g...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G4

IMO 1994 SL G4 Origin: AUS-ARM | Category: Geometry Problem N is an arbitrary point on the bisector of \angleBAC. P and O are points on the lines AB and AN, respectively, such that ∡ANP = 90◦= ∡APO. Q is an arbitrary point on NP, and an arbitrary line through Q meets the lines AB and AC at E and F respectively. Prove that ∡OQE = 90◦if and only if...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 13

IMO 1988 SL 13 Origin: GRE Problem In a right-angled triangle ABC, let AD be the altitude drawn to the hypotenuse and let the straight line joining the incenters of the triangles ABD, ACD intersect the sides AB, AC at the points K, L respectively. If E and E1 denote the areas of the triangles ABC and AKL respectively, show that E E1 \geq2. Solution Let AB = c, AC...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 20

IMO 1992 SL 20 Origin: FRA Problem In the plane, let there be given a circle C, a line l tangent to C, and a point M on l. Find the locus of points P that have the following property: There exist two points Q and R on l such that M is the midpoint of QR and C is the incircle of PQR. Solution Denote by U the point...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A2

IMO 1995 SL A2 Origin: SWE | Category: Algebra Problem Let a and b be nonnegative integers such that ab \geqc2, where c is an integer. Prove that there is a number n and integers x1, x2, . . . , xn, y1, y2, . . . , yn such that n i=1 x2 i = a, n i=1 y2 i = b, and n i=1 xiyi...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 9

IMO 1976 SL 9 Origin: FIN Problem Let P1(x) = x2 −2, Pj(x) = P1(Pj−1(x)), j = 2, 3, . . . . Show that for arbitrary n the roots of the equation Pn(x) = x are real and diﬀerent from one another. Solution The equation Pn(x) = x is of degree 2n, and has at most 2n distinct roots. If x > 2, then by simple induction Pn(x) >...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C5

IMO 2000 SL C5 Origin: RUS | Category: Combinatorics Problem In the plane we have n rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the n rectangles on its boundary. Prove that the sum of the numbers of the vertices...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C3

IMO 1994 SL C3 Origin: MCD | Category: Combinatorics Problem Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. (a) Prove that Peter can always transfer all his money into two accounts. (b) Can Peter always transfer all his money into one...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A6

IMO 1996 SL A6 Origin: IRE | Category: Algebra Problem Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^n+g(x)(x^n+1)$$ for some polynomials $f(x)$, $g(x)$ having integer coefficients. If $k_0$ denotes the least such $k$, determine $k_0$ as a function of $n$. $A6'$ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^n+g(x)(x^n+1)$$ for...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 21

IMO 1986 SL 21 Origin: TUR Problem Let ABCD be a tetrahedron having each sum of opposite sides equal to 1. Prove that rA + rB + rC + rD \leq \sqrt 3 , where rA, rB, rC, rD are the inradii of the faces, equality holding only if ABCD is regular. Solution Since the sum of all edges of ABCD is 3, the statement of the problem is an...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 9

IMO 1992 SL 9 Origin: IRN Problem Let f(x) be a polynomial with rational coeﬃcients and \alpha be a real number such that \alpha3 −\alpha = f(\alpha)3 −f(\alpha) = 331992. Prove that for each n \geq1, (f (n)(\alpha))3 −f (n)(\alpha) = 331992, where f (n)(x) = f(f(. . . f(x))), and n is a positive integer. Solution Since the equation x3 −x −c = 0 has only one real root...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 17

IMO 1978 SL 17 Origin: FRA Problem Prove that for any positive integers x, y, z with xy−z2 = 1 one can ﬁnd nonnegative integers a, b, c, d such that x = a2 + b2, y = c2 + d2, z = ac + bd. Set z = (2q)! to deduce that for any prime number p = 4q + 1, p can be represented as the sum of...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 3

IMO 1981 SL 3 Origin: CAN Problem Find the minimum value of max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g) subject to the constraints (i) a, b, c, d, e, f, g \geq0, (ii) a + b + c + d + e + f + g = 1. Solution Denote max(a +...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 28

IMO 1995 SL 28 Problem S6 (IND) Let N denote the set of all positive integers. Prove that there exists a unique function f : N \toN satisfying f(m + f(n)) = n + f(m + 95) for all m and n in N. What is the value of 19 k=1 f(k)? Solution Let F(x) = f(x) −95 for x \geq1. Writing k for m + 95, the given condition...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 8

IMO 1981 SL 8 Origin: FRG Problem Let f(n, r) be the arithmetic mean of the minima of all r- subsets of the set {1, 2, . . ., n}. Prove that f(n, r) = n+1 r+1 . Solution Since the number k, k = 1, 2, . . ., n −r + 1, is the minimum in exactly n−k r−1 r-element subsets of {1, 2, . . .,...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A4

IMO 1995 SL A4 Origin: USA | Category: Algebra Problem Let a, b, and c be given positive real numbers. Determine all positive real numbers x, y, and z such that x + y + z = a + b + c and 4xyz −(a2x + b2y + c2z) = abc. Solution The second equation is equivalent to a2 yz + b2 zx + c2 xy + abc xyz =...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 6

IMO 1993 SL 6 Origin: GER Problem Let N = {1, 2, 3, . . .}. Determine whether there exists a strictly increasing function f : N \toN with the following properties: f(1) = 2; (1) f(f(n)) = f(n) + n (n \inN). (2) Solution Notice that for \alpha = 1+ \sqrt , \alpha2n = \alphan+n for all n \inN. We shall show that f(n) = / \alphan + 1...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 11

IMO 1974 SL 11 Problem II 5 (BUL 1)IMO4 Consider a partition of an 8 \times 8 chessboard into p rectangles whose interiors are disjoint such that each of them has an equal number of white and black cells. Assume that a1 < a2 < \cdot \cdot \cdot < ap, where ai denotes the number of white cells in the ith rectangle. Find the maximal p for which such a...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G4

IMO 2002 SL G4 Origin: RUS | Category: Geometry Problem Circles S1 and S2 intersect at points P and Q. Distinct points A1 and B1 (not at P or Q) are selected on S1. The lines A1P and B1P meet S2 again at A2 and B2 respectively, and the lines A1B1 and A2B2 meet at C. Prove that as A1 and B1 vary, the circumcenters of triangles A1A2C all lie...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G1

IMO 2004 SL G1 Origin: ROM | Category: Geometry Problem Let ABC be an acute-angled triangle with AB ̸= AC. The circle with diameter BC intersects the sides AB and AC at M and N, respectively. Denote by O the midpoint of BC. The bisectors of the angles BAC and MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N2

IMO 1996 SL N2 Origin: RUS | Category: Number Theory Problem The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Solution Let $15a + 16b = x^2$ and $16a - 15b = y^2$, where $x, y \in...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 16

IMO 1971 SL 16 Origin: USS Problem Given a convex polyhedron P1 with 9 vertices A1, . . . , A9, let us denote by P2, P3, . . . , P9 the images of P1 under the translations mapping the vertex A1 to A2, A3, . . . , A9 respectively. Prove that among the polyhedra P1, . . . , P9 at least two have a common interior...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 12

IMO 1988 SL 12 Origin: GRE Problem In a triangle ABC, choose any points K \inBC, L \inAC, M \inAB, N \inLM, R \inMK, and F \inKL. If E1, E2, E3, E4, E5, E6, and E denote the areas of the triangles AMR, CKR, BKF, ALF, BNM, CLN, and ABC respectively, show that E \geq8 E1E2E3E4E5E6. Remark. Points K, L, M, N, R, F lie on segments BC, AC, AB,...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N2

IMO 2002 SL N2 Origin: ROM | Category: Number Theory Problem Let n \geq2 be a positive integer, with divisors 1 = d1 < d2 < \cdot \cdot \cdot < dk = n. Prove that d1d2 + d2d3 + \cdot \cdot \cdot + dk−1dk is always less than n2, and determine when it is a divisor of n2. Solution Set S = d1d2 + \cdot \cdot \cdot + dk−1dk. Since...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 10

IMO 1971 SL 10 Origin: POL Problem Prove that the sequence 2n −3 (n > 1) contains a subse- quence of numbers relatively prime in pairs. Solution We use induction. Suppose that every two of the numbers a1 = 2n1 − 3, a2 = 2n2 −3, . . . , ak = 2nk −3, where 2 = n1 < n2 < \cdot \cdot \cdot < nk, are coprime. Then one...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 18

IMO 1984 SL 18 Origin: USA Problem Inside triangle ABC there are three circles k1, k2, k3 each of which is tangent to two sides of the triangle and to its incircle k. The radii of k1, k2, k3 are 1, 4, and 9. Determine the radius of k. Solution Suppose that circles k1(O1, r1), k2(O2, r2), and k3(O3, r3) touch the edges of the angles \angleBAC, \angleABC, and \angleACB,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 9

IMO 1979 SL 9 Origin: FRG Problem Let S and F be two opposite vertices of a regular octagon. A counter starts at S and each second is moved to one of the two neigh- boring vertices of the octagon. The direction is determined by the toss of a coin. The process ends when the counter reaches F. We deﬁne an to be the number of distinct paths of duration...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 17

IMO 1998 SL 17 Origin: GBR Problem A sequence of integers a1, a2, a3, . . . is deﬁned as follows: a1 = 1, and for n \geq1, an+1 is the smallest integer greater than an such that ai + aj ̸= 3ak for any i, j, k in {1, 2, . . ., n + 1}, not necessarily distinct. Determine a1998. Solution Initially, we determine that the ﬁrst few...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 15

IMO 1998 SL 15 Origin: AUS Problem Determine all pairs (a, b) of real numbers such that a\lfloorbn\rfloor= b\lflooran\rfloor for all positive integers n. (Note that \lfloorx\rfloordenotes the greatest integer less than or equal to x.) Solution The condition is obviously satisﬁed if a = 0 or b = 0 or a = b or a, b are both integers. We claim that these are the only solutions. Suppose that...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 10

IMO 1973 SL 10 Origin: SWE Problem Let a1, a2, . . . , an be positive numbers and q a given real number, 0 < q < 1. Find n real numbers b1, b2, . . . , bn that satisfy: (1) ak < bk for all k = 1, 2, . . ., n; (2) q < bk+1 bk < 1 q for all k = 1, 2,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C4

IMO 2003 SL C4 Origin: IRN | Category: Combinatorics Problem Let x1, . . . , xn and y1, . . . , yn be real numbers. Let A = (aij)1\leqi,j\leqn be the matrix with entries aij = . 1, if xi + yj \geq0; 0, if xi + yj < 0. Suppose that B is an n \times n matrix whose entries are 0, 1 such that the sum...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 2

IMO 1978 SL 2 Origin: BUL Problem Two identically oriented equilateral triangles, ABC with center S and A′B′C, are given in the plane. We also have A′ ̸= S and B′ ̸= S. If M is the midpoint of A′B and N the midpoint of AB′, prove that the triangles SB′M and SA′N are similar. Solution Consider the transformation \varphi of the plane deﬁned as the homothety H with center...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 14

IMO 1997 SL 14 Origin: IND Problem Let b, m, n be positive integers such that b > 1 and m ̸= n. Prove that if bm −1 and bn −1 have the same prime divisors, then b + 1 is a power of 2. Solution We use the following nonstandard notation: (1◦) for x, y \inN, x ∼y means that x and y have the same prime divisors; (2◦)...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 7

IMO 1978 SL 7 Origin: FRA Problem We consider three distinct half-lines Ox, Oy, Oz in a plane. Prove the existence and uniqueness of three points A \inOx, B \inOy, C \inOz such that the perimeters of the triangles OAB, OBC, OCA are all equal to a given number 2p > 0. Solution Let x = OA, y = OB, z = OC, \alpha = \angleBOC, \beta = \angleCOA, \gamma...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 16

IMO 1978 SL 16 Origin: YUG Problem Determine all the triples (a, b, c) of positive real numbers such that the system ax + by −cz = 0, a 1 −x2 + b 1 −y2 −c 1 −z2 = 0, is compatible in the set of real numbers, and then ﬁnd all its real solutions. Solution Clearly |x| \leq1. As x runs over [−1, 1], the vector u = (ax,...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 8

IMO 1973 SL 8 Origin: ROM Problem Prove that there are exactly k [k/2] arrays a1, a2, . . . , ak+1 of nonnegative integers such that a1 = 0 and |ai−ai+1| = 1 for i = 1, 2, . . . , k. Solution Let fn be the required total number, and let fn(k) denote the number of sequences a1, . . . , an of nonnegative...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G7

IMO 2002 SL G7 Origin: BUL | Category: Geometry Problem The incircle Ωof the acute-angled triangle ABC is tangent to BC at K. Let AD be an altitude of triangle ABC and let M be the midpoint of AD. If N is the other common point of Ωand KM, prove that Ωand the circumcircle of triangle BCN are tangent at N. Solution Let k be the circle through B, C...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 5

IMO 1973 SL 5 Origin: FRA Problem A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles. Solution Let O be the vertex of the trihedron, Z the center of a circle k inscribed in the trihedron, and A, B, C points in which the plane of the circle meets the edges of the trihedron. We...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 8

IMO 1997 SL 8 Origin: GBR Problem Four diﬀerent points A, B, C, D are chosen on a circle \Gamma such that the triangle BCD is not right-angled. Prove that: (a) The perpendicular bisectors of AB and AC meet the line AD at cer- tain points W and V , respectively, and that the lines CV and BW meet at a certain point T . (b) The length of one...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 10

IMO 1981 SL 10 Origin: FRA Problem Determine the smallest natural number n having the following property: For every integer p, p \geqn, it is possible to subdivide (partition) a given square into p squares (not necessarily equal). Solution It is easy to see that partitioning into p = 2k squares is possible for k \geq2 (Fig. 1). Furthermore, whenever it is possible to partition the square into p squares,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N8

IMO 2003 SL N8 Origin: IRN | Category: Number Theory Problem Let p be a prime number and let A be a set of positive integers that satisﬁes the following conditions: (i) the set of prime divisors of the elements in A consists of p−1 elements; (ii) for any nonempty subset of A, the product of its elements is not a perfect pth power. What is the largest possible number...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 5

IMO 1977 SL 5 Origin: FRG Problem There are 2n words of length n over the alphabet {0, 1}. Prove that the following algorithm generates the sequence w0, w1, . . . , w2n−1 of all these words such that any two consecutive words diﬀer in exactly one digit. (1) w0 = 00 . . .0 (n zeros). (2) Suppose wm−1 = a1a2 . . . an, ai \in{0, 1}....

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 2

IMO 1975 SL 2 Origin: CZS Problem Let x1 \geqx2 \geq\cdot \cdot \cdot \geqxn and y1 \geqy2 \geq\cdot \cdot \cdot \geqyn be two n-tuples of numbers. Prove that n i=1 (xi −yi)2 \leq n i=1 (xi −zi)2 is true when z1, z2, . . . , zn denote y1, y2, . . . , yn taken in another order. Solution Since there are ﬁnitely many arrangements of the...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 4

IMO 1977 SL 4 Origin: FRG Problem Describe all closed bounded ﬁgures \Phi in the plane any two points of which are connectable by a semicircle lying in \Phi.

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 12

IMO 1975 SL 12 Origin: GRE Problem Consider on the ﬁrst quadrant of the trigonometric circle the arcs AM1 = x1, AM2 = x2, AM3 = x3, . . . , AM\nu = x\nu, such that x1 < x2 < x3 < \cdot \cdot \cdot < x\nu. Prove that \nu−1 i=0 sin 2xi − \nu−1 i=0 sin(xi −xi+1) < \pi 2 + \nu−1 i=0 sin(xi + xi+1)....

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G3

IMO 1999 SL G3 Origin: EST | Category: Geometry Problem A set S of points in space will be called completely sym- metric if it has at least three elements and satisﬁes the following condition: For every two distinct points A, B from S the perpendicular bisector of the segment AB is an axis of symmetry for S. Prove that if a completely symmetric set is ﬁnite, then it consists...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 15

IMO 1985 SL 15 Problem 5a.(FRA 3) Let K and K′ be two squares in the same plane, their sides of equal length. Is it possible to decompose K into a ﬁnite number of tri- angles T1, T2, . . . , Tp with mutually disjoint interiors and ﬁnd translations t1, t2, . . . , tp such that K′ = p2 i=1 ti(Ti)? Solution There is no loss of...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G5

IMO 2003 SL G5 Origin: KOR | Category: Geometry Problem Let ABC be an isosceles triangle with AC = BC, whose incenter is I. Let P be a point on the circumcircle of the triangle AIB lying inside the triangle ABC. The lines through P parallel to CA and CB meet AB at D and E, respectively. The line through P parallel to AB meets CA and CB at F...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N3

IMO 2004 SL N3 Origin: IRN | Category: Number Theory Problem A function f from the set of positive integers N into itself is such that for all m, n \inN the number (m2 + n)2 is divisible by f 2(m) + f(n). Prove that f(n) = n for each n \inN. Solution For m = n = 1 we obtain that f(1)2 + f(1) divides (12 + 1)2 =...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G8

IMO 2004 SL G8 Origin: SMN | Category: Geometry Problem A cyclic quadrilateral ABCD is given. The lines AD and BC intersect at E, with C between B and E; the diagonals AC and BD intersect at F. Let M be the midpoint of the side CD, and let N ̸= M be a point on the circumcircle of the triangle ABM such that AN/BN = AM/BM. Prove that the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 19

IMO 1998 SL 19 Origin: BLR Problem For any positive integer n, let \tau(n) denote the number of its positive divisors (including 1 and itself). Determine all positive integers m for which there exists a positive integer n such that \tau(n2) \tau(n) = m. Solution For n = p\alpha1 1 p\alpha2 2 \cdot \cdot \cdot p\alphar r , where pi are distinct primes and \alphai natural num- bers, we have...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A7

IMO 1996 SL A7 Origin: ARM | Category: Algebra Problem Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such that for all $x\in\mathbb{R}$, we have $|f(x)|\leq 1$ and $$f\left(x+\frac13\right)+f(x)=f\left(x+\frac16\right)+f\left(x+\frac17\right).$$ Prove that $f$ is a periodic function (that is, there exists a nonzero real number $c$ such that $f(x+c)=f(x)$ for all $x\in\mathbb{R}$). Solution We are given that $$f(x+a+b)-f(x+a)=f(x+b)-f(x),$$ where $$a=\frac16,\qquad b=\frac17.$$ Summing up these equations...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 26

IMO 1968 SL 26 Origin: GDR Problem Let $a > 0$ be a real number and $f(x)$ a real function defined on all of $\mathbb{R}$, satisfying for all $x \in \mathbb{R}$, $$f(x + a) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.$$ (a) Prove that the function $f$ is periodic; i.e., there exists $b > 0$ such that for all $x$, $f(x + b) = f(x)$. (b) Give an example of such...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 2

IMO 1981 SL 2 Origin: BUL Problem A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahe- dron ABCD at the points E, F, G, H respectively. The points E, F, G, H are the vertices of a square. Prove that if the sphere is tangent to the edge AC, then it is also tangent to the edge BD. Solution Lemma. Let E, F, G,...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 23

IMO 1995 SL 23 Problem S1 (UKR) Does there exist a sequence F(1), F(2), F(3), . . . of nonneg- ative integers that simultaneously satisﬁes the following three conditions? (a) Each of the integers 0, 1, 2, . . . occurs in the sequence. (b) Each positive integer occurs in the sequence inﬁnitely often. (c) For any n \geq2, F F n163 = F(F(n)) + F(F(361)). Solution By...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 26

IMO 1988 SL 26 Origin: GBR Problem A function f deﬁned on the positive integers (and taking positive integer values) is given by f(1) = 1, f(3) = 3, f(2n) = f(n), f(4n + 1) = 2f(2n + 1) −f(n), f(4n + 3) = 3f(2n + 1) −2f(n), for all positive integers n. Determine with proof the number of positive integers less than or equal to 1988 for which f(n)...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 19

IMO 1984 SL 19 Origin: CAN Problem The triangular array (an,k) of numbers is given by an,1 = 1/n, for n = 1, 2, . . ., an,k+1 = an−1,k −an,k, for 1 \leqk \leqn −1. Find the harmonic mean of the 1985th row. Solution First, we shall prove that the numbers in the nth row are exactly the numbers n n−1 , n n−1 , n n−1...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G2

IMO 2002 SL G2 Origin: KOR | Category: Geometry Problem Let ABC be a triangle for which there exists an interior point F such that \angleAFB = \angleBFC = \angleCFA. Let the lines BF and CF meet the sides AC and AB at D and E respectively. Prove that AB + AC \geq4DE. Solution Construct equilateral triangles ACP and ABQ outside the triangle ABC. Since \angleAPC + \angleAFC = 60◦+...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G2

IMO 1999 SL G2 Origin: JAP | Category: Geometry Problem A circle is called a separator for a set of ﬁve points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the ﬁfth point is outside the circle. Prove that every set of ﬁve points such that no three are collinear and no four are concyclic has exactly four...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 4

IMO 1987 SL 4 Origin: FRA Problem Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH. Prove the inequality AF + AH + AC \leqAB + AD + AE + AG. In what cases does equality hold? Solution Setting x1 = −−\to AB, x2 = −−\to AD, x3 = −\to AE, we have to prove that \parallelx1 + x2\parallel+ \parallelx2 + x3\parallel+ \parallelx3 + x1\parallel\leq\parallelx1\parallel+ \parallelx2\parallel+ \parallelx3\parallel+ \parallelx1 + x2 + x3\parallel....

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G5

IMO 2004 SL G5 Origin: SMN | Category: Geometry Problem Let A1A2 . . . An be a regular n-gon. The points B1, . . . , Bn−1 are deﬁned as follows: (i) If i = 1 or i = n −1, then Bi is the midpoint of the side AiAi+1. (ii) If i ̸= 1, i ̸= n −1, and S is the intersection point of A1Ai+1 and AnAi,...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 17

IMO 1979 SL 17 Origin: NET Problem Inside an equilateral triangle ABC one constructs points P, Q and R such that \angleQAB = \anglePBA = 15◦, \angleRBC = \angleQCB = 20◦, \anglePCA = \angleRAC = 25◦. Determine the angles of triangle PQR. Solution Let K, L, and M be intersections of CQ and BR, AR and CP, and AQ and BP, respectively. Let \angleX denote the angle of the hexagon...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 4

IMO 1992 SL 4 Origin: CHN Problem Given nine points in space, no four of which are coplanar, ﬁnd the minimal natural number n such that for any coloring with red or blue of n edges drawn between these nine points there always exists a triangle having all edges of the same color. Solution There are 36 possible edges in total. If not more than 3 edges are left undrawn,...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 18

IMO 1990 SL 18 Origin: NOR Problem Let a, b be natural numbers with 1 \leqa \leqb, and M = / a+b . Deﬁne the function f : Z \toZ by f(n) = . n + a, if n < M, n −b, if n \geqM. Let f 1(n) = f(n), f i+1(n) = f(f i(n)), i = 1, 2, . . . . Find the smallest natural number k...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 4

IMO 1974 SL 4 Problem I 4 (USS 4) The sum of the squares of ﬁve real numbers a1, a2, a3, a4, a5 equals 1. Prove that the least of the numbers (ai −aj)2, where i, j = 1, 2, 3, 4, 5 and i ̸= j, does not exceed 1/10. Solution Assume w.l.o.g. that a1 \leqa2 \leqa3 \leqa4 \leqa5. If m is the least value of |ai −aj|, i...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 17

IMO 1993 SL 17 Origin: NET Problem Let n be an integer greater than 1. In a circular arrange- ment of n lamps L0, . . . , Ln−1, each one of that can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, Step0,Step1,. . . . If Lj−1 (j is taken mod n) is ON,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N2

IMO 1994 SL N2 Origin: AUS | Category: Number Theory Problem Determine all pairs (m, n) of positive integers such that n3+1 mn−1 is an integer. Solution Since mn −1 and m3 are relatively prime, mn −1 divides n3 + 1 if and only if it divides m3(n3 + 1) = (m3n3 −1) + m3 + 1. Thus n3 + 1 mn −1 \inZ ⇔m3 + 1 mn −1 \inZ;...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 13

IMO 1977 SL 13 Origin: POL Problem Let B be a set of k sequences each having n terms equal to 1 or −1. The product of two such sequences (a1, a2, . . . , an) and (b1, b2, . . . , bn) is deﬁned as (a1b1, a2b2, . . . , anbn). Prove that there exists a sequence (c1, c2, . . . , cn) such that...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N4

IMO 2004 SL N4 Origin: POL | Category: Number Theory Problem Let k be a ﬁxed integer greater than 1, and let m = 4k2 −5. Show that there exist positive integers a and b such that the sequence (xn) deﬁned by x0 = a, x1 = b, xn+2 = xn+1 + xn for n = 0, 1, 2, . . . has all of its terms relatively prime to...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A4

IMO 2001 SL A4 Origin: LIT | Category: Algebra Problem Find all functions f : R \toR satisfying f(xy)(f(x) −f(y)) = (x −y)f(x)f(y) for all x, y. Solution Let (∗) denote the given functional equation. Substituting y = 1 we get f(x)2 = xf(x)f(1). If f(1) = 0, then f(x) = 0 for all x, which is the trivial solution. Suppose f(1) = C ̸= 0. Let G = {y...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 8

IMO 1989 SL 8 Origin: FRA Problem Let R be a rectangle that is the union of a ﬁnite number of rectangles Ri, 1 \leqi \leqn, satisfying the following conditions: (i) The sides of every rectangle Ri are parallel to the sides of R. (ii) The interiors of any two diﬀerent Ri are disjoint. (iii) Every Ri has at least one side of integral length. Prove that R has at...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 12

IMO 1984 SL 12 Origin: NET Problem Find two positive integers a, b such that none of the num- bers a, b, a + b is divisible by 7 and (a + b)7 −a7 −b7 is divisible by 77. Solution By the binomial formula we have (a + b)7 −a7 −b7 = 7ab[(a5 + b5) + 3ab(a3 + b3) + 5a2b2(a + b)] = 7ab(a + b)(a2 + ab +...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 13

IMO 1991 SL 13 Origin: POL Problem Given any integer n \geq2, assume that the integers a1, a2, . . . , an are not divisible by n and, moreover, that n does not divide a1 + a2 + \cdot \cdot \cdot+an. Prove that there exist at least n diﬀerent sequences (e1, e2, \cdot \cdot \cdot , en) consisting of zeros or ones such that e1a1 + e2a2 + \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 3

IMO 1992 SL 3 Origin: CHN Problem The diagonals of a quadrilateral ABCD are perpendicular: AC\perpBD. Four squares, ABEF, BCGH, CDIJ, DAKL, are erected ex- ternally on its sides. The intersection points of the pairs of straight lines CL, DF; DF, AH; AH, BJ; BJ, CL are denoted by P1, Q1, R1, S1, respec- tively, and the intersection points of the pairs of straight lines AI, BK; BK, CE; CE,...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C1

IMO 1996 SL C1 Origin: FIN | Category: Combinatorics Problem We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB| = 20$, $|BC| = 12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G5

IMO 1996 SL G5 Origin: ARM | Category: Geometry Problem Let $ABCDEF$ be a convex hexagon such that $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel AF$. Let $R_A$, $R_C$, $R_E$ be the circumradii of triangles $FAB$, $BCD$, $DEF$ respectively, and let $P$ denote the perimeter of the hexagon. Prove that $$R_A + R_C + R_E \geq \frac{P}{2}.$$ Solution Let $a, b, c, d, e, f$ denote the lengths...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 7

IMO 1989 SL 7 Origin: FIN Problem Show that any two points lying inside a regular n-gon E can be joined by two circular arcs lying inside E and meeting at an angle of at least 1 −2 n \pi. Solution Assume that P1 and P2 are points inside E, and that the line P1P2 inter- sects the perimeter of E at Q1 and Q2. If we prove...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 8

IMO 1985 SL 8 Problem 1b.(TUR 5) Find the smallest positive integer n such that (i) n has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of n. Solution Among ten consecutive integers that divide n, there must exist numbers divisible by 23, 32, 5, and 7. Thus the desired number has the form n = 2\alpha13\alpha25\alpha37\alpha411\alpha5 \cdot \cdot \cdot , where...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N2

IMO 2004 SL N2 Origin: RUS | Category: Number Theory Problem The function \psi from the set N of positive integers into itself is deﬁned by the equality \psi(n) = n k=1 (k, n), n \inN, where (k, n) denotes the greatest common divisor of k and n. (a) Prove that \psi(mn) = \psi(m)\psi(n) for every two relatively prime m, n \in N. (b) Prove that for each a...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 11

IMO 1992 SL 11 Origin: JAP Problem In a triangle ABC, let D and E be the intersections of the bisec- tors of \angleABC and \angleACB with the sides AC, AB, respectively. Determine the angles \angleA, \angleB, \angleC if ∡BDE = 24◦, ∡CED = 18◦. Solution Let I be the incenter of \triangleABC. Since 90◦+ \alpha/2 = \angleBIC = \angleDIE = 138◦, we obtain that \angleA = 96◦. A B...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 11

IMO 1987 SL 11 Origin: POL Problem Find the number of partitions of the set {1, 2, . . ., n} into three subsets A1, A2, A3, some of which may be empty, such that the following conditions are satisﬁed: (i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have diﬀerent parity. (ii) If A1, A2, A3 are all...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 15

IMO 1990 SL 15 Origin: MEX Problem Determine for which positive integers k the set X = {1990, 1990 + 1, 1990 + 2, . . . , 1990 + k} can be partitioned into two disjoint subsets A and B such that the sum of the elements of A is equal to the sum of the elements of B. Solution Let S(Z) denote the sum of all the elements...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 11

IMO 1993 SL 11 Origin: IRE Problem Let n > 1 be an integer and let f(x) = xn + 5xn−1 + 3. Prove that there do not exist polynomials g(x), h(x), each having integer coeﬃcients and degree at least one, such that f(x) = g(x)h(x). Solution Due to the extended Eisenstein criterion, f must have an irreducible factor of degree not less than n −1. Since f has no...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C5

IMO 2003 SL C5 Origin: ROM | Category: Combinatorics Problem Every point with integer coordinates in the plane is the center of a disk with radius 1/1000. (a) Prove that there exists an equilateral triangle whose vertices lie in diﬀerent disks. (b) Prove that every equilateral triangle with vertices in diﬀerent disks has side length greater than 96. Solution (a) By the pigeonhole principle there are two diﬀerent integers x1,...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 21

IMO 1998 SL 21 Origin: CAN Problem Let a0, a1, a2, . . . be an increasing sequence of nonnegative inte- gers such that every nonnegative integer can be expressed uniquely in the form ai + 2aj + 4ak, where i, j, k are not necessarily distinct. Determine a1998. Solution Such a sequence is obviously strictly increasing. We note that it must be unique. Indeed, given a0, a1, . ....

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 3

IMO 1975 SL 3 Origin: USA Problem Find the integer represented by 109 n=1 n−2/3 . Here [x] denotes the greatest integer less than or equal to x (e.g. [ \sqrt 2] = 1). Solution From (k + 1)2/3 + (k + 1)1/3k1/3 + k2/3 (k + 1)1/3 −k1/3 = 1 and 3k2/3 < (k + 1)2/3+ (k + 1)1/3k1/3 + k2/3 < 3(k + 1)2/3 we obtain...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 15

IMO 1997 SL 15 Origin: RUS Problem An inﬁnite arithmetic progression whose terms are positive in- tegers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer. Solution Let a + bt, t = 0, 1, 2, . . ., be a given arithmetic progression that contains a square and a cube (a, b > 0). We use...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 25

IMO 1998 SL 25 Origin: NZL Problem Let U = {1, 2, . . ., n}, where n \geq3. A subset S of U is said to be split by an arrangement of the elements of U if an element not in S occurs in the arrangement somewhere between two elements of S. For example, 13542 splits {1, 2, 3} but not {3, 4, 5}. Prove that for any n...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 24

IMO 1997 SL 24 Origin: LIT Problem For a positive integer n, let f(n) denote the number of ways to represent n as the sum of powers of 2 with nonnegative integer exponents. Representations that diﬀer only in the ordering in their summands are not considered to be distinct. (For instance, f(4) = 4 because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.)...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C6

IMO 1999 SL C6 Origin: GBR | Category: Combinatorics Problem Suppose that every integer has been given one of the colors red, blue, green, yellow. Let x and y be odd integers such that |x| ̸= |y|. Show that there are two integers of the same color whose diﬀerence has one of the following values: x, y, x + y, x −y. Solution We denote colors by capital initial letters....

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C6

IMO 1994 SL C6 Origin: FIN | Category: Combinatorics Problem On an inﬁnite square grid, two players alternately mark sym- bols on empty cells. The ﬁrst player always marks X’s, the second O’s. One symbol is marked per turn. The ﬁrst player wins if there are 11 con- secutive X’s in a row, column, or diagonal. Prove that the second player can prevent the ﬁrst from winning. Solution Tile the...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 1

IMO 1990 SL 1 Origin: AUS Problem The integer 9 can be written as a sum of two consecutive integers: 9 = 4+5. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: 9 = 4 + 5 = 2 + 3 + 4. Is there an integer that can be written as a sum of 1990 consecutive integers and that...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 4

IMO 1993 SL 4 Origin: SPA Problem In the triangle ABC, let D, E be points on the side BC such that \angleBAD = \angleCAE. If M, N are, respectively, the points of tangency with BC of the incircles of the triangles ABD and ACE, show that MB + MD = NC + NE . Solution We note that p does not exceed one half of the circumradius of \triangleA′B′C′....

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G5

IMO 1999 SL G5 Origin: FRA | Category: Geometry Problem Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three circles three circles orthogonal to Ωpassing through B and C, A and C, and A and B respectively. The circles Ωa, Ωb meet again in C′; in the same way we obtain the points B′ and A′. Prove that the radius of the circumcircle of A′B′C′ is half...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 11

IMO 1990 SL 11 Problem (IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let M be a point of chord AB other than E. Draw the circle through D, E, and M. The tangent line to the circle DEM at E meets the lines BC, AC at F, G, respectively. Given AM AB = \lambda, ﬁnd GE EF . Solution Assume B(A, E, M,...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 22

IMO 1988 SL 22 Origin: KOR Problem Let p be the product of two consecutive integers greater than Solution Set X = p i=1 xi and w.l.o.g. assume that X \geq0 (if (x1, . . . , xp) is a solution, then (−x1, . . . , −xp) is a solution too). Since x2 \geqx for all integers x, it follows that p i=1 x2 i \geqX. If the last...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 26

IMO 1995 SL 26 Problem S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which xn n = n−1 j=0 xj n for n = 1, 2, 3, . . .. Prove that for all n, 2 − 2n−1 \leqxn < 2 −1 2n . Solution For n = 1 the result is trivial, since x1 = 1. Suppose now that n \geq2...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 6

IMO 1972 SL 6 Origin: GDR Problem Show that for any n ̸\equiv0 (mod 10) there exists a multiple of n not containing the digit 0 in its decimal expansion. Solution Let n = 2\alpha5\betam, where \alpha = 0 or \beta = 0. These two cases are analogous, and we treat only \alpha = 0, n = 5\betam. The case m = 1 is settled by the following lemma. Lemma....

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 7

IMO 1970 SL 7 Origin: USS Problem For which digits a do exist integers n \geq4 such that each digit of n(n+1) equals a? Solution For a = 5 one can take n = 10, while for a = 6 one takes n = 11. Now assume a ̸\in{5, 6}. If there exists an integer n such that each digit of n(n + 1)/2 is equal to a, then there...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 17

IMO 1983 SL 17 Origin: GDR Problem Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove that max 1\leqi<j\leqn PiPj > \sqrt 2 (\sqrtn −1) min 1\leqi<j\leqn PiPj. Solution Set a = min PiPj, b = max PiPj. We use the following lemma. Lemma. There exists a disk of radius less than or equal to b/ \sqrt 3 containing all the Pi’s....

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G1

IMO 1999 SL G1 Origin: ARM | Category: Geometry Problem Let ABC be a triangle and M an interior point. Prove that min{MA, MB, MC} + MA + MB + MC < AB + AC + BC. Solution We use the following simple lemma. Lemma. Suppose that M is the interior point of a convex quadrilateral ABCD. Then it follows that MA + MB < AD + DC + CB....

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 12

IMO 1986 SL 12 Origin: GDR Problem To each vertex Pi (i = 1, . . . , 5) of a pentagon an integer xi is assigned, the sum s = xi being positive. The following operation is allowed, provided at least one of the xi’s is negative: Choose a negative xi, replace it by −xi, and add the former value of xi to the integers assigned to the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G4

IMO 1999 SL G4 Origin: GBR | Category: Geometry Problem For a triangle T = ABC we take the point X on the side (AB) such that AX/XB = 4/5, the point Y on the segment (CX) such that CY = 2Y X, and, if possible, the point Z on the ray (CA such that ∡CXZ = 180◦−∡ABC. We denote by \Sigma the set of all triangles T for which...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 5

IMO 1991 SL 5 Origin: SPA Problem In the triangle ABC, with ∡A = 60◦, a parallel IF to AC is drawn through the incenter I of the triangle, where F lies on the side AB. The point P on the side BC is such that 3BP = BC. Show that ∡BFP = ∡B/2. Solution Let P1 be the point on the side BC such that \angleBFP1 = \beta/2. Then...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C1

IMO 2002 SL C1 Origin: COL | Category: Combinatorics Problem Let n be a positive integer. Each point (x, y) in the plane, where x and y are nonnegative integers with x + y \leqn, is colored red or blue, subject to the following condition: If a point (x, y) is red, then so are all points (x′, y′) with x′ \leqx and y′ \leqy. Let A be the number...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C3

IMO 2001 SL C3 Origin: RUS | Category: Combinatorics Problem Deﬁne a k-clique to be a set of k people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining. Solution Consider...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 27

IMO 1988 SL 27 Origin: GBR Problem The triangle ABC is acute-angled. Let L be any line in the plane of the triangle and let u, v, w be the lengths of the perpendiculars from A, B, C respectively to L. Prove that u2 tan A + v2 tan B + w2 tan C \geq2∆, where ∆is the area of the triangle, and determine the lines L for which equality...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 2

IMO 1993 SL 2 Origin: CAN Problem Let triangle ABC be such that its circumradius R is equal to Solution It is well known that r \leq1 2R. Therefore 1 3(1 + r)2 \leq1 1 + 1 2 = 3 4. It remains only to show that p \leq

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C6

IMO 2002 SL C6 Origin: POL | Category: Combinatorics Problem Let n be an even positive integer. Show that there is a permutation x1, x2, . . . , xn of 1, 2, . . . , n such that for every 1 \leqi \leqn the number xi+1 is one of 2xi, 2xi −1, 2xi −n, 2xi −n −1 (where we take xn+1 = x1). Solution Write n = 2m....

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 15

IMO 1981 SL 15 Origin: GBR Problem Find the point P inside the triangle ABC for which BC PD + CA PE + AB PF is minimal, where PD, PE, PF are the perpendiculars from P to BC, CA, AB respectively. Solution Set BC = a, CA = b, AB = c, and denote the area of \triangleABC by P, and a/PD + b/PE + c/PF by S. Since a...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C6

IMO 1996 SL C6 Origin: CAN | Category: Combinatorics Problem A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one bean is chosen. Two beans are taken from this square; one of them is placed on the square immediately to the left, and the other is placed on the square...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 27

IMO 1995 SL 27 Problem S5 (FIN) For positive integers n, the numbers f(n) are deﬁned induc- tively as follows: f(1) = 1, and for every positive integer n, f(n+1) is the greatest integer m such that there is an arithmetic progression of positive integers a1 < a2 < \cdot \cdot \cdot < am = n for which f(a1) = f(a2) = \cdot \cdot \cdot = f(am). Prove that there...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G2

IMO 2001 SL G2 Origin: KOR | Category: Geometry Problem In acute triangle ABC with circumcenter O and altitude AP, ∡C \geq∡B + 30◦. Prove that ∡A + ∡COP < 90◦. Solution Since \angleOCP = 90◦−\angleA, we are led to showing that \angleOCP > \angleCOP, i.e., OP > CP. By the triangle inequality it suﬃces to prove CP < 1 2CO. Let CO = R. The law of sines yields...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C3

IMO 1996 SL C3 Origin: USA | Category: Combinatorics Problem Let $k, m, n$ be integers such that $1 < n \leq m-1 \leq k$. Determine the maximum size of a subset $S$ of the set ${1, 2, 3, \ldots, k}$ such that no $n$ distinct elements of $S$ add up to $m$. Solution Denote the required maximum size by $M_k(m, n)$. If $m < \frac{n(n+1)}{2}$, then trivially $M_k(m, n)...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 19

IMO 1981 SL 19 Origin: YUG Problem A ﬁnite set of unit circles is given in a plane such that the area of their union U is S. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that 2S 9 . Solution Consider the partition of plane \pi into regular hexagons, each having in- radius 2. Fix one of these...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 2

IMO 1983 SL 2 Origin: BEL Problem Let n be a positive integer. Let \sigma(n) be the sum of the natural divisors d of n (including 1 and n). We say that an integer m \geq1 is superabundant (P.Erd¨os, 1944) if \forallk \in{1, 2, . . ., m −1}, \sigma(m) m \sigma(k) k . Prove that there exists an inﬁnity of superabundant numbers. Solution By deﬁnition, \sigma(n) = d|n...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 16

IMO 1997 SL 16 Origin: BLR Problem In an acute-angled triangle ABC, let AD, BE be altitudes and AP, BQ internal bisectors. Denote by I and O the incenter and the cir- cumcenter of the triangle, respectively. Prove that the points D, E, and I are collinear if and only if the points P, Q, and O are collinear. Solution Let da(X), db(X), dc(X) denote the distances of a point...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 9

IMO 1991 SL 9 Origin: FRA Problem In the plane we are given a set E of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of E, there exist at least 1593 other points of E to which it is joined by a path. Show that there exist six points of E every pair of which are joined by...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 8

IMO 1986 SL 8 Origin: USA Problem From a collection of n persons q distinct two-member teams are selected and ranked 1, . . . , q (no ties). Let m be the least integer larger than or equal to 2q/n. Show that there are m distinct teams that may be listed so that (i) each pair of consecutive teams on the list have one member in common and (ii)...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 21

IMO 1988 SL 21 Origin: POL Problem Forty-nine students solve a set of three problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students A and B such that for each problem, A will score at least as many points as B. Solution Let X be the set of all ordered triples a = (a1, a2, a3) for...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 5

IMO 1988 SL 5 Origin: CZS Problem Let n be an even positive integer. Let A1, A2, . . . , An+1 be sets having n elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which n can one assign to every element of the union one of the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G8

IMO 2001 SL G8 Origin: ISR | Category: Geometry Problem Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect \angleBAC and let BQ bisect \angleABC, with P on BC and Q on AC. If AB + BP = AQ + QB, what are the angles of the triangle? Solution Let S and T respectively be the points on the extensions of AB and AQ over B and...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 22

IMO 1968 SL 22 Origin: CZS Problem Find all positive integers $x$ for which $p(x) = x^2 - 10x - 22$, where $p(x)$ denotes the product of the digits of $x$.

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 11

IMO 1991 SL 11 Origin: AUS Problem Prove that m=0 (−1)m 1991 −m 1991 −m m Solution To start with, observe that n−m n−m m = 1 n n−m m + n−m−1 m−1 . For n = 1, 2, . . . set Sn = [n/2] m=0(−1)mn−m m . Using the identity m k m−1 k + m−1 k−1 we obtain...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C2

IMO 1999 SL C2 Origin: CAN | Category: Combinatorics Problem (a) If a 5 \times n rectangle can be tiled using n pieces like those shown in the diagram, prove that n is even. (b) Show that there are more than 2\cdot3k−1 ways to tile a ﬁxed 5\times2k rect- angle (k \geq3) with 2k pieces. (Symmetric constructions are considered to be diﬀerent.) Solution (a) Color the ﬁrst, third, and ﬁfth...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 4

IMO 1973 SL 4 Origin: GBR Problem Let P be a set of 7 diﬀerent prime numbers and C a set of 28 diﬀerent composite numbers each of which is a product of two (not necessarily diﬀerent) numbers from P. The set C is divided into 7 disjoint four-element subsets such that each of the numbers in one set has a com- mon prime divisor with at least two other...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 8

IMO 1971 SL 8 Origin: NET Problem Determine whether there exist distinct real numbers a, b, c, t for which: (i) the equation ax2 + btx + c = 0 has two distinct real roots x1, x2, (ii) the equation bx2 + ctx + a = 0 has two distinct real roots x2, x3, (iii) the equation cx2 + atx + b = 0 has two distinct real roots x3,...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 5

IMO 1985 SL 5 Origin: ROM Problem Let D be the interior of the circle C and let A \inC. Show that the function f : D \toR, f(M) = |MA| |MM′|, where M ′ = (AM \capC, is strictly convex; i.e., f(P) < f(M1)+f(M2) , \forallM1, M2 \inD, M1 ̸= M2, where P is the midpoint of the segment M1M2. Solution Let w.l.o.g. circle C have unit radius. For...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 14

IMO 1986 SL 14 Origin: IRE Problem The circle inscribed in a triangle ABC touches the sides BC, CA, AB in D, E, F, respectively, and X, Y, Z are the midpoints of EF, FD, DE, respectively. Prove that the centers of the inscribed circle and of the circles around XY Z and ABC are collinear. Solution We shall use the following simple fact. Lemma. If <k is the image...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 4

IMO 1970 SL 4 Origin: CZS Problem For what natural numbers n can the product of some of the numbers n, n + 1, n + 2, n + 3, n + 4, n + 5 be equal to the product of the remaining ones? Solution Suppose that n is such a natural number. If a prime number p divides any of the numbers n, n+1, . . ., n+5,...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 14

IMO 1990 SL 14 Origin: JAP Problem In the coordinate plane a rectangle with vertices (0, 0), (m, 0), (0, n), (m, n) is given where both m and n are odd integers. The rectangle is partitioned into triangles in such a way that (i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form x...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C3

IMO 1999 SL C3 Origin: GBR | Category: Combinatorics Problem A biologist watches a chameleon. The chameleon catches ﬂies and rests after each catch. The biologist notices that: (i) the ﬁrst ﬂy is caught after a resting period of one minute; (ii) the resting period before catching the 2mth ﬂy is the same as the resting period before catching the mth ﬂy and one minute shorter than the resting period...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 22

IMO 1993 SL 22 Origin: GBR Problem A, B, C, D are four points in the plane, with C, D on the same side of the line AB, such that AC \cdot BD = AD \cdot BC and ∡ADB = 90◦+ ∡ACB. Find the ratio AB \cdot CD AC \cdot BD, and prove that circles ACD, BCD are orthogonal. (Intersecting circles are said to be orthogonal if at either common...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A2

IMO 2000 SL A2 Origin: GBR | Category: Algebra Problem Let a, b, c be positive integers satisfying the conditions b > 2a and c > 2b. Show that there exists a real number t with the property that all the three numbers ta, tb, tc have their fractional parts lying in the interval (1/3, 2/3]. Solution We note that {ta} lies in 1 3, 2 if and only...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N6

IMO 1994 SL N6 Origin: LAT | Category: Number Theory Problem Let x1 and x2 be relatively prime positive integers. For n \geq2, deﬁne xn+1 = xnxn−1 + 1. (a) Prove that for every i > 1, there exists j > i such that xi i divides xj j. (b) Is it true that x1 must divide xj j for some j > 1? Solution (a) Let p be a...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 6

IMO 1979 SL 6 Origin: CZS Problem Find the real values of p for which the equation 2p + 1 −x2 + 3x + p + 4 = x2 + 9x + 3p + 9 in x has exactly two real distinct roots ( \sqrt t means the positive square root of t). Solution Setting q = x2 + x −p, the given equation becomes (x + 1)2 −2q +...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A5

IMO 2001 SL A5 Origin: BUL | Category: Algebra Problem Find all positive integers a1, a2, . . . , an such that 100 = a0 a1 a1 a2 \cdot \cdot \cdot + an−1 an , where a0 = 1 and (ak+1 −1)ak−1 \geqa2 k(ak −1) for k = 1, 2, . . ., n −1. Solution Let a1, a2, . . . , an satisfy the conditions of the...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C7

IMO 1996 SL C7 Origin: IRE | Category: Combinatorics Problem Let $U$ be a finite set and let $f$, $g$ be bijective functions from $U$ onto itself. Let $$S={w\in U:f(f(w))=g(g(w))},$$ $$T={w\in U:f(g(w))=g(f(w))},$$ and suppose that $U=S\cup T$. Prove that for $w\in U$, $f(w)\in S$ if and only if $g(w)\in S$. Solution For convenience, we shall write $f^2$, $fg$, $\ldots$ for the functions $f\circ f$, $f\circ g$, $\ldots$. We need two...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 31

IMO 1989 SL 31 Origin: SWE Problem Let a1 \geqa2 \geqa3 be given positive integers and let N(a1, a2, a3) be the number of solutions (x1, x2, x3) of the equation a1 x1 a2 x2 a3 x3 = 1, where x1, x2, and x3 are positive integers. Show that N(a1, a2, a3) \leq6a1a2(3 + ln(2a1)). Solution Let us denote by Npqr the number of solutions for which ap/xp \geqaq/xq \geq...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 3

IMO 1988 SL 3 Origin: CAN Problem The triangle ABC is inscribed in a circle. The interior bi- sectors of the angles A, B, and C meet the circle again at A′, B′, and C′ respectively. Prove that the area of triangle A′B′C′ is greater than or equal to the area of triangle ABC. Solution Let R be the circumradius, r the inradius, s the semiperimeter, ∆the area of ABC...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N3

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N6

IMO 2001 SL N6 Origin: RUS | Category: Number Theory Problem Is it possible to ﬁnd 100 positive integers not exceeding 25,000 such that all pairwise sums of them are diﬀerent? Solution Yes. The desired result is an immediate consequence of the following fact applied on p = 101. Lemma. For any odd prime number p, there exist p nonnegative integers less than 2p2 with all pairwise sums mutually distinct....

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 17

IMO 1990 SL 17 Origin: NET Problem Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let A be the beginning vertex and B be the end vertex. Let there be p \times q...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 13

IMO 1973 SL 13 Origin: YUG Problem Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1. Solution Let S1, S2, S3, S4 denote the areas of the faces of the tetrahedron, V its volume, h1, h2, h3, h4 its altitudes, and r the radius of its inscribed sphere. Since 3V = S1h1 = S2h2...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 5

IMO 1989 SL 5 Origin: COL Problem Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an having all real roots. If r16 1 + r16 2 + \cdot \cdot \cdot + r16 n = n, where the rj are the roots of p(x), ﬁnd all such roots. Solution According to the Cauchy–Schwarz inequality, n i=1 ai \leq n i=1 a2 i n i=1 = n n ...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G1

IMO 2001 SL G1 Origin: UKR | Category: Geometry Problem Let A1 be the center of the square inscribed in acute triangle ABC with two vertices of the square on side BC. Thus one of the two re- maining vertices of the square is on side AB and the other is on AC. Points B1, C1 are deﬁned in a similar way for inscribed squares with two vertices on sides...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N7

IMO 1995 SL N7 Origin: BLR | Category: Number Theory Problem Does there exist an integer n > 1 that satisﬁes the following condition? The set of positive integers can be partitioned into n nonempty subsets such that an arbitrary sum of n −1 integers, one taken from each of any n −1 of the subsets, lies in the remaining subset. Solution We shall show that there is no such...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A6

IMO 2001 SL A6 Origin: KOR | Category: Algebra Problem Prove that for all positive real numbers a, b, c, a \sqrt a2 + 8bc + a \sqrt b2 + 8ca + c \sqrt c2 + 8ab \geq1. Solution We shall determine a constant k > 0 such that a \sqrt a2 + 8bc \geq ak ak + bk + ck for all a, b, c > 0. (1) This...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N3

IMO 1996 SL N3 Origin: BUL | Category: Number Theory Problem A finite sequence of integers $a_0, a_1, \ldots, a_n$ is called quadratic if for each $i \in {1, 2, \ldots, n}$ we have the equality $|a_i - a_{i-1}| = i^2$. (a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence with $a_0 = b$ and $a_n = c$. (b)...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G8

IMO 2000 SL G8 Origin: RUS | Category: Geometry Problem A1A2A3 is an acute-angled triangle. The foot of the altitude from Ai is Ki, and the incircle touches the side opposite Ai at Li. The line K1K2 is reﬂected in the line L1L2. Similarly, the line K2K3 is reﬂected in L2L3, and K3K1 is reﬂected in L3L1. Show that the three new lines form a triangle with vertices on the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A1

IMO 1999 SL A1 Origin: POL | Category: Algebra Problem Let n \geq2 be a ﬁxed integer. Find the least constant C such that the inequality i<j xixj(x2 i + x2 j) \leqC i xi holds for every x1, . . . , xn \geq0 (the sum on the left consists of n sum- mands). For this constant C, characterize the instances of equality. Solution For all...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 7

IMO 1977 SL 7 Origin: GBR Problem Let a, b, A, B be given constant real numbers and f(x) = 1 −a cos x −b sin x −A cos 2x −B sin 2x. Prove that if f(x) \geq0 for all real x, then a2 + b2 \leq2 and A2 + B2 \leq1.

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 29

IMO 1988 SL 29 Origin: USA Problem A number of signal lights are equally spaced along a one-way railroad track, labeled in order 1, 2, . . ., N (N \geq2). As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C6

IMO 2000 SL C6 Origin: FRA | Category: Combinatorics Problem Let p and q be relatively prime positive integers. A subset S of {0, 1, 2, . . .} is called ideal if 0 \inS and for each element n \inS, the integers n + p and n + q belong to S. Determine the number of ideal subsets of {0, 1, 2 . . .}. Solution Every integer z...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 17

IMO 1971 SL 17 Origin: YUG Problem Prove the inequality a1 + a3 a1 + a2 a2 + a4 a2 + a3 a3 + a1 a3 + a4 a4 + a2 a4 + a1 \geq4, where ai > 0, i = 1, 2, 3, 4. Solution We use the following obvious consequences of (a + b)2 \geq4ab: (a1 + a2)(a3 + a4) \geq (a1 + a2 + a3 + a4)2...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N3

IMO 2002 SL N3 Origin: MON | Category: Number Theory Problem Let p1, p2, . . . , pn be distinct primes greater than 3. Show that 2p1p2\cdot\cdot\cdotpn + 1 has at least 4n divisors. Solution We observe that if a, b are coprime odd numbers, then gcd(2a+1, 2b+1) = 3. In fact, this g.c.d. divides gcd(22a−1, 22b−1) = 2gcd(2a,2b)−1 = 22−1 = 3, while 3 obviously divides both 2a...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 15

IMO 1984 SL 15 Origin: LUX Problem Angles of a given triangle ABC are all smaller than 120◦. Equilateral triangles AFB, BDC and CEA are constructed in the exterior of \triangleABC. (a) Prove that the lines AD, BE, and CF pass through one point S. (b) Prove that SD + SE + SF = 2(SA + SB + SC). Solution (a) Since rotation by 60◦around A transforms the triangle CAF...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A4

IMO 1999 SL A4 Origin: BLR | Category: Algebra Problem Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that for any two integers x, y taken from two diﬀerent subsets, the number x2 −xy +y2 belongs to the third subset. Solution Deﬁne f(x, y) = x2 −xy + y2. Let us assume that three such sets A, B, and C do exist and...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 10

IMO 1990 SL 10 Origin: ICE Problem A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone. Solution Let r be the radius of the base and h the height of the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A3

IMO 1999 SL A3 Origin: FIN | Category: Algebra Problem A game is played by n girls (n \geq2), everybody having a ball. Each of the n pairs of players, in an arbitrary order, exchange the balls they have at that moment. The game is called nice if at the end nobody has her own ball, and it is called tiresome if at the end everybody has her initial...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A6

IMO 2000 SL A6 Origin: IRE | Category: Algebra Problem A nonempty set A of real numbers is called a B3-set if the conditions a1, a2, a3, a4, a5, a6 \inA and a1+a2+a3 = a4+a5+a6 imply that the sequences (a1, a2, a3) and (a4, a5, a6) are identical up to a permutation. Let A = {a0 = 0 < a1 < a2 < \cdot \cdot \cdot }, B = {b0...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 3

IMO 1976 SL 3 Origin: CZS Problem In a convex quadrangle with area 32 cm2, the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to 16 cm. (a) What is the length of the other diagonal? (b) What are the lengths of the edges of the quadrangle if the perimeter is a minimum? (c) Is it possible to choose the edges...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C3

IMO 2004 SL C3 Origin: AUS | Category: Combinatorics Problem The following operation is allowed on a ﬁnite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a ﬁxed integer n \geq4, ﬁnd the least number of edges of a graph that can be obtained by repeated ap- plications of this operation from...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 22

IMO 1979 SL 22 Origin: USS Problem There are two circles in the plane. Let a point A be one of the points of intersection of these circles. Two points begin moving simultaneously with constant speeds from the point A, each point along its own circle. The two points return to the point A at the same time. Prove that there is a point P in the plane such that...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 12

IMO 1982 SL 12 Problem B6 (FIN 3) Four distinct circles C, C1, C2, C3 and a line L are given in the plane such that C and L are disjoint and each of the circles C1, C2, C3 touches the other two, as well as C and L. Assuming the radius of C to be 1, determine the distance between its center and L. Solution Let y be the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G5

IMO 1995 SL G5 Origin: NZL | Category: Geometry Problem Let ABCDEF be a convex hexagon with AB = BC = CD, DE = EF = FA, and ∡BCD = ∡EFA = \pi/3 (that is, 60◦). Let G and H be two points interior to the hexagon such that angles AGB and DHE are both 2\pi/3 (that is, 120◦). Prove that AG + GB + GH + DH + HE...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C4

IMO 2002 SL C4 Origin: BUL | Category: Combinatorics Problem Let T be the set of ordered triples (x, y, z), where x, y, z are integers with 0 \leqx, y, z \leq9. Players A and B play the following guessing game: Player A chooses a triple (x, y, z) in T , and Player B has to discover A’s triple in as few moves as possible. A move consists...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 25

IMO 1990 SL 25 Origin: TUR Problem Let Q+ be the set of positive rational numbers. Construct a function f : Q+ \toQ+ such that f(xf(y)) = f(x) y , for all x, y in Q+. Solution Plugging in x = 1 we get f(f(y)) = f(1)/y and hence f(y1) = f(y2) implies y1 = y2 i.e. that the function is bijective. Plugging in y = 1 gives us f(xf(1))...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 14

IMO 1973 SL 14 Origin: YUG Problem A soldier has to investigate whether there are mines in an area that has the form of an equilateral triangle. The radius of his detector is equal to one-half of an altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path that the soldier has to traverse in order to check the whole region. Solution Suppose...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 14

IMO 1975 SL 14 Origin: YUG Problem Let x0 = 5 and xn+1 = xn + xn (n = 0, 1, 2, . . .). Prove that 45 < x1000 < 45, 1. Solution We shall prove that for every n \inN, \sqrt2n + 25 \leqxn \leq\sqrt2n + 25 + 0.1. Note that for n = 1000 this gives us exactly the desired inequalities. First, notice that the recurrent relation...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 13

IMO 1985 SL 13 Problem 4a.(BUL 1) Let m boxes be given, with some balls in each box. Let n < m be a given integer. The following operation is performed: choose n of the boxes and put 1 ball in each of them. Prove: (a) If m and n are relatively prime, then it is possible, by performing the operation a ﬁnite number of times, to arrive at the...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 24

IMO 1998 SL 24 Origin: SWE Problem Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 916532748 may be changed to 913562748. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 31

IMO 1988 SL 31 Origin: USS Problem Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a diﬀerent order. Prove that there are at least two people such that the number of participants sitting between them before and after the break is the same. Solution Denote the number of participants by 2n, and assign to each...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N7

IMO 2003 SL N7 Origin: BRA | Category: Number Theory Problem The sequence a0, a1, a2, . . . is deﬁned as follows: a0 = 2, ak+1 = 2a2 k −1 for k \geq0. Prove that if an odd prime p divides an, then 2n+3 divides p2 −1. Solution Deﬁne the sequence xk of positive reals by ak = cosh xk (cosh is the hyperbolic cosine deﬁned by cosh t...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 21

IMO 1993 SL 21 Origin: GBR Problem A circle S is said to cut a circle \Sigma diametrally if their common chord is a diameter of \Sigma. Let SA, SB, SC be three circles with distinct centers A, B, C respectively. Prove that A, B, C are collinear if and only if there is no unique circle S that cuts each of SA, SB, SC diametrally. Prove further that if...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 4

IMO 1989 SL 4 Origin: BUL Problem Prove that for every integer n > 1 the equation xn n! + xn−1 (n −1)! + \cdot \cdot \cdot + x2 2! + x 1! + 1 = 0 has no rational roots. Solution First we note that for every integer k > 0 and prime number p, pk doesn’t divide k!. This follows from the fact that the highest exponent r...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 15

IMO 1988 SL 15 Origin: ICE Problem Let ABC be an acute-angled triangle. Three lines LA, LB, and LC are constructed through the vertices A, B, and C respectively according to the following prescription: Let H be the foot of the altitude drawn from the vertex A to the side BC; let SA be the circle with diameter AH; let SA meet the sides AB and AC at M and...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 12

IMO 1990 SL 12 Origin: IRE Problem Let ABC be a triangle and L the line through C parallel to the side AB. Let the internal bisector of the angle at A meet the side BC at D and the line L at E and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If GF = DE,...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N5

IMO 2001 SL N5 Origin: BUL | Category: Number Theory Problem Let a > b > c > d be positive integers and suppose ac + bd = (b + d + a −c)(b + d −a + c). Prove that ab + cd is not prime. Solution The given equality is equivalent to a2 −ac + c2 = b2 + bd + d2. Hence (ab + cd)(ad + bc)...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 7

IMO 1997 SL 7 Origin: RUS Problem Let ABCDEF be a convex hexagon such that AB = BC, CD = DE, EF = FA. Prove that BC BE + DE DA + FA FC \geq3 2. When does equality occur? Solution Let us set AC = a, CE = b, EA = c. Applying Ptolemy’s inequality for the quadrilateral ACEF we get AC \cdot EF + CE \cdot AF \geqAE...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 12

IMO 1987 SL 12 Origin: POL Problem Given a nonequilateral triangle ABC, the vertices listed coun- terclockwise, ﬁnd the locus of the centroids of the equilateral triangles A′B′C′ (the vertices listed counterclockwise) for which the triples of points A, B′, C′; A′, B, C′; and A′, B′, C are collinear. Solution Here all angles will be oriented and measured counterclockwise. Note that ∡CA′B = ∡AB′C = ∡BC′A = \pi/3. Let...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N1

IMO 1999 SL N1 Origin: TWN | Category: Number Theory Problem Find all pairs of positive integers (x, p) such that p is a prime, x \leq2p , and xp−1 is a divisor of (p −1)x + 1. Solution Obviously (1, p) (where p is an arbitrary prime) and (2, 2) are solutions and the only solutions to the problem for x < 3 or p < 3. Let us...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 21

IMO 1989 SL 21 Origin: NET Problem Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than 120◦. Solution In order to obtain a triangle as the intersection we must have three points P, Q, R on three sides of the tetrahedron passing through one vertex, say T . It is...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 15

IMO 1975 SL 15 Origin: USS Problem Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)? Solution Assume that the center of the circle is at the origin O(0, 0), and that the points A1, A2, . . . , A1975 are arranged on the upper...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 20

IMO 1982 SL 20 Problem C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri- angles ABM, CDP, BCN, ADQ, the ﬁrst two outward and the other two inward. Prove that MN = AC. What can be said about the quadrilateral MNPQ? Solution Since MN is the image of AC under rotation about B for 60◦, we have MN = AC. Similarly, PQ is the image of...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 11

IMO 1988 SL 11 Origin: GDR Problem The lock on a safe consists of three wheels, each of which may be set in eight diﬀerent positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations that must be tried if one is to guarantee being able to open...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N2

IMO 1995 SL N2 Origin: RUS | Category: Number Theory Problem Let Z denote the set of all integers. Prove that for any integers A and B, one can ﬁnd an integer C for which M1 = {x2 +Ax+B : x \inZ} and M2 = {2x2 + 2x + C : x \inZ} do not intersect. Solution If A is odd, then every number in M1 is of the form...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 24

IMO 1988 SL 24 Origin: SWE Problem Let {ak}\infty be a sequence of nonnegative real numbers such that ak −2ak+1 + ak+2 \geq0 and k j=1 aj \leq1 for all k = 1, 2, . . . . Prove that 0 \leq(ak −ak+1) < k2 for all k = 1, 2, . . . . Solution The ﬁrst condition means in fact that ak−ak+1 is decreasing. In particular, if ak...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 23

IMO 1991 SL 23 Origin: IND Problem Let f and g be two integer-valued functions deﬁned on the set of all integers such that (a) f(m + f(f(n))) = −f(f(m + 1) −n for all integers m and n; (b) g is a polynomial function with integer coeﬃcients and g(n) = g(f(n)) for all integers n. Determine f(1991) and the most general form of g. Solution From (i), replacing m...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 16

IMO 1982 SL 16 Problem C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the equation x3 −3xy2 + y3 = n has a solution in integers (x, y), then it has at least three such solutions. Show that the equation has no solution in integers when n = 2891. Solution It is easy to verify that whenever (x, y) is a solution of the equation...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 2

IMO 1974 SL 2 Problem I 2 (POL 1) Prove that the squares with sides 1/1, 1/2, 1/3, . . . may be put into the square with side 3/2 in such a way that no two of them have any interior point in common. Solution We denote by qi the square with side 1 i . Let us divide the big square into rectangles ri by parallel lines, where...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 14

IMO 1985 SL 14 Problem 4b.(IRE 4) A set of 1985 points is distributed around the circumference of a circle and each of the points is marked with 1 or −1. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 2

IMO 1960 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m41s Problem For which real numbers $x$ does the following inequality hold: $$\frac{4x^2}{\left(1 - \sqrt{1+2x}\right)^2} < 2x + 9\,?$$ Exploration The expression contains the denominator $$\left(1-\sqrt{1+2x}\right)^2,$$ so the first task is to determine the domain. The square root requires $$1+2x\ge 0,$$ hence $$x\ge -\frac12.$$ However, the denominator also vanishes when $$1-\sqrt{1+2x}=0.$$ This gives $$\sqrt{1+2x}=1,$$ hence...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 1

IMO 1960 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 27m41s Problem Find all three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the digits of the initial number. Problem Understanding We seek all three-digit integers whose quotient upon division by $11$ equals the sum of the squares of their digits. Let the number be $$N=100a+10b+c,$$...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 1

IMO 1991 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 18m14s Problem Given a triangle $,ABC,,$ let $,I,$ be the center of its inscribed circle. The internal bisectors of the angles $,A,B,C,$ meet the opposite sides in $,A^{\prime },B^{\prime },C^{\prime },$ respectively. Prove that $$ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. $$ Problem Understanding We must...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 1

IMO 1990 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m49s Problem Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If $$ \frac {AM}{AB}...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 1

IMO 1986 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m25s Problem Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set ${2,5,13,d}$ such that $ab-1$ is not a perfect square. Exploration The problem asks us to show that for any positive integer $d \neq 2,5,13$, there exist distinct $a,b \in {2,5,13,d}$...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 3

IMO 1974 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m13s Problem Prove that the number $\textstyle \sum_{k = 0}^n \tbinom{2n+1}{2k+1} 2^{3k}$ is not divisible by 5 for any integer $n \geq 0$. Exploration For $n=0$ the sum equals $$\binom11 2^0=1,$$ which is not divisible by $5$. For $n=1$, $$\binom31 2^0+\binom33 2^3=3+8=11,$$ and $11\not\equiv0\pmod5$. For $n=2$, $$\binom51+\binom53 2^3+\binom55 2^6 =5+10\cdot8+64 =149,$$ and $$149\equiv4\pmod5.$$ For $n=3$,...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 5

IMO 1979 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 4m30s Problem Determine all real numbers a for which there exists non-negative reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\sum_{k=1}^{5} kx_{k}=a,$ $\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$ Exploration The three equalities resemble moment conditions. If one defines $$m_{1}=\sum_{k=1}^{5}kx_{k}, \qquad m_{3}=\sum_{k=1}^{5}k^{3}x_{k}, \qquad m_{5}=\sum_{k=1}^{5}k^{5}x_{k},$$ then the problem asks for all real numbers $a$ such that $$m_{1}=a,\qquad m_{3}=a^{2},\qquad m_{5}=a^{3}.$$...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 2

IMO 2024 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 33m38s Problem Find all positive integer pairs $(a,b),$ such that there exists positive integers $g,N,$ such that $$ \gcd (a^n+b,b^n+a)=g $$ holds for all integer $n\ge N$. Exploration Define $$d_n=\gcd(a^n+b,b^n+a).$$ The previous solution failed because it incorrectly asserted $$a+b\mid d_n \quad\text{for odd }n.$$ Before proceeding, we test this carefully. Take $(a,b)=(2,3)$. For $n=1$, $$d_1=\gcd(5,5)=5.$$ For...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 4

IMO 1979 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 16m50s Problem We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which$$ \frac{QP+PR}{QR} $$is maximum. Exploration The quantity to maximize is $$\frac{QP+PR}{QR},$$ where $P$ and $R$ lie in the plane $p$, while $Q\notin p$. Since the numerator contains $PR$ and the...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 2

IMO 1974 Problem 2 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 32m46s Problem In the triangle $ABC$, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $$ \sin{A} \sin{B} \leq \sin^2 \frac{C}{2}. $$ Exploration The reviewers correctly identified two independent problems. The first problem is the claimed factorization of the...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 3

IMO 1979 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m28s Problem Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 4

IMO 1968 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 11m45s Problem Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle. Exploration Consider a tetrahedron with vertices $A, B, C, D$ and edges $AB, AC, AD, BC, BD, CD$, all positive. The claim is that there exists a vertex...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 2

IMO 1979 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 43m03s Problem We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side....

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 3

IMO 1961 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 39m28s Problem Solve the equation $\cos^n{x} - \sin^n{x} = 1$ where n is a given positive integer. Exploration We seek all real numbers $x$ satisfying $$\cos^n x-\sin^n x=1,$$ where $n\in\mathbb N$ is fixed. Since $\sin x,\cos x\in[-1,1]$, the left-hand side is bounded. Equality with $1$ suggests an extremal phenomenon. Writing $$\cos^n x=1+\sin^n x$$ immediately gives...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 1

IMO 1979 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 10m50s Problem If $p$ and $q$ are natural numbers so that$$ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, $$prove that $p$ is divisible by $1979$. Exploration For small values, the alternating harmonic sums reduce to fractions whose numerators often contain large prime factors. Computing the first few cases helps identify the correct structural decomposition. For $n=1$, $$1=\frac11.$$ For $n=2$,...

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 6

IMO 1973 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 22m47s Problem Let $a_1, a_2, \ldots, a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0 < q < 1$. Find $n$ numbers $b_1, b_2, \ldots, b_n$ for which (a) $a_k < b_k$ for $k = 1, 2, \ldots, n$, (b) $q < \tfrac{b_{k+1}}{b_k} < \tfrac {1}{q}$ for $k...

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 4

IMO 1973 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 33m43s Problem A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to...

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 5

IMO 1973 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 36m23s Problem $G$ is a set of non-constant functions of the real variable $x$ of the form $$ f(x) = ax + b, a \text{ and } b \text{ are real numbers,} $$ and $G$ has the following properties: (a) If $f$ and $g$ are in $G$, then $g \circ f$ is in $G$; here...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IND25

IMO 1992 LL IND25 Origin: IND Problem (a) Show that the set N of all natural numbers can be parti- tioned into three disjoint subsets A, B, and C satisfying the following conditions: A2 = A, B2 = C, C2 = B, AB = B, AC = C, BC = A, where HK stands for {hk | h \inH, k \inK} for any two subsets H, K of N, and...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL CZS9

IMO 1977 LL CZS9 Origin: CZS Problem Let ABCD be a regular tetrahedron and Z an isometry map- ping A, B, C, D into B, C, D, A, respectively. Find the set M of all points X of the face ABC whose distance from Z(X) is equal to a given number t. Find necessary and suﬃcient conditions for the set M to be nonempty. Solution A necessary and suﬃcient condition...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USA49

IMO 1982 LL USA49 Origin: USA Problem Simplify n k=0 (2n)! (k!)2((n −k)!)2 .

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG30

IMO 1988 LL HKG30 Origin: HKG Problem Find the total number of diﬀerent integers that the function f(x) = [x] + [2x] + 5x [3x] + [4x] takes for 0 \leqx \leq100.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IRE55

IMO 1989 LL IRE55 Origin: IRE Problem Let [x] denote the greatest integer less than or equal to x. Let \alpha be the positive root of the equation x2 −1989x −1 = 0. Prove that there exist inﬁnitely many natural numbers n that satisfy the equation [\alphan + 1989\alpha[\alphan]] = 1989n + (19892 + 1)[\alphan].

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL37

IMO 1967 LL POL37 Origin: POL Problem Prove that for arbitrary positive numbers the following in- equality holds: a + 1 b + 1 c \leqa8 + b8 + c8 a3b3c3 . Solution Using the A–G mean inequality we obtain 8a2b3c3 \leq2a8 + 3b8 + 3c8, 8a3b2c3 \leq3a8 + 2b8 + 3c8, 8a3b3c2 \leq3a8 + 3b8 + 2c8. By adding these inequalities and dividing by 3a3b3c3 we obtain the desired...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS17

IMO 1969 LL CZS17 Origin: CZS Problem Let d and p be two real numbers. Find the ﬁrst term of an arith- metic progression a1, a2, a3, . . . with diﬀerence d such that a1a2a3a4 = p. Find the number of solutions in terms of d and p.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL ICE36

IMO 1987 LL ICE36 Origin: ICE Problem A game consists in pushing a ﬂat stone along a sequence of squares S0, S1, S2, . . . that are arranged in linear order. The stone is ini- tially placed on square S0. When the stone stops on a square Sk it is pushed again in the same direction and so on until it reaches S1987 or goes beyond it; then the...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BRA9

IMO 1982 LL BRA9 Origin: BRA Problem Let n be a natural number, n \geq2, and let \varphi be Euler’s function; i.e., \varphi(n) is the number of positive integers not exceeding n and coprime to n. Given any two real numbers \alpha and \beta, 0 \leq\alpha < \beta \leq1, prove that there exists a natural number m such that \alpha < \varphi(m) m < \beta.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ROM90

IMO 1989 LL ROM90 Origin: ROM Problem Prove that the sequence (an)n\geq0, an = [n \sqrt 2], contains an inﬁnite number of perfect squares.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL3

IMO 1972 LL BUL3 Origin: BUL Problem On a line a set of segments is given of total length less than n. Prove that every set of n points of the line can be translated in some direction along the line for a distance smaller than n/2 so that none of the points remain on the segments.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL GBR17

IMO 1974 LL GBR17 Origin: GBR Problem Show that there exists a set S of 15 distinct circles on the surface of a sphere, all having the same radius and such that 5 touch exactly 5 others, 5 touch exactly 4 others, and 5 touch exactly 3 others.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GDR28

IMO 1987 LL GDR28 Origin: GDR Problem In a chess tournament there are n \geq5 players, and they have already played n2 2 games (each pair have played each other at most once). (a) Prove that there are ﬁve players a, b, c, d, e for which the pairs ab, ac, bc, ad, ae, de have already played. (b) Is the statement also valid for the n2...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GBR27

IMO 1989 LL GBR27 Origin: GBR Problem Integers cm,n (m \geq0, n \geq0) are deﬁned by cm,0 = 1 for all m \geq0, c0,n = 1 for all n \geq0, and cm,n = cm−1,n −ncm−1,n−1 for all m > 0, n > 0. Prove that cm,n = cn,m for all m \geq0, n \geq0.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL ICE37

IMO 1987 LL ICE37 Origin: ICE Problem Five distinct numbers are drawn successively and at random from the set {1, . . . , n}. Show that the probability of a draw in which the ﬁrst three numbers as well as all ﬁve numbers can be arranged to form an arithmetic progression is greater than (n−2)3 .

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GBR24

IMO 1987 LL GBR24 Origin: GBR Problem Prove that if the equation x4 + ax3 + bx + c = 0 has all its roots real, then ab \leq0.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL22

IMO 1966 LL BUL22 Origin: BUL Problem Assume that two parallelograms P, P ′ of equal areas have sides a, b and a′, b′ respectively such that a′ \leqa \leqb \leqb′ and a segment of length b′ can be placed inside P. Prove that P and P ′ can be partitioned into four pairwise congruent parts.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL ROM39

IMO 1977 LL ROM39 Origin: ROM Problem Consider 37 distinct points in space, all with integer coordi- nates. Prove that we may ﬁnd among them three distinct points such that their barycenter has integers coordinates. Solution By the pigeonhole principle, we can ﬁnd 5 distinct points among the given 37 such that their x-coordinates are congruent and their y-coordinates are congruent modulo 3. Now among these 5 points either there...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA19

IMO 1969 LL FRA19 Origin: FRA Problem Let n be an integer that is not divisible by any square greater than 1. Denote by xm the last digit of the number xm in the number system with base n. For which integers x is it possible for xm to be 0? Prove that the sequence xm is periodic with period t independent of x. For which x do we have...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL TUR40

IMO 1978 LL TUR40 Origin: TUR Problem If Cp n = n! p!(n−p)! (p \geq1), prove the identity Cp n = Cp−1 n−1 + Cp−1 n−2 + \cdot \cdot \cdot + Cp−1 p Cp−1 p−1 and then evaluate the sum S = 1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdot \cdot \cdot + 97 \cdot 98 \cdot 99.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE61

IMO 1969 LL SWE61 Origin: SWE Problem Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n. Prove that if n is power of 2, then so is an.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL CZS6

IMO 1977 LL CZS6 Origin: CZS Problem Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi, j = 1, 2, . . ., n. Prove that if n j=1(cos xj + 1) is an odd integer, then n j=1 sin xj \geq1. Solution Let ⟨y⟩denote the distance from y \inR to the closest even integer. We claim that ⟨1 + cos...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL HUN25

IMO 1977 LL HUN25 Origin: HUN Problem Prove the identity (z + a)n = zn + a n k=1 n k (a −kb)k−1(z + kb)n−k. Solution Let fn(z) = zn + a n k=1 n k (a −kb)k−1(z + kb)n−k. We shall prove by induction on n that fn(z) = (z + a)n. This is trivial for n = 1. Suppose that the statement is true...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET48

IMO 1969 LL NET48 Origin: NET Problem Let x1, x2, x3, x4, and x5 be positive integers satisfying x1 +x2 +x3 +x4 +x5 = 1000, x1 −x2 +x3 −x4 +x5 > 0, x1 +x2 −x3 +x4 −x5 > 0, −x1 +x2 +x3 −x4 +x5 > 0, x1 −x2 +x3 +x4 −x5 > 0, −x1 +x2 −x3 +x4 +x5 > 0. (a) Find the maximum of (x1 + x3)x2+x4. (b) In...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL USS46

IMO 1972 LL USS46 Origin: USS Problem Numbers 1, 2, . . . , 16 are written in a 4\times4 square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers 1 and 16 lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SWE80

IMO 1988 LL SWE80 Origin: SWE Problem Let S be an inﬁnite set of integers containing zero and such that the distance between successive numbers never exceeds a given ﬁxed number. Consider the following procedure: Given a set X of integers, we construct a new set consisting of all numbers x \pm s, where x belongs to X and s belongs to S. Starting from S0 = {0} we successively...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL USA81

IMO 1988 LL USA81 Origin: USA Problem There are n \geq3 job openings at a factory, ranked 1 to n in order of increasing pay. There are n job applicants, ranked 1 to n in order of increasing ability. Applicant i is qualiﬁed for job j if and only if i \geqj. The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS55

IMO 1967 LL USS55 Origin: USS Problem Find all x for which for all n, sin x + sin 2x + sin 3x + \cdot \cdot \cdot + sin nx \leq \sqrt 2 . Solution It is enough to ﬁnd all x from (0, 2\pi] such that the given inequality holds for all n. Suppose 0 < x < 2\pi/3. If n is the maximum integer for which nx \leq...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IRE34

IMO 1992 LL IRE34 Origin: IRE Problem Let a, b, c be integers. Prove that there are integers p1, q1, r1, p2, q2, r2 such that a = q1r2 −q2r1, b = r1p2 −r2p1, c = p1q2 −p2q1.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS5

IMO 1966 LL USS5 Origin: USS Problem Prove the inequality tan \pi sin x 4 sin \alpha + tan \pi cos x 4 cos \alpha > 1 for any x, \alpha with 0 \leqx \leq\pi/2 and \pi/6 < y < \pi/3.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL USA44

IMO 1978 LL USA44 Origin: USA Problem In \triangleABC with \angleC = 60o, prove that c a + c b \geq2.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR86

IMO 1989 LL POR86 Origin: POR Problem Given two natural numbers w and n, the tower of n w’s is the natural number Tn(w) deﬁned by Tn(w) = ww...w , with n w’s on the right side. More precisely, T1(w) = w and Tn+1(w) = wTn(w). For example, T3(2) = 222 = 16, T4(2) = 216 = 65536, and T2(3) = 33 = 27. Find the smallest tower of 3’s...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MON52

IMO 1985 LL MON52 Origin: MON Problem In the triangle ABC, let B1 be on AC, E on AB, G on BC, and let EG be parallel to AC. Furthermore, let EG be tangent to the inscribed circle of the triangle ABB1 and intersect BB1 at F. Let r, r1, and r2 be the inradii of the triangles ABC, ABB1, and BFG, respectively. Prove that r = r1 + r2.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL POL38

IMO 1970 LL POL38 Origin: POL Problem Find the greatest integer A for which in any permutation of the numbers 1, . . . , 100 there exist ten consecutive numbers whose sum is at least A.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL FIN14

IMO 1978 LL FIN14 Origin: FIN Problem Let p(x, y) and q(x, y) be polynomials in two variables such that for x \geq0, y \geq0 the following conditions hold: (i) p(x, y) and q(x, y) are increasing functions of x for every ﬁxed y. (ii) p(x, y) is an increasing and q(x) is a decreasing function of y for every ﬁxed x. (iii) p(x, 0) = q(x, 0) for every...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRG29

IMO 1986 LL FRG29 Origin: FRG Problem We deﬁne a binary operation ⋆in the plane as follows: Given two points A and B in the plane, C = A ⋆B is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points I, M, O in the plane if I ⋆(M ⋆O) = (O ⋆I) ⋆M holds?

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL MON66

IMO 1988 LL MON66 Origin: MON Problem Suppose \alphai > 0, \betai > 0 for 1 \leqi \leqn (n > 1) and that n i=1 \alphai = n i=1 \betai = \pi. Prove that n i=1 cos \betai sin \alphai \leq n i=1 cot \alphai.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL GRE36

IMO 1979 LL GRE36 Origin: GRE Problem A regular tetrahedron A1B1C1D1 is inscribed in a regular tetrahedron ABCD, where A1 lies in the plane BCD, B1 in the plane ACD, etc. Prove that A1B1 \geqAB/3.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SWE57

IMO 1984 LL SWE57 Origin: SWE Problem Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let n = (10a + b)10c+d. Find the probability that 1984! is divisible by n.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL ISR37

IMO 1983 LL ISR37 Origin: ISR Problem The points A1, A2, . . . , A1983 are set on the circumference of a circle and each is given one of the values \pm1. Show that if the number of points with the value +1 is greater than 1789, then at least 1207 of the points will have the property that the partial sums that can be formed by taking the...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL CAN15

IMO 1983 LL CAN15 Origin: CAN Problem Find all possible ﬁnite sequences {n0, n1, n2, . . . , nk} of integers such that for each i, i appears in the sequence ni times (0 \leqi \leqk).

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL BUL5

IMO 1989 LL BUL5 Origin: BUL Problem The sequences a0, a1, . . . and b0, b1, . . . are deﬁned by the equal- ities a0 = \sqrt 2 , an+1 = \sqrt 1 − 1 −a2n, n = 0, 1, 2, . . . and b0 = 1, bn+1 = 1 + b2n −1 bn , n = 0, 1, 2, . . .. Prove the inequalities 2n+2an...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL VIE73

IMO 1979 LL VIE73 Origin: VIE Problem In a plane a ﬁnite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of diﬀerent colors. What is the smallest number of circles that requires four colors?

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET37

IMO 1970 LL NET37 Origin: NET Problem Solve the set of simultaneous equations v2+ w2+ x2+ y2 = 6 −2u, u2+ w2+ x2+ y2 = 6 −2v, u2+ v2+ x2+ y2 = 6 −2w, u2+ v2+ w2+ y2 = 6 −2x, u2+ v2+ w2+ x2 = 6 −2y.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL YUG53

IMO 1971 LL YUG53 Origin: YUG Problem Denote by xn(p) the multiplicity of the prime p in the canonical representation of the number n! as a product of primes. Prove that xn(p) n < p−1 and limn\to\infty xn(p) n p−1.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USS51

IMO 1982 LL USS51 Origin: USS Problem Let n numbers x1, x2, . . . , xn be chosen in such a way that 1 \geqx1 \geqx2 \geq\cdot \cdot \cdot \geqxn \geq0. Prove that (1 + x1 + x2 + \cdot \cdot \cdot + xn)\alpha \leq1 + x\alpha 1 + 2\alpha−1x\alpha 2 + \cdot \cdot \cdot + n\alpha−1x\alpha n if 0 \leq\alpha \leq1.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USA37

IMO 1974 LL USA37 Origin: USA Problem Let a, b, and c denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side a and then propelled toward side b with direction deﬁned by the angle \theta. For what values of \theta will the ball strike the sides b, c, a in that order?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL HUN39

IMO 1979 LL HUN39 Origin: HUN Problem A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of n days of walking from the camp, under the following conditions: (i) Each member of the expedition can pick up at most 3 liters of water. (ii) Each member must drink one liter...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FIN11

IMO 1987 LL FIN11 Origin: FIN Problem Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph of a real function f : [0, 1] \to[0, 1] is continuous and increasing, and it is linear on every subinterval I in [0, 1] such that the endpoints but no interior points of I are in S. We want to compute, using a computer, the extreme...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BUL7

IMO 1969 LL BUL7 Origin: BUL Problem Prove that the equation x3 + y3 + z3 = 1969 has no integral solutions.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL KOR64

IMO 1989 LL KOR64 Origin: KOR Problem Let a regular (2n + 1)-gon be inscribed in a circle of radius r. We consider all the triangles whose vertices are from those of the regular (2n + 1)-gon. (a) How many triangles among them contain the center of the circle in their interior? (b) Find the sum of the areas of all those triangles that contain the center of the circle...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL FIN13

IMO 1974 LL FIN13 Origin: FIN Problem Prove that 2147 −1 is divisible by 343.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL POL51

IMO 1979 LL POL51 Origin: POL Problem Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be circles satisfying the following conditions: S1 is tangent to CA and AB, S2 is tangent to S1, AB, and BC, S3 is tangent to S2, BC, and CA, \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot S7...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL ROM38

IMO 1977 LL ROM38 Origin: ROM Problem Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1 \leq\cdot \cdot \cdot \leq bn < c1 \leq\cdot \cdot \cdot \leqcn be real numbers. Prove: ⎡ ⎣ n j=1 mj(aj + bj + cj) ⎤ ⎦ 3 ⎛ ⎝ n j=1 mj ⎞ ⎠ ⎡ ⎣ n j=1 mj(ajbj...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL CAN6

IMO 1992 LL CAN6 Origin: CAN Problem Suppose that n numbers x1, x2, . . . , xn are chosen randomly from the set {1, 2, 3, 4, 5}. Prove that the probability that x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \equiv0 (mod 5) is at least 1/5.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL CUB22

IMO 1983 LL CUB22 Origin: CUB Problem Does there exist an inﬁnite number of sets C consisting of 1983 consecutive natural numbers such that each of the numbers is divisible by some number of the form a1983, with a \inN, a ̸= 1?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL BUL9

IMO 1971 LL BUL9 Origin: BUL Problem The base of an inclined prism is a triangle ABC. The per- pendicular projection of B1, one of the top vertices, is the midpoint of BC. The dihedral angle between the lateral faces through BC and AB is \alpha, and the lateral edges of the prism make an angle \beta with the base. If r1, r2, r3 are exradii of a perpendicular section...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRG17

IMO 1988 LL FRG17 Origin: FRG Problem Show that if n runs through all positive integers, f(n) = / n + \sqrt 3n + 1/2 runs through all positive integers skipping the terms of the sequence an = n2+2n .

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL TUR60

IMO 1987 LL TUR60 Origin: TUR Problem It is given that x = −2272, y = 103 +102c+10b+a, and z = 1 satisfy the equation ax + by + cz = 1, where a, b, c are positive integers with a < b < c. Find y.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL SWE36

IMO 1974 LL SWE36 Origin: SWE Problem Consider inﬁnite diagrams D = ... ... ... n20 n21 n22 . . . n10 n11 n12 . . . n00 n01 n02 . . . where all but a ﬁnite number of the integers nij, i = 0, 1, 2, . . ., j = 0, 1, 2, . . ., are equal to 0. Three elements of a diagram are called...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL TUR39

IMO 1978 LL TUR39 Origin: TUR Problem A is a 2m-digit positive integer each of whose digits is 1. B is an m-digit positive integer each of whose digits is 4. Prove that A+B + 1 is a perfect square.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL USS80

IMO 1986 LL USS80 Origin: USS Problem Let ABCD be a tetrahedron and O its incenter, and let the line OD be perpendicular to AD. Find the angle between the planes DOB and DOC.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL USS45

IMO 1972 LL USS45 Origin: USS Problem Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect at point O. Let a line through O intersect segment AB at M and segment CD at N. Prove that the segment MN is not longer than at least one of the segments AC and BD.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS1

IMO 1966 LL CZS1 Origin: CZS Problem We are given n > 3 points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MON41

IMO 1987 LL MON41 Origin: MON Problem Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MON40

IMO 1987 LL MON40 Origin: MON Problem The perpendicular line issued from the center of the circum- circle to the bisector of angle C in a triangle ABC divides the segment of the bisector inside ABC into two segments with ratio of lengths \lambda. Given b = AC and a = BC, ﬁnd the length of side c.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL FRA26

IMO 1982 LL FRA26 Origin: FRA Problem Let (an)n\geq0 and (bn)n\geq0 be two sequences of natural numbers. Determine whether there exists a pair (p, q) of natural numbers that satisfy p < q and ap \leqaq, bp \leqbq.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IRN36

IMO 1992 LL IRN36 Origin: IRN Problem Find all rational solutions of a2 + c2 + 17(b2 + d2) = 21, ab + cd = 2.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CHN15

IMO 1986 LL CHN15 Origin: CHN Problem Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all positive integers and let an integer l \inN be given. Prove that there exist a set X \subsetN of cardinality l, an inﬁnite set T \subsetN, and an integer k with 1 \leqk \leqq such that for any t \inT and any ﬁnite set Y \subsetX, the...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL KOR43

IMO 1992 LL KOR43 Origin: KOR Problem Find the number of positive integers n satisfying \varphi(n) | n such that \infty m=1 n m −n −1 m = 1992. What is the largest number among them? As usual, \varphi(n) is the number of positive integers less than or equal to n and relatively prime to n.6

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GBR26

IMO 1989 LL GBR26 Origin: GBR Problem Let a, b, c, d be positive integers such that ab = cd and a + b = c −d. Prove that there exists a right-angled triangle the measures of whose sides (in some unit) are integers and whose area measure is ab square units.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GRE30

IMO 1989 LL GRE30 Origin: GRE Problem In a triangle ABC for which 6(a + b + c)r2 = abc, we consider a point M on the inscribed circle and the projections D, E, F of M on the sides BC, AC, and AB respectively. Let S, S1 denote the areas of the triangles ABC and DEF respectively. Find the maximum and minimum values of the quotient S S1 (here...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MON55

IMO 1986 LL MON55 Origin: MON Problem Given an integer n \geq2, determine all n-digit numbers M0 = a1a2 . . . an (ai ̸= 0, i = 1, 2, . . ., n) divisible by the numbers M1 = a2a3 . . . ana1, M2 = a3a4 . . . ana1a2, . . . , Mn−1 = ana1a2 . . . an−1.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL GRE34

IMO 1979 LL GRE34 Origin: GRE Problem Notice that in the fraction 16 64 we can perform a simpliﬁcation as 1̸6 ̸64 = 1 4 obtaining a correct equality. Find all fractions whose numer- ators and denominators are two-digit positive integers for which such a simpliﬁcation is correct.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL FRG12

IMO 1977 LL FRG12 Origin: FRG Problem Let z be an integer > 1 and let M be the set of all numbers of the form zk = 1 + z + \cdot \cdot \cdot + zk, k = 0, 1, . . . . Determine the set T of divisors of at least one of the numbers zk from M. Solution According to part (a) of the previous problem...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GDR27

IMO 1978 LL GDR27 Origin: GDR Problem Determine the sixth number after the decimal point in the number \sqrt 1978 + /\sqrt 1978 020.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON41

IMO 1969 LL MON41 Origin: MON Problem Given two numbers x0 and x1, let \alpha and \beta be coeﬃcients of the equation 1 −\alphay −\betay2 = 0. Under the given conditions, ﬁnd an expression for the solution of the system xn+2 −\alphaxn+1 −\betaxn = 0, n = 0, 1, 2, . . . .

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA41

IMO 1988 LL INA41 Origin: INA Problem (a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the midpoint of side BC and points E and F are chosen on sides AC and AB respectively, and suppose that the lines EF and AM intersect at G. If AE = 2AF then ﬁnd the ratio EG/GF. (b) Let E be a point external to...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL POL33

IMO 1977 LL POL33 Origin: POL Problem A circle K centered at (0, 0) is given. Prove that for every vector (a1, a2) there is a positive integer n such that the circle K translated by the vector n(a1, a2) contains a lattice point (i.e., a point both of whose coordinates are integers). Solution Let r be the radius of K and s > \sqrt 2/r an integer. Consider the...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL GDR25

IMO 1966 LL GDR25 Origin: GDR Problem Show that tan 7◦30′ = \sqrt 6 + \sqrt 2 − \sqrt 3 −2.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SPA75

IMO 1988 LL SPA75 Origin: SPA Problem Let ABC be a triangle with inradius r and circumradius R. Show that sin A 2 sin B 2 + sin B 2 sin C 2 + sin C 2 sin A 2 \leq5 8 + r 4R.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL POL69

IMO 1988 LL POL69 Origin: POL Problem For a convex polygon P in the plane let P ′ denote the convex polygon with vertices at the midpoints of the sides of P. Given an integer n \geq3, determine sharp bounds for the ratio area(P ′) area(P) over all convex n-gons P.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BRA12

IMO 1982 LL BRA12 Origin: BRA Problem Let there be 3399 numbers arbitrarily chosen among the ﬁrst 6798 integers 1, 2, . . ., 6798 in such a way that none of them divides an- other. Prove that there are exactly 1982 numbers in {1, 2, . . ., 6798} that must end up being chosen.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BUL13

IMO 1979 LL BUL13 Origin: BUL Problem The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let M be an arbitrary set of n squares of this net. Prove that it is possible to choose no fewer than n/4 squares of M in such a way that no two of them have a common point.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BEL5

IMO 1979 LL BEL5 Origin: BEL Problem Describe which natural numbers do not belong to the set E = {[n + \sqrtn + 1/2] | n \inN}.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL FIN24

IMO 1982 LL FIN24 Origin: FIN Problem Prove that if a person a has inﬁnitely many descendants (chil- dren, their children, etc.), then a has an inﬁnite sequence a0, a1, . . . of descendants (i.e., a = a0 and for all n \geq1, an+1 is always a child of an). It is assumed that no-one can have inﬁnitely many children. Variant 1. Prove that if a has inﬁnitely many...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL POL25

IMO 1974 LL POL25 Origin: POL Problem Let f : R \toR be of the form f(x) = x + \epsilon sin x, where 0 < |\epsilon| \leq1. Deﬁne for any x \inR, xn = f ◦\cdot \cdot \cdot ◦f n times (x). Show that for every x \inR there exists an integer k such that limn\to\inftyxn = k\pi.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GBR25

IMO 1989 LL GBR25 Origin: GBR Problem Let ABC be a triangle. Prove that there is a unique point U in the plane of ABC such that there exist real numbers \lambda, µ, \nu, \kappa, not all zero, such that \lambdaPL2 + µPM 2 + \nuPN 2 −\kappaUP 2 is constant for all points P of the plane, where L, M, N are the feet of the perpendiculars from P...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL VIE75

IMO 1983 LL VIE75 Origin: VIE Problem Find the sum of the ﬁftieth powers of all sides and diagonals of a regular 100-gon inscribed in a circle of radius R.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ISR49

IMO 1986 LL ISR49 Origin: ISR Problem Let C1, C2 be circles of radius 1/2 tangent to each other and both tangent internally to a circle C of radius 1. The circles C1 and C2 are the ﬁrst two terms of an inﬁnite sequence of distinct circles Cn deﬁned as follows: Cn+2 is tangent externally to Cn and Cn+1 and internally to C. Show that the radius of each Cn...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GBR15

IMO 1976 LL GBR15 Origin: GBR Problem Let ABC and A′B′C′ be any two coplanar triangles. Let L be a point such that AL\parallelBC, A′L\parallelB′C′, and M, N similarly deﬁned. The line BC meets B′C′ at P, and similarly deﬁned are Q and R. Prove that PL, QM, RN are concurrent.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS7

IMO 1967 LL CZS7 Origin: CZS Problem Find all real solutions of the system of equations x1 + x2 + \cdot \cdot \cdot + xn = a, x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = a2, . . . . . . . . . . . . . . . . . . xn 1 + xn 2 + \cdot \cdot \cdot + xn...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL VIE75

IMO 1987 LL VIE75 Origin: VIE Problem Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1 (k = 1, 2, . . . ). Prove that for every n \inN, n k=1 ak+1 1987\sqrtak < 1987.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ISR58

IMO 1989 LL ISR58 Origin: ISR Problem Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coeﬃcients. Show that there exist real polynomials Ar(x), Br(x) (r = 1, 2, 3) such that n s=1(Ps(x))2 = (A1(x))2 + (B1(x))2 = (A2(x))2 + x(B2(x))2 = (A3(x))2 −x(B3(x))2.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL TUR37

IMO 1978 LL TUR37 Origin: TUR Problem Simplify loga(abc) + logb(abc) + logc(abc), where a, b, c are positive real numbers.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FRG18

IMO 1992 LL FRG18 Origin: FRG Problem Fibonacci numbers are deﬁned as follows: F1 = F2 = 1, Fn+2 = Fn+1+Fn, n \geq1. Let an be the number of words that consist of n letters 0 or 1 and contain no two letters 1 at distance two from each other. Express an in terms of Fibonacci numbers.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL ROM57

IMO 1987 LL ROM57 Origin: ROM Problem The bisectors of the angles B, C of a triangle ABC intersect the opposite sides in B′, C′ respectively. Prove that the straight line B′C′ intersects the inscribed circle in two diﬀerent points.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL HUN20

IMO 1966 LL HUN20 Origin: HUN Problem We are given three equal rectangles with the same center in three mutually perpendicular planes, with the long sides also mutually perpendicular. Consider the polyhedron with vertices at the vertices of these rectangles. (a) Find the volume of this polyhedron; (b) can this polyhedron be regular, and under what conditions?

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GBR32

IMO 1985 LL GBR32 Origin: GBR Problem A collection of 2n letters contains 2 each of n diﬀerent letters. The collection is partitioned into n pairs, each pair containing 2 letters, which may be the same or diﬀerent. Denote the number of distinct parti- tions by un. (Partitions diﬀering in the order of the pairs in the partition or in the order of the two letters in the pairs are...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ROM56

IMO 1979 LL ROM56 Origin: ROM Problem Show that for every natural number n, n \sqrt 2 −[n \sqrt 2] > 2n \sqrt and that for every \epsilon > 0 there exists a natural number n with n \sqrt 2 − [n \sqrt 2] < 2n \sqrt 2 + \epsilon.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRA13

IMO 1988 LL FRA13 Origin: FRA Problem Let T be a triangle with inscribed circle C. A square with sides of length a is circumscribed about the same circle C. Show that the total length of the parts of the edges of the square interior to the triangle T is at least 2a.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ISR48

IMO 1986 LL ISR48 Origin: ISR Problem Let P be a convex 1986-gon in the plane. Let A, D be interior points of two distinct sides of P and let B, C be two distinct interior points of the line segment AD. Starting with an arbitrary point Q1 on the boundary of P, deﬁne recursively a sequence of points Qn as follows: given Qn extend the directed line segment QnB...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL NET24

IMO 1976 LL NET24 Origin: NET Problem Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A \geq1 there exists an interval I of length 2 n\sqrt A such that for all x \inI, |(x −x1)(x −x2) \cdot \cdot \cdot (x −xn)| \leqA.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL SWE53

IMO 1970 LL SWE53 Origin: SWE Problem A square ABCD is divided into (n −1)2 congruent squares, with sides parallel to the sides of the given square. Consider the grid of all n2 corners obtained in this manner. Determine all integers n for which it is possible to construct a nondegenerate parabola with its axis parallel to one side of the square and that passes through exactly n points of...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL SWE42

IMO 1977 LL SWE42 Origin: SWE Problem The sequence an,k, k = 1, 2, 3, . . ., 2n, n = 0, 1, 2, . . ., is deﬁned by the following recurrence formula: a1 = 2, an,k = 2a3 n−1,k, an,k+2n−1 = 1 2a3 n−1,k for k = 1, 2, 3, . . ., 2n−1, n = 0, 1, 2, . . . . Prove that the numbers an,k...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS53

IMO 1966 LL USS53 Origin: USS Problem Prove that in every convex hexagon of area S one can draw a diagonal that cuts oﬀa triangle of area not exceeding 1 6S.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL KOR58

IMO 1988 LL KOR58 Origin: KOR Problem For each pair of positive integers k and n, let Sk(n) be the base-k digit sum of n. Prove that there are at most two primes p less than 20,000 for which S31(p) is a composite number.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL AUS2

IMO 1982 LL AUS2 Origin: AUS Problem Given a ﬁnite number of angular regions A1, . . . , Ak in a plane, each Ai being bounded by two half-lines meeting at a vertex and provided with a + or −sign, we assign to each point P of the plane and not on a bounding half-line the number k −l, where k is the number of + regions and l...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BEL10

IMO 1970 LL BEL10 Origin: BEL Problem Let A, B, C be angles of a triangle. Prove that 1 < cos A + cos B + cos C \leq3 2.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL HUN42

IMO 1979 LL HUN42 Origin: HUN Problem Let a quadratic polynomial g(x) = ax2 + bx + c be given and an integer n \geq1. Prove that there exists at most one polynomial f(x) of nth degree such that f(g(x)) = g(f(x)).

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL GDR30

IMO 1979 LL GDR30 Origin: GDR Problem Let M be a set of points in a plane with at least two elements. Prove that if M has two axes of symmetry g1 and g2 intersecting at an angle \alpha = q\pi, where q is irrational, then M must be inﬁnite.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL THA71

IMO 1992 LL THA71 Origin: THA Problem Let P1(x, y) and P2(x, y) be two relatively prime polynomials with complex coeﬃcients. Let Q(x, y) and R(x, y) be polynomials with complex coeﬃcients and each of degree not exceeding d. Prove that there exist two integers A1, A2 not simultaneously zero with |Ai| \leqd + 1 (i = 1, 2) and such that the polynomial A1P1(x, y) + A2P2(x, y) is...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL AUS1

IMO 1989 LL AUS1 Origin: AUS Problem In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is deﬁned with the following properties: (i) c = a ∗b is in Sn for any a \inSn, b \inSn. (ii) If the ordinary product a\cdotb is less than or equal to n, then a∗b = a\cdotb. (iii) The ordinary rules of multiplication hold for ∗, i.e., (1)...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USS47

IMO 1977 LL USS47 Origin: USS Problem A square ABCD is given. A line passing through A intersects CD at Q. Draw a line parallel to AQ that intersects the boundary of the square at points M and N such that the area of the quadrilateral AMNQ is maximal. Solution M lies on AB and N lies on BC. If CQ \leq2CD/3, then BM = CQ/2. If CQ > 2CD/3,...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL FRA15

IMO 1978 LL FRA15 Origin: FRA Problem Prove that for every positive integer n coprime to 10 there exists a multiple of n that does not contain the digit 1 in its decimal representation.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL POL53

IMO 1979 LL POL53 Origin: POL Problem An inﬁnite increasing sequence of positive integers nj (j = 1, 2, . . . ) has the property that for a certain c, N nj\leqN nj \leqc, for every N > 0 Prove that there exist ﬁnitely many sequences m(i) j (i = 1, 2, . . . , k) such that {n1, n2, . . . } = %k i=1{m(i)...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL THA98

IMO 1989 LL THA98 Origin: THA Problem Let f : N \toN be such that (i) f is strictly increasing; (ii) f(mn) = f(m)f(n) \forallm, n \inN; and (iii) if m ̸= n and mn = nm, then f(m) = n or f(n) = m. Determine f(30).

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL CUB11

IMO 1989 LL CUB11 Origin: CUB Problem Given the equation y4 + 4y2x −11y2 + 4xy −8y + 8x2 −40x + 52 = 0, ﬁnd all real solutions.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN23

IMO 1967 LL HUN23 Origin: HUN Problem Prove that for an arbitrary pair of vectors f and g in the plane, the inequality af 2 + bfg + cg2 \geq0 holds if and only if the following conditions are fulﬁlled: a \geq0, c \geq0, 4ac \geqb2. Solution Suppose that a \geq0, c \geq0, 4ac \geqb2. If a = 0, then b = 0, and the inequality reduces to the obvious...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BUL15

IMO 1970 LL BUL15 Origin: BUL Problem Given a triangle ABC, let R be the radius of its circumcir- cle, O1, O2, O3 the centers of its exscribed circles, and q the perimeter of \triangleO1O2O3. Prove that q \leq6 \sqrt 3 R.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL HUN19

IMO 1966 LL HUN19 Origin: HUN Problem Construct a triangle given the three exradii.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL TUR82

IMO 1985 LL TUR82 Origin: TUR Problem Find all cubic polynomials x3 + ax2 + bx + c admitting the rational numbers a, b, and c as roots.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL SPA57

IMO 1983 LL SPA57 Origin: SPA Problem In the system of base n2 + 1 ﬁnd a number N with n diﬀerent digits such that: (i) N is a multiple of n. Let N = nN ′. (ii) The number N and N ′ have the same number n of diﬀerent digits in base n2 + 1, none of them being zero. (iii) If s(C) denotes the number in base...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FRA17

IMO 1987 LL FRA17 Origin: FRA Problem Consider the number \alpha obtained by writing one after another the decimal representations of 1, 1987, 19872, . . . to the right the decimal point. Show that \alpha is irrational.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BUL9

IMO 1969 LL BUL9 Origin: BUL Problem One hundred convex polygons are placed on a square with edge of length 38 cm. The area of each of the polygons is smaller than \pi cm2, and the perimeter of each of the polygons is smaller than 2\pi cm. Prove that there exists a disk with radius 1 in the square that does not intersect any of the polygons.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS41

IMO 1966 LL CZS41 Origin: CZS Problem If A1A2 . . . An is a regular n-gon (n \geq3), how many diﬀerent obtuse triangles AiAjAk exist?

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USS45

IMO 1976 LL USS45 Origin: USS Problem We are given n (n \geq5) circles in a plane. Suppose that every three of them have a common point. Prove that all n circles have a common point.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ROM64

IMO 1986 LL ROM64 Origin: ROM Problem Let (an)n\inN be the sequence of integers deﬁned recursively by a1 = a2 = 1, an+2 = 7an+1 −an −2 for n \geq1. Prove that an is a perfect square for every n.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BUL14

IMO 1970 LL BUL14 Origin: BUL Problem Let \alpha + \beta + \gamma = \pi. Prove that sin 2\alpha + sin 2\beta + sin 2\gamma = 2(sin \alpha + sin \beta + sin \gamma)(cos \alpha + cos \beta + cos \gamma) −2(sin \alpha + sin \beta + sin \gamma).

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS42

IMO 1966 LL CZS42 Origin: CZS Problem Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that there is a subsequence ak1, ak2, . . . , akm, where 1 \leqk1 < k2 < \cdot \cdot \cdot < km \leq n, such that a2 k1 + a2 k2 + \cdot \cdot \cdot + a2 km is divisible by n.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX43

IMO 1983 LL LUX43 Origin: LUX Problem Given a square ABCD, let P, Q, R, and S be four variable points on the sides AB, BC, CD, and DA, respectively. Determine the positions of the points P, Q, R, and S for which the quadrilateral PQRS is a parallelogram, a rectangle, a square, or a trapezoid.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL CAN15

IMO 1982 LL CAN15 Origin: CAN Problem Show that the set S of natural numbers n for which 3/n cannot be written as the sum of two reciprocals of natural numbers (S = {n | 3/n ̸= 1/p + 1/q for any p, q \inN}) is not the union of ﬁnitely many arithmetic progressions.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL NET21

IMO 1974 LL NET21 Origin: NET Problem Let M be a nonempty subset of Z+ such that for every element x in M, the numbers 4x and [\sqrtx] also belong to M. Prove that M = Z+.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL ROM36

IMO 1977 LL ROM36 Origin: ROM Problem Consider a sequence of numbers (a1, a2, . . . , a2n). Deﬁne the operation S((a1, a2, . . . , a2n)) = (a1a2, a2a3, . . . , a2n−1a2n, a2na1). Prove that whatever the sequence (a1, a2, . . . , a2n) is, with ai \in{−1, 1} for i = 1, 2, . . . , 2n, after ﬁnitely many applications of...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USS44

IMO 1974 LL USS44 Origin: USS Problem We are given n mass points of equal mass in space. We deﬁne a sequence of points O1, O2, O3, . . . as follows: O1 is an arbitrary point (within the unit distance of at least one of the n points); O2 is the center of gravity of all the n given points that are inside the unit sphere centered at O1;...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL KOR61

IMO 1989 LL KOR61 Origin: KOR Problem Let A be a set of positive integers such that no positive integer greater than 1 divides all the elements of A. Prove that any suﬃciently large positive integer can be written as a sum of elements of A. (Elements may occur several times in the sum.)

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL ROM63

IMO 1992 LL ROM63 Origin: ROM Problem Let a and b be integers. Prove that 2a2−1 b2+2 is not an integer.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL CZS19

IMO 1970 LL CZS19 Origin: CZS Problem Let n > 1 be a natural number, a \geq1 a real number, and x1, x2, . . . , xn numbers such that x1 = 1, xk+1 xk = a+\alphak for k = 1, 2, . . . , n− 1, where \alphak are real numbers with \alphak \leq k(k+1). Prove that n−1\sqrtxn < a + n −1.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM42

IMO 1967 LL ROM42 Origin: ROM Problem Decompose into real factors the expression 1 −sin5 x−cos5 x. Solution E = (1−sin x)(1−cos x)[3+2(sin x+cos x)+2 sin x cos x+sin x cos x(sin x+ cos x)].

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BEL6

IMO 1979 LL BEL6 Origin: BEL Problem Prove that 1 4 sin2 36◦−1 = cos 72◦.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL COL21

IMO 1983 LL COL21 Origin: COL Problem Prove that there are inﬁnitely many positive integers n for which it is possible for a knight, starting at one of the squares of an n \times n chessboard, to go through each of the squares exactly once.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL AUT5

IMO 1970 LL AUT5 Origin: AUT Problem Prove that n n+1 + n+1 + \cdot \cdot \cdot + n n+1 \geq1 for n \geq2.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GBR26

IMO 1987 LL GBR26 Origin: GBR Problem Prove that if x, y, z are real numbers such that x2+y2+z2 = 2, then x + y + z \leqxyz + 2.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL USS65

IMO 1969 LL USS65 Origin: USS Problem Prove that for a > b2, ! a −b a + b a −b \sqrt a + \cdot \cdot \cdot = a −3 4b2 −1 2b.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL USS49

IMO 1971 LL USS49 Origin: USS Problem Diagonals of a convex quadrilateral ABCD intersect at a point O. Find all angles of this quadrilateral if ∡OBA = 30◦, ∡OCB = 45◦, ∡ODC = 45◦, and ∡OAD = 30◦.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BRA11

IMO 1983 LL BRA11 Origin: BRA Problem A boy at point A wants to get water at a circular lake and carry it to point B. Find the point C on the lake such that the distance walked by the boy is the shortest possible given that the line AB and the lake are exterior to each other.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL VIE97

IMO 1985 LL VIE97 Origin: VIE Problem In a plane a circle with radius R and center w and a line \Lambda are given. The distance between w and \Lambda is d, d > R. The points M and N are chosen on \Lambda in such a way that the circle with diameter MN is externally tangent to the given circle. Show that there exists a point A in the...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MOR38

IMO 1984 LL MOR38 Origin: MOR Problem Determine all continuous functions f such that \forall(x, y) \inR2 f(x + y)f(x −y) = (f(x)f(y))2 .

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS13

IMO 1969 LL CZS13 Origin: CZS Problem Let p be a prime odd number. Is it possible to ﬁnd p−1 natural numbers n + 1, n + 2, . . . , n + p −1 such that the sum of the squares of these numbers is divisible by the sum of these numbers?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IND51

IMO 1989 LL IND51 Origin: IND Problem Let t(n), for n = 3, 4, 5, . . ., represent the number of distinct, incongruent, integer-sided triangles whose perimeter is n; e.g., t(3) = 1. Prove that t(2n −1) −t(2n) = n or n 6 + 1 .

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL ROM49

IMO 1984 LL ROM49 Origin: ROM Problem Let n > 1 and xi \inR for i = 1, . . . , n. Set Sk = xk 1 + xk 2 + \cdot \cdot \cdot + xk n for k \geq1. If S1 = S2 = \cdot \cdot \cdot = Sn+1, show that xi \in{0, 1} for every i = 1, 2, . . ., n.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL COL11

IMO 1992 LL COL11 Origin: COL Problem Let \varphi(n, m), m ̸= 1, be the number of positive integers less than or equal to n that are coprime with m. Clearly, \varphi(m, m) = \varphi(m), where \varphi(m) is Euler’s phi function. Find all integers m that satisfy the following inequality: \varphi(n, m) n \geq\varphi(m) m for every positive integer n.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL BUL1

IMO 1977 LL BUL1 Origin: BUL Problem A pentagon ABCDE inscribed in a circle for which BC < CD and AB < DE is the base of a pyramid with vertex S. If AS is the longest edge starting from S, prove that BS > CS. Solution Let P be the projection of S onto the plane ABCDE. Obviously BS > CS is equivalent to BP > CP. The conditions...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET50

IMO 1969 LL NET50 Origin: NET Problem The bisectors of the exterior angles of a pentagon B1B2B3B4B5 form another pentagon A1A2A3A4A5. Construct B1B2B3B4B5 from the given pentagon A1A2A3A4A5.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL POL70

IMO 1988 LL POL70 Origin: POL Problem In 3-dimensional space a point O is given and a ﬁnite set A of segments with the sum of the lengths equal to 1988. Prove that there exists a plane disjoint from A such that the distance from it to O does not exceed 574.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BUL13

IMO 1970 LL BUL13 Origin: BUL Problem A triangle ABC is given. Each side of ABC is divided into equal parts, and through each of the division points are drawn lines parallel to AB, BC, and CA, thus cutting ABC into small triangles. To each of the vertices of these triangles is assigned 1, 2, or 3, so that: (1) to A, B, C are assigned 1, 2 and 3...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GDR37

IMO 1986 LL GDR37 Origin: GDR Problem Prove that the set {1, 2, . . ., 1986} can be partitioned into 27 disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE59

IMO 1969 LL SWE59 Origin: SWE Problem For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .) construct a continuous function f such that there do not exist x, y with 0 < \lambda < y = x + \lambda \leq1 for which f(x) = f(y).

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CAN14

IMO 1985 LL CAN14 Origin: CAN Problem Let k be a positive integer. Deﬁne u0 = 0, u1 = 1, and un = kun−1 −un−2, n \geq2. Show that for each integer n, the number u3 1 + u3 2 + \cdot \cdot \cdot + u3 n is a multiple of u1 + u2 + \cdot \cdot \cdot + un.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL VIE76

IMO 1979 LL VIE76 Origin: VIE Problem Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter 6.25. Find the sides of the triangle.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA18

IMO 1969 LL FRA18 Origin: FRA Problem Let a and b be two nonnegative integers. Denote by H(a, b) the set of numbers n of the form n = pa + qb, where p and q are positive integers. Determine H(a) = H(a, a). Prove that if a ̸= b, it is enough to know all the sets H(a, b) for coprime numbers a, b in order to know all...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL USS56

IMO 1970 LL USS56 Origin: USS Problem A square hole of depth h whose base is of length a is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length L > \sqrt 2a2 + h2, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL TUR62

IMO 1987 LL TUR62 Origin: TUR Problem Let l, l′ be two lines in 3-space and let A, B, C be three points taken on l with B as midpoint of the segment AC. If a, b, c are the distances of A, B, C from l′, respectively, show that b \leq a2+c2 , equality holding if l, l′ are parallel.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL VIE58

IMO 1977 LL VIE58 Origin: VIE Problem Prove that for every triangle the following inequality holds: ab + bc + ca 4S \geqcot \pi 6 , where a, b, c are lengths of the sides and S is the area of the triangle. Solution The following inequality (Finsler and Hadwiger, 1938) is sharper than the one we have to prove: 2ab + 2bc + 2ca −a2 −b2 −c2 \geq4S \sqrt...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL SWE51

IMO 1970 LL SWE51 Origin: SWE Problem Let p be a prime number. A rational number x, with 0 < x < 1, is written in lowest terms. The rational number obtained from x by adding p to both the numerator and the denominator diﬀers from x by 1/p2. Determine all rational numbers x with this property.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL36

IMO 1967 LL POL36 Origin: POL Problem Prove that the center of the sphere circumscribed around a tetrahedron ABCD coincides with the center of a sphere inscribed in that tetrahedron if and only if AB = CD, AC = BD, and AD = BC. Solution Suppose that the skew edges of the tetrahedron ABCD are equal. Let K, L, M, P, Q, R be the midpoints of edges AB, AC,...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL IRE50

IMO 1988 LL IRE50 Origin: IRE Problem Let g(n) be deﬁned as follows: g(1) = 0, g(2) = 1, g(n + 2) = g(n) + g(n + 1) + 1 (n \geq1). Prove that if n > 5 is a prime, then n divides g(n)(g(n) + 1).

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL USA64

IMO 1987 LL USA64 Origin: USA Problem Let r > 1 be a real number, and let n be the largest integer smaller than r. Consider an arbitrary real number x with 0 \leqx \leq n r−1. By a base-r expansion of x we mean a representation of x in the form x = a1 r + a2 r2 + a3 r3 + \cdot \cdot \cdot , where the ai...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL KOR47

IMO 1992 LL KOR47 Origin: KOR Problem Find the largest integer not exceeding $1992 n=1 3n+2 3n+1.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ICE40

IMO 1989 LL ICE40 Origin: ICE Problem A sequence of real numbers x0, x1, x2, . . . is deﬁned as follows: x0 = 1989 and for each n \geq1 xn = −1989 n n−1 k=0 xk. Calculate the value of 1989 n=0 2nxn.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FRG26

IMO 1979 LL FRG26 Origin: FRG Problem Let n be a natural number. If 4n + 2n + 1 is a prime, prove that n is a power of three.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN25

IMO 1967 LL HUN25 Origin: HUN Problem Three disks of diameter d are touching a sphere at their centers. Moreover, each disk touches the other two disks. How do we choose the radius R of the sphere so that the axis of the whole ﬁgure makes an angle 1 The statement so formulated is false. It would be trivially true under the addi- tional assumption that the polygonal line is...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL BUL7

IMO 1971 LL BUL7 Origin: BUL Problem In a triangle ABC, let H be its orthocenter, O its circumcenter, and R its circumradius. Prove that: (a) |OH| = R\sqrt1 −8 cos \alpha cos \beta cos \gamma, where \alpha, \beta, \gamma are angles of the tri- angle ABC; (b) O \equivH if and only if ABC is equilateral.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL POL42

IMO 1982 LL POL42 Origin: POL Problem Let F be the family of all k-element subsets of the set {1, 2, . . ., 2k + 1}. Prove that there exists a bijective function f : F \toF such that for every A \inF, the sets A and f(A) are disjoint.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SPA77

IMO 1988 LL SPA77 Origin: SPA Problem Consider h + 1 chessboards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of h?

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS6

IMO 1966 LL USS6 Origin: USS Problem A convex planar polygon M with perimeter l and area S is given. Let M(R) be the set of all points in space that lie a distance at most R from a point of M. Show that the volume V (R) of this set equals V (R) = 4 3\piR3 + \pi 2 lR2 + 2SR.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL IRE39

IMO 1985 LL IRE39 Origin: IRE Problem Given a triangle ABC and external points X, Y , and Z such that ∡BAZ = ∡CAY , ∡CBX = ∡ABZ, and ∡ACY = ∡BCX, prove that AX, BY , and CZ are concurrent.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL USA80

IMO 1992 LL USA80 Origin: USA Problem Given a graph with n vertices and a positive integer m that is less than n, prove that the graph contains a set of m + 1 vertices in which the diﬀerence between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most m −1.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL41

IMO 1967 LL POL41 Origin: POL Problem A line l is drawn through the intersection point H of the altitudes of an acute-angled triangle. Prove that the symmetric images la, lb, lc of l with respect to sides BC, CA, AB have one point in common, which lies on the circumcircle of ABC. Solution It is well known that the points K, L, M, symmetric to H with respect to...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL MON62

IMO 1988 LL MON62 Origin: MON Problem The positive integer n has the property that in any set of n integers chosen from the integers 1, 2, . . . , 1988, twenty-nine of them form an arithmetic progression. Prove that n > 1788.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL SWE35

IMO 1978 LL SWE35 Origin: SWE Problem A sequence (an)N 0 of real numbers is called concave if 2an \geq an−1 + an+1 for all integers n, 1 \leqn \leqN −1. (a) Prove that there exists a constant C > 0 such that N n=0 an \geqC(N −1) N n=0 a2 n (1) for all concave positive sequences (an)N 0 . (b) Prove that (1) holds with C...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GBR30

IMO 1985 LL GBR30 Origin: GBR Problem A plane rectangular grid is given and a “rational point” is deﬁned as a point (x, y) where x and y are both rational numbers. Let A, B, A′, B′ be four distinct rational points. Let P be a point such that A′B′ AB = B′P BC = PA′ PA . In other words, the triangles ABP, A′B′P are directly or oppositely similar....

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE92

IMO 1988 LL VIE92 Origin: VIE Problem Let p \geq2 be a natural number. Prove that there exists an integer n0 such that n0 i=1 i p\sqrti + 1 > p.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GDR29

IMO 1969 LL GDR29 Origin: GDR Problem Find all real numbers \lambda such that the equation sin4 x −cos4 x = \lambda(tan4 x −cot4 x) (a) has no solution, (b) has exactly one solution, (c) has exactly two solutions, (d) has more than two solutions (in the interval (0, \pi/4)).

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CZS10

IMO 1974 LL CZS10 Origin: CZS Problem A regular octagon P is given whose incircle k has diameter 1. About k is circumscribed a regular 16-gon, which is also inscribed in P, cutting from P eight isosceles triangles. To the octagon P, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every 11-gon so obtained...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL BUL4

IMO 1977 LL BUL4 Origin: BUL Problem We are given n points in space. Some pairs of these points are connected by line segments so that the number of segments equals [n2/4], and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle. Solution Consider any vertex vn from which the maximal number d of seg- ments...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL GBR16

IMO 1972 LL GBR16 Origin: GBR Problem Consider the set S of all the diﬀerent odd positive integers that are not multiples of 5 and that are less than 30m, m being a positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two integers one of which divides the other? Prove your result.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USA38

IMO 1974 LL USA38 Origin: USA Problem Consider the binomial coeﬃcients n k n! k!(n−k)! (k = 1, 2, . . . , n−1). Determine all positive integers n for which n , n , . . . , n n−1 are all even numbers.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ISR43

IMO 1979 LL ISR43 Origin: ISR Problem Let a, b, c denote the lengths of the sides BC, CA, AB, respec- tively, of a triangle ABC. If P is any point on the circumference of the circle inscribed in the triangle, show that aPA2+bPB2+cPC2 is constant.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USA88

IMO 1985 LL USA88 Origin: USA Problem Determine the range of w(w + x)(w + y)(w + z), where x, y, z, and w are real numbers such that x + y + z + w = x7 + y7 + z7 + w7 = 0.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FIN18

IMO 1979 LL FIN18 Origin: FIN Problem Show that for no integers a \geq1, n \geq1 is the sum 1 + 1 + a + 1 + 2a + \cdot \cdot \cdot + 1 + na an integer.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL NET52

IMO 1992 LL NET52 Origin: NET Problem Let n be an integer > 1. In a circular arrangement of n lamps L0, . . . , Ln−1, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, Step0, Step1, . . . . If Lj−1 (j is taken mod n) is ON,...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SWE56

IMO 1984 LL SWE56 Origin: SWE Problem Let a, b, c be nonnegative integers such that a \leqb \leqc, 2b ̸= a + c and a+b+c is an integer. Is it possible to ﬁnd three nonnegative integers d, e, and f such that d \leqe \leqf, f ̸= c, and such that a2+b2+c2 = d2 + e2 + f 2?

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA46

IMO 1988 LL INA46 Origin: INA Problem (a) Calculate x = (11+6 \sqrt 2)\sqrt 11−6 \sqrt 2−(11−6 \sqrt 2)\sqrt 11+6 \sqrt (\sqrt\sqrt 5+2+\sqrt\sqrt 5−2)−(\sqrt\sqrt 5+1) . (b) For each positive number x, let k = (x+1/x)6−(x6+1/x6)−2 (x+1/x)3+(x3+1/x3) . Calculate the minimum value of k.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRA22

IMO 1986 LL FRA22 Origin: FRA Problem Let (an)n\inN be the sequence of integers deﬁned recursively by a0 = 0, a1 = 1, an+2 = 4an+1 + an for n \geq0. Find the common divisors of a1986 and a6891.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GDR31

IMO 1969 LL GDR31 Origin: GDR Problem Find the number of permutations a1, . . . , an of the set {1, 2, . . ., n} such that |ai −ai+1| ̸= 1 for all i = 1, 2, . . ., n −1. Find a recurrence formula and evaluate the number of such permutations for n \leq6.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL35

IMO 1966 LL POL35 Origin: POL Problem If a, b, c, d are integers such that ad is odd and bc is even, prove that at least one root of the polynomial ax3 + bx2 + cx + d is irrational.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CAN13

IMO 1985 LL CAN13 Origin: CAN Problem Find the average of the quantity (a1 −a2)2 + (a2 −a3)2 + \cdot \cdot \cdot + (an−1 −an)2 taken over all permutations (a1, a2, . . . , an) of (1, 2, . . . , n).

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GBR17

IMO 1967 LL GBR17 Origin: GBR Problem Let k, m, and n be positive integers such that m+k + 1 is a prime number greater than n + 1. Write cs for s(s + 1). Prove that the product (cm+1 −ck)(cm+2 −ck) \cdot \cdot \cdot (cm+n −ck) is divisible by the product c1c2 \cdot \cdot \cdot cn. Solution Using cr −cs = (r −s)(r + s + 1) we can...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GDR18

IMO 1976 LL GDR18 Origin: GDR Problem Prove that the number 191976 + 761976: (a) is divisible by the (Fermat) prime number F4 = 224 + 1; (b) is divisible by at least four distinct primes other than F4.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL HKG23

IMO 1992 LL HKG23 Origin: HKG Problem An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1 2 + 1 3 + 1 9 + 18 = 1. Prove that all integers greater than 23 are...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS52

IMO 1966 LL USS52 Origin: USS Problem A ﬁgure of area 1 is cut out from a sheet of paper and divided into 10 parts, each of which is colored in one of 10 colors. Then the ﬁgure is turned to the other side and again divided into 10 parts (not necessarily in the same way). Show that it is possible to color these parts in the 10 colors so...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CUB7

IMO 1974 LL CUB7 Origin: CUB Problem Let P be a prime number and n a natural number. Prove that the product N = pn2 2n−1 i=1; 2∤i ((p −1)i)! p2i pi is a natural number that is not divisible by p.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL NOR61

IMO 1985 LL NOR61 Origin: NOR Problem Consider the set A = {0, 1, 2, . . ., 9} and let (B1, B2, . . . , Bk) be a collection of nonempty subsets of A such that Bi \capBj has at most two elements for i ̸= j. What is the maximal value of k?

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON32

IMO 1967 LL MON32 Origin: MON Problem Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are \alpha, \beta, \gamma. Solution Let us denote by V the volume of the given body, and by Va, Vb, Vc the volumes of the parts of the given ball that lie inside the...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USS50

IMO 1982 LL USS50 Origin: USS Problem Let O be the midpoint of the axis of a right circular cylinder. Let A and B be diametrically opposite points of one base, and C a point of the other base circle that does not belong to the plane OAB. Prove that the sum of dihedral angles of the trihedral OABC is equal to 2\pi.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL USA103

IMO 1989 LL USA103 Origin: USA Problem An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not diﬃcult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET35

IMO 1970 LL NET35 Origin: NET Problem Find for every value of n a set of numbers p for which the fol- lowing statement is true: Any convex n-gon can be divided into p isosceles triangles. Alternative version. The same about division into p polygons with axis of symmetry.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL SWE33

IMO 1976 LL SWE33 Origin: SWE Problem A ﬁnite set of points P in the plane has the following prop- erty: Every line through two points in P contains at least one more point belonging to P. Prove that all points in P lie on a straight line.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL POL64

IMO 1985 LL POL64 Origin: POL Problem Let p be a prime. For which k can the set {1, 2, . . ., k} be partitioned into p subsets with equal sums of elements?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HUN38

IMO 1989 LL HUN38 Origin: HUN Problem Connecting the vertices of a regular n-gon we obtain a closed (not necessarily convex) n-gon. Show that if n is even, then there are two parallel segments among the connecting segments and if n is odd then there cannot be exactly two parallel segments.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL FIN44

IMO 1977 LL FIN44 Origin: FIN Problem Let E be a ﬁnite set of points in space such that E is not contained in a plane and no three points of E are collinear. Show that E contains the vertices of a tetrahedron T = ABCD such that T \capE = {A, B, C, D} (including interior points of T ) and such that the projection of A onto the...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL SWE40

IMO 1971 LL SWE40 Origin: SWE Problem Prove that 1 −1 1 −1 1 −1 \cdot \cdot \cdot 1 −1 n3 1 2, n = 2, 3, . . ..

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL YUG46

IMO 1966 LL YUG46 Origin: YUG Problem Let f(a, b, c) = |b −a| |ab| b + a ab −2 c |b −a| |ab| b + a ab 2 c . Prove that f(a, b, c) = 4 max{1/a, 1/b, 1/c}.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN36

IMO 1969 LL HUN36 Origin: HUN Problem In the plane 4000 points are given such that each line passes through at most 2 of these points. Prove that there exist 1000 disjoint quadrilaterals in the plane with vertices at these points.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL THA97

IMO 1989 LL THA97 Origin: THA Problem Let n be a positive integer, X = {1, 2, . . ., n}, and k a positive integer such that n/2 \leqk \leqn. Determine, with proof, the number of all functions f : X \toX that satisfy the following conditions: (i) f 2 = f; (ii) the number of elements in the image of f is k; (iii) for each y in...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL HUN38

IMO 1979 LL HUN38 Origin: HUN Problem Prove the following statement: If a polynomial f(x) with real coeﬃcients takes only nonnegative values, then there exists a positive integer n and polynomials g1(x), g2(x), . . . , gn(x) such that f(x) = g1(x)2 + g2(x)2 + \cdot \cdot \cdot + gn(x)2.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GDR37

IMO 1985 LL GDR37 Origin: GDR Problem Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and area A satisﬁes tan \alpha 2 + tan \beta 2 + tan \gamma 2 \leq9R2 4A .

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SWE58

IMO 1984 LL SWE58 Origin: SWE Problem Let (an)\infty 1 be a sequence such that an \leqan+m \leqan + am for all positive integers n and m. Prove that an n has a limit as n approaches inﬁnity.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL GRE26

IMO 1988 LL GRE26 Origin: GRE Problem Let AB and CD be two perpendicular chords of a circle with center O and radius r, and let X, Y, Z, W denote in cyclical order the four parts into which the disk is thus divided. Find the maximum and minimum of the quantity A(Z) A(Y )+A(W), where A(U) denotes the area of U.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL21

IMO 1966 LL BUL21 Origin: BUL Problem Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality 6V \pi 2 \leq 2S \pi \sqrt . When does equality occur?

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL24

IMO 1966 LL POL24 Origin: POL Problem There are n \geq2 people in a room. Prove that there exist two among them having equal numbers of friends in that room. (Friendship is always mutual.)

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRA12

IMO 1988 LL FRA12 Origin: FRA Problem Show that there do not exist more than 27 half-lines (or rays) emanating from the origin in 3-dimensional space such that the angle between each pair of rays is greater than of equal to \pi/4.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IND27

IMO 1992 LL IND27 Origin: IND Problem Let ABC be an arbitrary scalene triangle. Deﬁne \Sigma to be the set of all circles y that have the following properties: (i) y meets each side of \triangleABC in two (possibly coincident) points; (ii) if the points of intersection of y with the sides of the triangle are la- beled by P, Q, R, S, T , U, with the points occurring...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRG15

IMO 1988 LL FRG15 Origin: FRG Problem Let 1 \leqk < n. Consider all ﬁnite sequences of positive integers with sum n. Find T (n, k), the total number of terms of size k in all of these sequences.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL POL31

IMO 1976 LL POL31 Origin: POL Problem Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL BUL3

IMO 1977 LL BUL3 Origin: BUL Problem In a company of n persons, each person has no more than d acquaintances, and in that company there exists a group of k persons, k \geqd, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than [n2/4]. Solution Let v1, v2, . . . , vk be k persons who are not acquainted with...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL BEL9

IMO 1987 LL BEL9 Origin: BEL Problem In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z} we have made a random sequence of 28 throws. What is the probability that the sequence CUBA JULY 1987 appears in this order in the sequence already thrown?

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ROM70

IMO 1985 LL ROM70 Origin: ROM Problem Let C be a class of functions f : N \toN that contains the functions S(x) = x + 1 and E(x) = x −[\sqrtx]2 for every x \inN. ([x] is the integer part of x.) If C has the property that for every f, g \inC, f + g, fg, f ◦g \inC, show that the function max(f(x) −g(x), 0) is in...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL GDR19

IMO 1971 LL GDR19 Origin: GDR Problem In a triangle P1P2P3 let PiQi be the altitude from Pi for i = 1, 2, 3 (Qi being the foot of the altitude). The circle with diameter PiQi meets the two corresponding sides at two points diﬀerent from Pi. Denote the length of the segment whose endpoints are these two points by li. Prove that l1 = l2 = l3.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USS67

IMO 1984 LL USS67 Origin: USS Problem With the medians of an acute-angled triangle another triangle is constructed. If R and Rm are the radii of the circles circumscribed about the ﬁrst and the second triangle, respectively, prove that Rm > 5 6R.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ROM68

IMO 1985 LL ROM68 Origin: ROM Problem Show that the sequence {an}n\geq1 deﬁned by an = [n \sqrt 2] contains an inﬁnite number of integer powers of 2. ([x] is the integer part of x.)

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL CZS12

IMO 1972 LL CZS12 Origin: CZS Problem A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed in it. The lengths of sides of the hexagon satisfy AA′ = A′B, BB′ = B′C, CC′ = C′A. Prove that the area P of triangle ABC is not greater than the area P ′ of triangle A′B′C′. When does P = P ′ hold?

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL CZS10

IMO 1972 LL CZS10 Origin: CZS Problem Given ﬁve points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL GDR24

IMO 1988 LL GDR24 Origin: GDR Problem Let Zm,n be the set of all ordered pairs (i, j) with i \in {1, . . . , m} and j \in{1, . . ., n}. Also let am,n be the number of all those subsets of Zm,n that contain no two ordered pairs (i1, j1), (i2, j2) with |i1 −i2| + |j1 −j2| = 1. Show that for all positive integers...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GRE32

IMO 1987 LL GRE32 Origin: GRE Problem Solve the equation 28x = 19y + 87z, where x, y, z are integers.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL VIE49

IMO 1976 LL VIE49 Origin: VIE Problem Determine whether there exist 1976 nonsimilar triangles with angles \alpha, \beta, \gamma, each of them satisfying the relations sin \alpha + sin \beta + sin \gamma cos \alpha + cos \beta + cos \gamma = 12 and sin \alpha sin \beta sin \gamma = 12 25.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM39

IMO 1966 LL ROM39 Origin: ROM Problem In a plane, a circle with center O and radius R and two points A, B are given. (a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P, say P1, P2, and ﬁnd the distance between them if OA = a,...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ROM61

IMO 1986 LL ROM61 Origin: ROM Problem Given a positive integer n, ﬁnd the greatest integer p with the property that for any function f : P(X) \toC, where X and C are sets of cardinality n and p, respectively, there exist two distinct sets A, B \inP(X) such that f(A) = f(B) = f(A \cupB). (P(X) is the family of all subsets of X.)

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM9

IMO 1966 LL ROM9 Origin: ROM Problem Find x such that sin 3x cos(60◦−4x) + 1 sin(60◦−7x) −cos(30◦+ x) + m = 0, where m is a ﬁxed real number.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL1

IMO 1969 LL BEL1 Origin: BEL Problem A parabola P1 with equation x2 −2py = 0 and parabola P2 with equation x2 + 2py = 0, p > 0, are given. A line t is tangent to P2. Find the locus of pole M of the line t with respect to P1.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON42

IMO 1969 LL MON42 Origin: MON Problem Let Ak (1 \leqk \leqh) be n-element sets such that each two of them have a nonempty intersection. Let A be the union of all the sets Ak, and let B be a subset of A such that for each k (1 \leqk \leqh) the intersection of Ak and B consists of exactly two diﬀerent elements ak and bk. Find all subsets X...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS50

IMO 1966 LL USS50 Origin: USS Problem Given a quadrangle of sides a, b, c, d and area S, show that S \leq a+c \cdot b+d 2 .

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL SWE40

IMO 1977 LL SWE40 Origin: SWE Problem The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size 2 \times 2. Prove that there are at least three such squares for which the sum of the 4 numbers contained exceeds 100. Solution Let us divide the chessboard into 16 squares Q1, Q2, . ....

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BRA12

IMO 1983 LL BRA12 Origin: BRA Problem The number 0 or 1 is to be assigned to each of the n vertices of a regular polygon. In how many diﬀerent ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GBR25

IMO 1969 LL GBR25 Origin: GBR Problem Let a, b, x, y be positive integers such that a and b have no common divisor greater than 1. Prove that the largest number not ex- pressible in the form ax + by is ab −a −b. If N(k) is the largest number not expressible in the form ax + by in only k ways, ﬁnd N(k).

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL FIN46

IMO 1977 LL FIN46 Origin: FIN Problem Let f be a strictly increasing function deﬁned on the set of real numbers. For x real and t positive, set g(x, t) = f(x + t) −f(x) f(x) −f(x −t). Assume that the inequalities 2−1 < g(x, t) < 2 hold for all positive t if x = 0, and for all t \leq|x| otherwise. Show that 14−1 < g(x, t) <...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL GDR17

IMO 1977 LL GDR17 Origin: GDR Problem A ball K of radius r is touched from the outside by mutually equal balls of radius R. Two of these balls are tangent to each other. Moreover, for two balls K1 and K2 tangent to K and tangent to each other there exist two other balls tangent to K1, K2 and also to K. How many balls are tangent to K? For...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA47

IMO 1989 LL INA47 Origin: INA Problem Let log2 2 x −4 log2 x −m2 −2m −13 = 0 be an equation in x. Prove: (a) For any real value of m the equation has has two distinct solutions. (b) The product of the solutions of the equation does not depend on m. (c) One of the solutions of the equation is less than 1, while the other solution is...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL GDR35

IMO 1982 LL GDR35 Origin: GDR Problem If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL NET31

IMO 1978 LL NET31 Origin: NET Problem Let the polynomials P(x) = xn + an−1xn−1 + \cdot \cdot \cdot + a1x + a0, Q(x) = xm + bm−1xm−1 + \cdot \cdot \cdot + b1x + b0, be given satisfying the identity P(x)2 = (x2 −1)Q(x)2 + 1. Prove the identity P ′(x) = nQ(x).

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL PHI76

IMO 1989 LL PHI76 Origin: PHI Problem Let k and s be positive integers. For sets of real numbers {\alpha1, \alpha2, . . . , \alphas} and {\beta1, \beta2, . . . , \betas} that satisfy s i=1 \alphaj i = s i=1 \betaj i for each j = 1, 2, . . ., k, we write {\alpha1, \alpha2, . . . , \alphas} =k {\beta1, \beta2, . . ....

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE58

IMO 1969 LL SWE58 Origin: SWE Problem Six points P1, . . . , P6 are given in 3-dimensional space such that no four of them lie in the same plane. Each of the line segments PjPk is colored black or white. Prove that there exists one triangle PjPkPl whose edges are of the same color.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON34

IMO 1967 LL MON34 Origin: MON Problem The faces of a convex polyhedron are six squares and eight equilateral triangles, and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge are equal. Prove that it is possible to circumscribe a sphere around this polyhedron and compute the ratio of the squares of the volumes...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL FRA26

IMO 1970 LL FRA26 Origin: FRA Problem Consider a ﬁnite set of vectors in space {a1, a2, . . . , an} and the set E of all vectors of the form x = \lambda1a1 +\lambda2a2 +\cdot \cdot \cdot+\lambdanan, where \lambdai are nonnegative numbers. Let F be the set consisting of all the vectors in E and vectors parallel to a given plane P. Prove that there exists a set...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS54

IMO 1967 LL USS54 Origin: USS Problem Is it possible to put 100 (or 200) points on a wooden cube such that by all rotations of the cube the points map into themselves? Justify your answer. Solution Let S be the given set of points on the cube. Let x, y, z denote the numbers of points from S lying at a vertex, at the midpoint of an edge, at...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL BRA7

IMO 1985 LL BRA7 Origin: BRA Problem A convex quadrilateral is inscribed in a circle of radius 1. Prove that the diﬀerence between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than 2.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS26

IMO 1966 LL CZS26 Origin: CZS Problem (a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2 1 +\cdot \cdot \cdot+a2 k), where k \geq1 is a natural number and a1, . . . , ak are arbitrary real numbers. (b) If real numbers a1, . . . , an satisfy a1 + a2 + \cdot \cdot \cdot + an \geq (n −1)(a2 1 + \cdot \cdot \cdot + a2n), show...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL KUW39

IMO 1983 LL KUW39 Origin: KUW Problem If \alpha is the real root of the equation E(x) = x3 −5x −50 = 0 such that xn+1 = (5xn + 50)1/3 and x1 = 5, where n is a positive integer, prove that: (a) x3 n+1 −\alpha3 = 5(xn −\alpha) (b) \alpha < xn+1 < xn

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GDR36

IMO 1985 LL GDR36 Origin: GDR Problem Determine whether there exist 100 distinct lines in the plane having exactly 1985 distinct points of intersection.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MON43

IMO 1987 LL MON43 Origin: MON Problem Let 2n + 3 points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly n interior points is not less than 1 2n+3 .

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL KOR60

IMO 1989 LL KOR60 Origin: KOR Problem A real-valued function f on Q satisﬁes the following conditions for arbitrary \alpha, \beta \inQ: (i) f(0) = 0, (ii) f(\alpha) > 0 if \alpha ̸= 0, (iii) f(\alpha\beta) = f(\alpha)f(\beta), (iv) f(\alpha + \beta) \leqf(\alpha) + f(\beta), (v) f(m) \leq1989 for all m \inZ. Prove that f(\alpha + \beta) = max{f(\alpha), f(\beta)} if f(\alpha) ̸= f(\beta). Here, Z, Q denote the sets...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL TUR102

IMO 1989 LL TUR102 Origin: TUR Problem If in a convex quadrilateral ABCD, E and F are the midpoints of the sides BC and DA respectively. Show that the sum of the areas of the triangles EDA and FBC is equal to the area of the quadrangle.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL KOR45

IMO 1992 LL KOR45 Origin: KOR Problem Let n be a positive integer. Prove that the number of ways to express n as a sum of distinct positive integers (up to order) and the number of ways to express n as a sum of odd positive integers (up to order) are the same.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CZS17

IMO 1986 LL CZS17 Origin: CZS Problem We call a tetrahedron right-faced if each of its faces is a right- angled triangle. (a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra. (b) Prove that a tetrahedron with vertices A1, A2, A3, A4 is ﬁght-faced if and only if there exist four distinct real numbers c1, c2, c3, and c4 such that the edges AjAk have lengths...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE49

IMO 1967 LL SWE49 Origin: SWE Problem Let n and k be positive integers such that 1 \leqn \leqN + 1, 1 \leqk \leqN + 1. Show that min n̸=k | sin n −sin k| < 2 N . Solution Since sin 1, sin 2, . . . , sin(N +1) \in(−1, 1), two of these N +1 numbers have distance less than 2/N. Therefore | sin n −sin k|...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM30

IMO 1966 LL ROM30 Origin: ROM Problem If n is a natural number, prove that (a) log10(n + 1) > 10n + log10 n; (b) log n! > 3n 1 2 + 1 3 + \cdot \cdot \cdot + 1 n −1 .

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL BUL2

IMO 1974 LL BUL2 Origin: BUL Problem Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1, un = un−1 + un−2 for n > 1. Prove that there exist inﬁnitely many prime numbers p that divide up−1.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CHN13

IMO 1986 LL CHN13 Origin: CHN Problem Let N = {1, 2, . . ., n}, n \geq3. To each pair i, j of elements of N, i ̸= j, there is assigned a number fij \in{0, 1} such that fij + fji = 1. Let r(i) = j̸=i fij and write M = maxi\inN r(i), m = mini\inN r(i). Prove that for any w \inN with r(w) =...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL MON23

IMO 1972 LL MON23 Origin: MON Problem Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an arbitrary n) for which the following equality holds: a2n . . . a1 = (an . . . a1)2?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL HUN41

IMO 1979 LL HUN41 Origin: HUN Problem Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL POL51

IMO 1987 LL POL51 Origin: POL Problem The function F is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continu- ity is not assumed). Prove that F maps squares into squares.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL AUS1

IMO 1982 LL AUS1 Origin: AUS Problem It is well known that the binomial coeﬃcients n k n! k!(n−k)!, 0 \leqk \leqn, are positive integers. The factorial n! is deﬁned inductively by 0! = 1, n! = n \cdot (n −1)! for n \geq1. (a) Prove that n+1 2n n is an integer for n \geq0. (b) Given a positive integer k, determine the smallest integer Ck with...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USS68

IMO 1984 LL USS68 Origin: USS Problem In the Martian language every ﬁnite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the ﬁrst volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume....

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL CUB12

IMO 1989 LL CUB12 Origin: CUB Problem Let P(x) be a polynomial such that the following inequalities are satisﬁed: P(0) > 0; P(1) > P(0); P(2) > 2P(1) −P(0); P(3) > 3P(2) −3P(1) + P(0); and also for every natural number n, P(n + 4) > 4P(n + 3) −6P(n + 2) + 4P(n + 1) −P(n). Prove that for every positive natural number n, P(n) is positive.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL PRK61

IMO 1992 LL PRK61 Origin: PRK Problem There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards numbered with diﬀerent natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL CZS9

IMO 1972 LL CZS9 Origin: CZS Problem Given natural numbers k and n, k \leqn, n \geq3, ﬁnd the set of all values in the interval (0, \pi) that the kth-largest among the interior angles of a convex ngon can take.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG31

IMO 1988 LL HKG31 Origin: HKG Problem The circle x2 + y2 = r2 meets the coordinate axes at A = (r, 0), B = (−r, 0), C = (0, r), and D = (0, −r). Let P = (u, v) and Q = (−u, v) be two points on the circumference of the circle. Let N be the point of intersection of PQ and the y-axis, and let M...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL ROM27

IMO 1974 LL ROM27 Origin: ROM Problem Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary points on C1 and C2 respectively, and Q the midpoint of segment P1P2. Find the locus of points Q as P1 and P2 go through all possible positions. Alternative version. Let C1, C2, C3 be three circles in the same plane. Find the locus of the centroid of triangle...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CZS20

IMO 1985 LL CZS20 Origin: CZS Problem Let T be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points (x, y, z) and (u, v, w) are called neighbors if |x −u| + |y −v| + |z −w| = 1. Show that there exists a subset S of T such that for each p \inT , there is exactly one point...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL YUG79

IMO 1979 LL YUG79 Origin: YUG Problem Let S be a unit circle and K a subset of S consisting of several closed arcs. Let K satisfy the following properties: (i) K contains three points A, B, C, that are the vertices of an acute- angled triangle; (ii) for every point A that belongs to K its diametrically opposite point A′ and all points B on an arc of length...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM45

IMO 1970 LL ROM45 Origin: ROM Problem Let M be an interior point of tetrahedron V ABC. Denote by A1, B1, C1 the points of intersection of lines MA, MB, MC with the planes V BC, V CA, V AB, and by A2, B2, C2 the points of intersection of lines V A1, V B1, V C1 with the sides BC, CA, AB. (a) Prove that the volume of the...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USA37

IMO 1976 LL USA37 Origin: USA Problem From a square board 11 squares long and 11 squares wide, the central square is removed. Prove that the remaining 120 squares cannot be covered by 15 strips each 8 units long and one unit wide.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ROM57

IMO 1979 LL ROM57 Origin: ROM Problem Let M be a set, and A, B, C given subsets of M. Find a necessary and suﬃcient condition for the existence of a set X \subsetM for which (X \cupA) \ (X \capB) = C. Describe all such sets X.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE50

IMO 1967 LL SWE50 Origin: SWE Problem The function ϕ(x, y, z), deﬁned for all triples (x, y, z) of real numbers, is such that there are two functions f and g deﬁned for all pairs of real numbers such that ϕ(x, y, z) = f(x + y, z) = g(x, y + z) for all real x, y, and z. Show that there is a function h of one...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BUL8

IMO 1969 LL BUL8 Origin: BUL Problem Find all functions f deﬁned for all x that satisfy the condition xf(y) + yf(x) = (x + y)f(x)f(y), for all x and y. Prove that exactly two of them are continuous.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BUL10

IMO 1984 LL BUL10 Origin: BUL Problem Assume that the bisecting plane of the dihedral angle at edge AB of the tetrahedron ABCD meets the edge CD at point E. Denote by S1, S2, S3, respectively the areas of the triangles ABC, ABE, and ABD. Prove that no tetrahedron exists for which S1, S2, S3 (in this order) form an arithmetic or geometric progression.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL COL12

IMO 1992 LL COL12 Origin: COL Problem Given a triangle ABC such that the circumcenter is in the interior of the incircle, prove that the triangle ABC is acute-angled.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MOR39

IMO 1984 LL MOR39 Origin: MOR Problem Let ABC be an isosceles triangle, AB = AC, \angleA = 20◦. Let D be a point on AB, and E a point on AC such that \angleACD = 20◦and \angleABE = 30◦. What is the measure of the angle \angleCDE?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL HUN25

IMO 1971 LL HUN25 Origin: HUN Problem Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments A2B1, B2C1, C2A1 by P, Q, R in this order. Prove that the triangle PQR is equilateral.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL VIE96

IMO 1985 LL VIE96 Origin: VIE Problem Determine all functions f : R \toR satisfying the following two conditions: (a) f(x + y) + f(x −y) = 2f(x)f(y) for all x, y \inR, (b) limx\to\inftyf(x) = 0.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL GBR19

IMO 1974 LL GBR19 Origin: GBR Problem (Alternative to GBR 2) Prove that there exists, for n \geq4, a set S of 3n equal circles in spacethat can be partitioned into three subsets s5, s4, and s3, each containing n circles, such that each circle in sr touches exactly r circles in S.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ROM69

IMO 1985 LL ROM69 Origin: ROM Problem Let A and B be two ﬁnite disjoint sets of points in the plane such that any three distinct points in A\cupB are not collinear. Assume that at least one of the sets A, B contains at least ﬁve points. Show that there exists a triangle all of whose vertices are contained in A or in B that does not contain in its...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL USA65

IMO 1983 LL USA65 Origin: USA Problem Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect in a point P. Prove that AP PC = cot \angleBAC + cot \angleDAC cot \angleBCA + cot \angleDCA.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL BEL4

IMO 1985 LL BEL4 Origin: BEL Problem Let x, y, and z be real numbers satisfying x + y + z = xyz. Prove that x(1 −y2)(1 −z2) + y(1 −z2)(1 −x2) + z(1 −x2)(1 −y2) = 4xyz.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM47

IMO 1966 LL ROM47 Origin: ROM Problem Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the triangle. Compute this minimum length in terms of the sides a, b, c of the triangle.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SWE76

IMO 1985 LL SWE76 Origin: SWE Problem Are there integers m and n such that 5m2 −6mn + 7n2 = 1985?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL USS48

IMO 1971 LL USS48 Origin: USS Problem A sequence of real numbers x1, x2, . . . , xn is given such that xi+1 = xi + 30000 1 −x2 i , i = 1, 2, . . ., and x1 = 0. Can n be equal to 50000 if xn < 1?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS15

IMO 1969 LL CZS15 Origin: CZS Problem Let K1, . . . , Kn be nonnegative integers. Prove that K1!K2! \cdot \cdot \cdot Kn! \geq[K/n]!n, where K = K1 + \cdot \cdot \cdot + Kn.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL56

IMO 1969 LL POL56 Origin: POL Problem Let a and b be two natural numbers that have an equal number n of digits in their decimal expansions. The ﬁrst m digits (from left to right) of the numbers a and b are equal. Prove that if m > n/2, then a1/n −b1/n < 1 n.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRG27

IMO 1986 LL FRG27 Origin: FRG Problem In an urn there are n balls numbered 1, 2, . . . , n. They are drawn at random one by one one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL CAN5

IMO 1992 LL CAN5 Origin: CAN Problem Let I, H, O be the incenter, centroid, and circumcenter of the nonisosceles triangle ABC. Prove that AI\parallelHO if and only if ∡BAC = 120◦.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL CAN18

IMO 1982 LL CAN18 Origin: CAN Problem You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that (a + ab−1a)−1 + (a + b)−1 = a−1, where x−1 is the element for which x−1x = xx−1 = e, where e is the element of the system such that for all a the equality ea...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN38

IMO 1969 LL HUN38 Origin: HUN Problem Let r and m (r \leqm) be natural numbers and Ak = 2k−1 2m \pi. Evaluate m2 m k=1 m l=1 sin(rAk) sin(rAl) cos(rAk −rAl).

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MOR37

IMO 1984 LL MOR37 Origin: MOR Problem Denote by [x] the greatest integer not exceeding x. For all real k > 1, deﬁne two sequences: an(k) = [nk] and bn(k) = nk k −1 . If A(k) = {an(k) : n \inN} and B(k) = {bn(k) : n \inN}, prove that A(k) and B(k) form a partition of N if and only if k is irrational.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL GBR19

IMO 1972 LL GBR19 Origin: GBR Problem Let S be a subset of the real numbers with the following properties: (i) If x \inS and y \inS, then x −y \inS; (ii) If x \inS and y \inS, then xy \inS; (iii) S contains an exceptional number x′ such that there is no number y in S satisfying x′y + x′ + y = 0; (iv) If x \inS and...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CYP19

IMO 1985 LL CYP19 Origin: CYP Problem Solve the system of simultaneous equations \sqrtx −1/y − 2w + 3z = 1, x + 1/y2 −4w2 −9z2 = 3, x\sqrtx −1/y3 −8w3 + 27z3 = −5, x2 + 1/y4 −16w4 −81z4 = 15.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL5

IMO 1969 LL BEL5 Origin: BEL Problem Let G be the centroid of the triangle OAB. (a) Prove that all conics passing through the points O, A, B, G are hyper- bolas. (b) Find the locus of the centers of these hyperbolas.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL YUG51

IMO 1974 LL YUG51 Origin: YUG Problem There are n points on a ﬂat piece of paper, any two of them at a distance of at least 2 from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals 3/2. Prove that there exist two vectors of equal length less than 1 and with their sum having...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USA89

IMO 1985 LL USA89 Origin: USA Problem Given that n elements a1, a2, . . . , an are organized into n pairs P1, P2, . . . , Pn in such a way that two pairs Pi, Pj share exactly one el- ement when (ai, aj) is one of the pairs, prove that every element is in exactly two of the pairs.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GBR22

IMO 1978 LL GBR22 Origin: GBR Problem Two nonzero integers x, y (not necessarily positive) are such that x + y is a divisor of x2 + y2, and the quotient x2+y2 x+y is a divisor of

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BEL11

IMO 1970 LL BEL11 Origin: BEL Problem Let ABCD and A′B′C′D′ be two squares in the same plane and oriented in the same direction. Let A′′, B′′, C′′, and D′′ be the midpoints of AA′, BB′, CC′, and DD′. Prove that A′′B′′C′′D′′ is also a square.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL SPA59

IMO 1983 LL SPA59 Origin: SPA Problem Solve the equation tan2(2x) + 2 tan(2x) \cdot tan(3x) −1 = 0.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BRA7

IMO 1979 LL BRA7 Origin: BRA Problem M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four. Given that: (i) for each i = 1, 2, 3, 4 and for each k = 5, 6, 7, ai,k = ai,k−4; Pi = a1,i + a2,i+1 + a3,i+2 + a4,i+3; Si = a4,i + a3,i+1 + a2,i+2 + a1,i+3; Li = ai,1 + ai,2 +...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL NET42

IMO 1984 LL NET42 Origin: NET Problem Triangle ABC is given for which BC = AC + 1 2AB. The point P divides AB such that RP : PA = 1 : 3. Prove that \angleCAP = 2\angleCPA.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL2

IMO 1969 LL BEL2 Origin: BEL Problem (a) Find the equations of regular hyperbolas passing through the points A(\alpha, 0), B(\beta, 0), and C(0, \gamma). (b) Prove that all such hyperbolas pass through the orthocenter H of the triangle ABC. (c) Find the locus of the centers of these hyperbolas. (d) Check whether this locus coincides with the nine-point circle of the triangle ABC.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL GRE29

IMO 1988 LL GRE29 Origin: GRE Problem Find positive integers x1, x2, . . . , x29, at least one of which is greater than 1988, such that x2 1 + x2 2 + \cdot \cdot \cdot + x2 29 = 29x1x2 . . . x29.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL55

IMO 1969 LL POL55 Origin: POL Problem Find the conditions on the positive real number a such that there exists a tetrahedron k of whose edges (k = 1, 2, 3, 4, 5) have length a, and the other 6 −k edges have length 1.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FIN14

IMO 1992 LL FIN14 Origin: FIN Problem Integers a1, a2, . . . , an satisfy |ak| = 1 and n k=1 akak+1ak+2ak+3 = 2, where an+j = aj. Prove that n ̸= 1992.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL POL38

IMO 1982 LL POL38 Origin: POL Problem Numbers un,k (1 \leqk \leqn) are deﬁned as follows: u1,1 = 1, un,k = n k − d|n, d|k, d>1 un/d,k/d (the empty sum is deﬁned to be equal to zero). Prove that n | un,k for every natural number n and for every k (1 \leqk \leqn).

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL TUN43

IMO 1982 LL TUN43 Origin: TUN Problem (a) What is the maximal number of acute angles in a convex polygon? (b) Consider m points in the interior of a convex n-gon. The n-gon is partitioned into triangles whose vertices are among the n + m given points (the vertices of the n-gon and the given points). Each of the m points in the interior is a vertex of at least...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GDR28

IMO 1978 LL GDR28 Origin: GDR Problem Let c, s be real functions deﬁned on R{0} that are nonconstant on any interval and satisfy c x y = c(x)c(y) −s(x)s(y) for any x ̸= 0, y ̸= 0. Prove that: (a) c(1/x) = c(x), s(1/x) = −s(x) for any x ̸= 0, and also c(1) = 1, s(1) = s(−1) = 0; (b) c and s are either both...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CZS12

IMO 1974 LL CZS12 Origin: CZS Problem A circle K with radius r, a point D on K, and a convex angle with vertex S and rays a and b are given in the plane. Construct a parallelogram ABCD such that A and B lie on a and b respectively, SA + SB = r, and C lies on K.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL YUG81

IMO 1979 LL YUG81 Origin: YUG Problem Let P be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped P0 can be decomposed into parallelepipeds P1, P2, . . . , Pn \inP, prove that P0 \inP.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL USS77

IMO 1986 LL USS77 Origin: USS Problem Find all integers x, y, z that satisfy x3 + y3 + z3 = x + y + z = 8.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON43

IMO 1969 LL MON43 Origin: MON Problem Let p and q be two prime numbers greater than 3. Prove that if their diﬀerence is 2n, then for any two integers m and n, the number S = p2m+1 + q2m+1 is divisible by 3.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USS46

IMO 1976 LL USS46 Origin: USS Problem For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality a4 + b4 + c4 + d4 + 2abcd \geqa2b2 + a2c2 + a2d2 + b2c2 + b2d2 + c2d2.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL USA64

IMO 1979 LL USA64 Origin: USA Problem From point P on arc BC of the circumcircle about triangle ABC, PX is constructed perpendicular to BC, PY is perpendicular to AC, and PZ perpendicular to AB (all extended if necessary). Prove that BC PX = AC PY + AB PZ .

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USS49

IMO 1977 LL USS49 Origin: USS Problem Find all pairs of integers (p, q) for which all roots of the trino- mials x2 + px + q and x2 + qx + p are integers. Solution If one of p, q, say p, is zero, then −q is a perfect square. Conversely, (p, q) = (0, −t2) and (p, q) = (−t2, 0) satisfy the conditions for t \inZ. We...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BEL7

IMO 1982 LL BEL7 Origin: BEL Problem Find all solutions (x, y) \inZ2 of the equation x3 −y3 = 2xy + 8.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL IRE46

IMO 1986 LL IRE46 Origin: IRE Problem We wish to construct a matrix with 19 rows and 86 columns, with entries xij \in{0, 1, 2} (1 \leqi \leq19, 1 \leqj \leq86), such that: (i) in each column there are exactly k terms equal to 0; (ii) for any distinct j, k \in{1, . . . , 86} there is i \in{1, . . . , 19} with xij + xik...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL LUX50

IMO 1986 LL LUX50 Origin: LUX Problem Let D be the point on the side BC of the triangle ABC such that AD is the bisector of \angleCAB. Let I be the incenter of \triangleABC. (a) Construct the points P and Q on the sides AB and AC, respectively, such that PQ is parallel to BC and the perimeter of the triangle APQ is equal to k \cdot BC, where...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL TUR71

IMO 1986 LL TUR71 Origin: TUR Problem Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ITA45

IMO 1985 LL ITA45 Origin: ITA Problem Two persons, X and Y , play with a die. X wins a game if the outcome is 1 or 2; Y wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within 5 games. Determine the probabilities of winning in an unlimited number of games. If...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL POL40

IMO 1982 LL POL40 Origin: POL Problem We consider a game on an inﬁnite chessboard similar to that of solitaire: If two adjacent ﬁelds are occupied by pawns and the next ﬁeld is empty (the three ﬁelds lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third ﬁeld. Prove that if in the initial position pawns ﬁll a...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USS64

IMO 1984 LL USS64 Origin: USS Problem For a matrix (pij) of the format m \times n with real entries, set ai = n j=1 pij for i = 1, . . . , m and bj = m i=1 pij for j = 1, . . . , n. (1) By integering a real number we mean replacing the number with the in- teger closest to it....

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GBR16

IMO 1976 LL GBR16 Origin: GBR Problem Prove that there is a positive integer n such that the decimal representation of 7n contains a block of at least m consecutive zeros, where m is any given positive integer.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL FIN14

IMO 1974 LL FIN14 Origin: FIN Problem Let n and k be natural numbers and a1, a2, . . . , an positive real numbers satisfying a1 + a2 + \cdot \cdot \cdot + an = 1. Prove that a−k a−k \cdot \cdot \cdot + a−k n \geqnk+1.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ISR42

IMO 1985 LL ISR42 Origin: ISR Problem Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL1

IMO 1972 LL BUL1 Origin: BUL Problem Find all integer solutions of the equation 1 + x + x2 + x3 + x4 = y4.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL GDR10

IMO 1966 LL GDR10 Origin: GDR Problem How many real solutions are there to the equation x = 1964 sinx −189?

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CAN15

IMO 1985 LL CAN15 Origin: CAN Problem Superchess is played on on a 12 \times 12 board, and it uses su- perknights, which move between opposite corner cells of any 3\times4 subboard. Is it possible for a superknight to visit every other cell of a superchessboard exactly once and return to its starting cell?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL SWE45

IMO 1971 LL SWE45 Origin: SWE Problem Let m and n denote integers greater than 1, and let \nu(n) be the number of primes less than or equal to n. Show that if the equation n \nu(n) = m has a solution, then so does the equation n \nu(n) = m −1.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL CZS8

IMO 1978 LL CZS8 Origin: CZS Problem For two given triangles A1A2A3 and B1B2B3 with areas ∆A and ∆B, respectively, AiAk \geqBiBk, i, k = 1, 2, 3. Prove that ∆A \geq∆B if the triangle A1A2A3 is not obtuse-angled.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET34

IMO 1970 LL NET34 Origin: NET Problem In connection with a convex pentagon ABCDE we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MON50

IMO 1985 LL MON50 Origin: MON Problem From each of the vertices of a regular n-gon a car starts to move with constant speed along the perimeter of the n-gon in the same direction. Prove that if all the cars end up at a vertex A at the same time, then they never again meet at any other vertex of the n-gon. Can they meet again at A?

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL VIE56

IMO 1982 LL VIE56 Origin: VIE Problem Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1, |f(1)| \leq1, |f(−1)| \leq1, prove that for |x| \leq1, (a) |f(x)| \leq5/4, (b) |g(x)| \leq2.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL AUS3

IMO 1982 LL AUS3 Origin: AUS Problem Given n points X1, X2, . . . , Xn in the interval 0 \leqXi \leq1, i = 1, 2, . . . , n, show that there is a point y, 0 \leqy \leq1, such that n n i=1 |y −Xi| = 1 2.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL SPA67

IMO 1992 LL SPA67 Origin: SPA Problem In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle ABC, the median ma meets BC at A′ and the circumcircle again at A1. The symmedian sa meets BC at M and the circumcircle again at A2. Given that the...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL CZS6

IMO 1976 LL CZS6 Origin: CZS Problem For each point X of a given polytope, denote by f(X) the sum of the distances of the point X from all the planes of the faces of the polytope. Prove that if f attains its maximum at an interior point of the polytope, then f is constant.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL ROM28

IMO 1974 LL ROM28 Origin: ROM Problem Let M be a ﬁnite set and P = {M1, M2, . . . , Mk} a partition of M (i.e., %k i=1 Mi = M, Mi ̸= \emptyset, Mi \capMj = \emptysetfor all i, j \in{1, 2, . . ., k}, i ̸= j). We deﬁne the following elementary operation on P: Choose i, j \in{1, 2, . . ., k}, such...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BUL13

IMO 1982 LL BUL13 Origin: BUL Problem A regular n-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by S1, S2, and S, respectively. Let \sigma be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \sigmaS = 4S1S2 cos2 \pi n.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM44

IMO 1970 LL ROM44 Origin: ROM Problem If a, b, c are side lengths of a triangle, prove that (a + b)(b + c)(c + a) \geq8(a + b −c)(b + c −a)(c + a −b).

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL CZS9

IMO 1976 LL CZS9 Origin: CZS Problem Find all (real) solutions of the system 3x1 −x2 −x3 −x5 = 0, −x1 + 3x2 −x4 −x6 = 0, −x1 3x3 −x4 −x7 = 0, −x2 −x3 + 3x4 −x8 = 0, −x1 3x5 −x6 −x7 = 0, −x2 −x5 + 3x6 −x8 = 0, −x3 −x5 3x7 −x8 = 0, −x4 −x6 −x7 + 3x8 = 0.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ROM62

IMO 1986 LL ROM62 Origin: ROM Problem Determine all pairs of positive integers (x, y) satisfying the equation px −y3 = 1, where p is a given prime number.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL AUS3

IMO 1984 LL AUS3 Origin: AUS Problem The opposite sides of the reentrant hexagon AFBDCE in- tersect at the points K, L, M (as shown in the ﬁgure). It is given that AL = AM = a, BM = BK = b, CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f. (a) Given length a and the three angles...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE91

IMO 1988 LL VIE91 Origin: VIE Problem A regular 14-gon with side length a is inscribed in a circle of radius one. Prove that 2 −a 2a 3 cos \pi 7 .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM43

IMO 1967 LL ROM43 Origin: ROM Problem The equation x5 + 5\lambdax4 −x3 + (\lambda\alpha −4)x2 −(8\lambda + 3)x + \lambda\alpha −2 = 0 is given. (a) Determine \alpha such that the given equation has exactly one root inde- pendent of \lambda. (b) Determine \alpha such that the given equation has exactly two roots inde- pendent of \lambda. Solution We can write the given equation in the form x5 −x3...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FIN19

IMO 1979 LL FIN19 Origin: FIN Problem For k = 1, 2, . . . consider the k-tuples (a1, a2, . . . , ak) of positive integers such that a1 + 2a2 + \cdot \cdot \cdot + kak = 1979. Show that there are as many such k-tuples with odd k as there are with even k.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ROM93

IMO 1989 LL ROM93 Origin: ROM Problem For \Phi : N \toZ let us deﬁne M\Phi = {f : N \toZ; f(x) > F(\Phi(x)), \forallx \inN}. (a) Prove that if M\Phi1 = M\Phi2 ̸= \emptyset, then \Phi1 = \Phi2. (b) Does this property remain true if M\Phi = {f : N \toN; f(x) > F(\Phi(x)), \forallx \inN}?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR87

IMO 1989 LL POR87 Origin: POR Problem A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight L is placed in the left pan and another of weight R in the right pan, the pointer stops at the number R −L on the graduated ruler. There are n...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL NET26

IMO 1977 LL NET26 Origin: NET Problem Let p be a prime number greater than 5. Let V be the collection of all positive integers n that can be written in the form n = kp + 1 or n = kp −1 (k = 1, 2, . . .). A number n \inV is called indecomposable in V if it is impossible to ﬁnd k, l \inV such that...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL CZS21

IMO 1970 LL CZS21 Origin: CZS Problem Find necessary and suﬃcient conditions on given positive num- bers u, v for the following claim to be valid: there exists a right-angled triangle \triangleABC with CD = u, CE = v, where D, E are points of the segments AB such that AD = DE = EB = 1 3AB.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL FRA28

IMO 1982 LL FRA28 Origin: FRA Problem Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn, deﬁne vk = k\sqrtu1u2 \cdot \cdot \cdot uk. Prove that n k=1 vk \leqe \cdot n k=1 uk. (e is the base of the natural logarithm).

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HKG34

IMO 1989 LL HKG34 Origin: HKG Problem Given an acute triangle ﬁnd a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BRA8

IMO 1979 LL BRA8 Origin: BRA Problem The sequence (an) of real numbers is deﬁned as follows: a1 = 1, a2 = 2 and an = 3an−1 −an−2, n \geq3. Prove that for n \geq3, an = a2 n−1 an−2 1, where [x] denotes the integer p such that p \leqx < p + 1.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON35

IMO 1967 LL MON35 Origin: MON Problem Prove the identity n k=0 n k tan x 2k 1 + 2k 1 −tan2(x/2) k = sec2n x 2 + secn x. Solution The given sum can be rewritten as n k=0 n k tan2 x k + n k=0 n k 2 tan2 x 1 −tan2 x k . Since 2...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA44

IMO 1988 LL INA44 Origin: INA Problem (a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the polynomial g(x12) is divided by the polynomial g(x)? (b) If k is a positive integer and f is a function such that for every positive number x, f(x2 +1) \sqrtx = k, ﬁnd the value of f 9+y2 y2 \sqrt 12/y...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL POL56

IMO 1992 LL POL56 Origin: POL Problem A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If x, u, and v are three distinct vertices such that x \tou and x \tov, then u \tow and v \tow for some vertex w. Suppose that x \tou \toy \to\cdot \cdot \cdot \toz is a path of length n, that cannot be extended...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL KOR46

IMO 1992 LL KOR46 Origin: KOR Problem Prove that the sequence 5, 12, 19, 26, 33, . . . contains no term of the form 2n −1.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA23

IMO 1969 LL FRA23 Origin: FRA Problem Consider the integer d = ab−1 c , where a, b, and c are positive integers and c \leqa. Prove that the set G of integers that are between 1 and d and relatively prime to d (the number of such integers is denoted by ϕ(d)) can be partitioned into n subsets, each of which consists of b elements. What can be said...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL ITA39

IMO 1992 LL ITA39 Origin: ITA Problem Let n \geq2 be an integer. Find the minimum k for which there exists a partition of {1, 2, . . ., k} into n subsets X1, X2, . . . , Xn such that the following condition holds: for any i, j, 1 \leqi < j \leqn, there exist x1 \inX1, x2 \inX2 such that |xi −xj| = 1.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL BUL5

IMO 1974 LL BUL5 Origin: BUL Problem A straight cone is given inside a rectangular parallelepiped B, with the apex at one of the vertices, say T , of the parallelepiped, and the base touching the three faces opposite to T . Its axis lies at the long diagonal through T . If V1 and V2 are the volumes of the cone and the parallelepiped respectively, prove that V1 \leq...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL USS71

IMO 1987 LL USS71 Origin: USS Problem To every natural number k, k \geq2, there corresponds a sequence an(k) according to the following rule: a0 = k, an = \tau(an−1) for n \geq1, in which \tau(a) is the number of diﬀerent divisors of a. Find all k for which the sequence an(k) does not contain the square of an integer.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CZS9

IMO 1974 LL CZS9 Origin: CZS Problem Solve the following system of linear equations with unknown x1, . . . , xn (n \geq2) and parameters c1, . . . , cn: 2x1 −x2 c1; −x1 +2x2 −x3 c2; −x2 +2x3 −x4 c3; . . . . . . . . . . . . −xn−2 +2xn−1 −xn = cn−1; −xn−1 +2xn = cn.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL34

IMO 1966 LL BUL34 Origin: BUL Problem Determine all pairs of positive integers (x, y) satisfying the equa- tion 2x = 3y + 5.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TWN76

IMO 1992 LL TWN76 Origin: TWN Problem Given any triangle ABC and any positive integer n, we say that n is a decomposable number for triangle ABC if there exists a de- composition of the triangle ABC into n subtriangles with each subtriangle similar to \triangleABC. Determine the positive integers that are decomposable numbers for every triangle.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL TUR81

IMO 1985 LL TUR81 Origin: TUR Problem Given the side a and the corresponding altitude ha of a triangle ABC, ﬁnd a relation between a and ha such that it is possible to construct, with straightedge and compass, triangle ABC such that the altitudes of ABC form a right triangle admitting ha as hypotenuse.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL MON64

IMO 1988 LL MON64 Origin: MON Problem Given n points A1, A2, . . . , An, no three collinear, show that the n-gon A1A2 . . . An can be inscribed in a circle if and only if A1A2 \cdot A3An \cdot \cdot \cdot An−1An + A2A3 \cdot A4An \cdot \cdot \cdot An−1An \cdot A1An + \cdot \cdot \cdot +An−1An−2 \cdot A1An \cdot \cdot \cdot An−3An = A1An−1 \cdot...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL SWE41

IMO 1977 LL SWE41 Origin: SWE Problem A wheel consists of a ﬁxed circular disk and a mobile circular ring. On the disk the numbers 1, 2, 3, . . ., N are marked, and on the ring N integers a1, a2, . . . , aN of sum 1 are marked (see the ﬁgure). The ring can be turned into N diﬀer- ent positions in which the numbers on...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL VIE74

IMO 1983 LL VIE74 Origin: VIE Problem In a plane we are given two distinct points A, B and two lines a, b passing through B and A respectively (a ∋B, b ∋A) such that the line AB is equally inclined to a and b. Find the locus of points M in the plane such that the product of distances from M to A and a equals the product of...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL USA63

IMO 1979 LL USA63 Origin: USA Problem If a1, a2, . . . , an denote the lengths of the sides of an arbitrary n-gon, prove that 2 \geq a1 s −a1 + a2 s −a2 \cdot \cdot \cdot + an s −an \geq n n −1, where s = a1 + a2 + \cdot \cdot \cdot + an.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL FRA24

IMO 1970 LL FRA24 Origin: FRA Problem Let n and p be two integers such that 2p \leqn. Prove the inequality (n −p)! p! \leq n + 1 n−2p . For which values does equality hold?

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS49

IMO 1966 LL USS49 Origin: USS Problem Two mirror walls are placed to form an angle of measure \alpha. There is a candle inside the angle. How many reﬂections of the candle can an observer see?

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM32

IMO 1972 LL ROM32 Origin: ROM Problem If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n, show that max n1+\cdot\cdot\cdot+nk=n n1n2 \cdot \cdot \cdot nk = (t + 1)rtk−r, where t = [n/k] and r is the remainder of n upon division by k; i.e., n = tk + r, 0 \leqr \leqk −1.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL4

IMO 1967 LL BUL4 Origin: BUL Problem Suppose medians ma and mb of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled triangle. (b) The inequality 5(a2 + b2 −c2) \geq8ab is valid, where a, b, and c are side lengths of the given triangle. Solution (a) Let ABCD be a parallelogram, and K, L the midpoints of segments BC...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL CZS14

IMO 1979 LL CZS14 Origin: CZS Problem Let S be a set of n2 + 1 closed intervals (n a positive integer). Prove that at least one of the following assertions holds: (i) There exists a subset S′ of n + 1 intervals from S such that the inter- section of the intervals in S′ is nonempty. (ii) There exists a subset S′′ of n + 1 intervals from S...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM33

IMO 1972 LL ROM33 Origin: ROM Problem A rectangle ABCD is given whose sides have lengths 3 and 2n, where n is a natural number. Denote by U(n) the number of ways in which one can cut the rectangle into rectangles of side lengths 1 and 2. (a) Prove that U(n + 1) + U(n −1) = 4U(n); (b) Prove that U(n) = \sqrt 3[( \sqrt 3 + 1)(2 +...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL5

IMO 1967 LL BUL5 Origin: BUL Problem Solve the system x2 + x −1 = y, y2 + y −1 = z, z2 + z −1 = x. Solution If one of x, y, z is equal to 1 or −1, then we obtain solutions (−1, −1, −1) and (1, 1, 1). We claim that these are the only solutions to the system. Let f(t) = t2 + t −1....

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL BUL3

IMO 1978 LL BUL3 Origin: BUL Problem Find all numbers \alpha for which the equation x2 −2x[x] + x −\alpha = 0 has two nonnegative roots. ([x] denotes the largest integer less than or equal to x.)

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POL77

IMO 1989 LL POL77 Origin: POL Problem Given that cos x + cos y + cos z cos(x + y + z) = sin x + sin y + sin z sin(x + y + z) = a, show that cos(y + z) + cos(z + x) + cos(x + y) = a.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL POL52

IMO 1979 LL POL52 Origin: POL Problem Let a real number \lambda > 1 be given and a sequence (nk) of positive integers such that nk+1 nk \lambda for k = 1, 2, . . . . Prove that there exists a positive integer c such that no positive integer n can be represented in more than c ways in the form n = nk + nj or n =...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL AUS1

IMO 1984 LL AUS1 Origin: AUS Problem The fraction 10 can be written as the sum of two positive fractions with numerator 1 as follows: 10 = 1 5 + 1 10 and also 10 = 1 4 + 1 20. There are the only two ways in which this can be done. In how many ways can 1984 be written as the sum of two positive fractions with numerator...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GDR32

IMO 1969 LL GDR32 Origin: GDR Problem Find the maximal number of regions into which a sphere can be partitioned by n circles.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL GDR27

IMO 1966 LL GDR27 Origin: GDR Problem We are given a circle K and a point P lying on a line g. Construct a circle that passes through P and touches K and g.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL4

IMO 1972 LL BUL4 Origin: BUL Problem Given a triangle, prove that the points of intersection of three pairs of trisectors of the inner angles at the sides lying closest to those sides are vertices of an equilateral triangle.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL GBR28

IMO 1983 LL GBR28 Origin: GBR Problem Show that if the sides a, b, c of a triangle satisfy the equation 2(ab2 + bc2 + ca2) = a2b + b2c + c2a + 3abc, then the triangle is equilateral. Show also that the equation can be satisﬁed by positive real numbers that are not the sides of a triangle.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BUL14

IMO 1983 LL BUL14 Origin: BUL Problem Let l be tangent to the circle k at B. Let A be a point on k and P the foot of perpendicular from A to l. Let M be symmetric to P with respect to AB. Find the set of all such points M.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL CZS7

IMO 1977 LL CZS7 Origin: CZS Problem Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real numbers such that (n −1)(c2 1 + c2 2 + \cdot \cdot \cdot + c2 n) = (c1 + c2 + \cdot \cdot \cdot + cn)2, then either all these numbers are nonnegative or all these numbers are nonpositive. Solution Let us suppose that c1 \leqc2 \leq\cdot...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET51

IMO 1969 LL NET51 Origin: NET Problem A curve determined by y = x2 −10x + 52, 0 \leqx \leq100, is constructed in a rectangular grid. Determine the number of squares cut by the curve.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL NET30

IMO 1977 LL NET30 Origin: NET Problem A triangle ABC with \angleA = 30◦and \angleC = 54◦is given. On BC a point D is chosen such that \angleCAD = 12◦. On AB a point E is chosen such that \angleACE = 6◦. Let S be the point of intersection of AD and CE. Prove that BS = BC. Solution Suppose \angleSBA = x. By the trigonometric form of Ceva’s theorem...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL IRE49

IMO 1988 LL IRE49 Origin: IRE Problem Let −1 < x < 1. Show that k=0 1 −x2 1 −2x cos(2\pik/7) + x2 = 7(1 + x7) 1 −x7 . Deduce that csc2 \pi 7 + csc2 2\pi 7 + csc2 3\pi 7 = 8.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FIN15

IMO 1992 LL FIN15 Origin: FIN Problem Prove that there exist 78 lines in the plane such that they have exactly 1992 points of intersection.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL POL26

IMO 1974 LL POL26 Origin: POL Problem Let g(k) be the number of partitions of a k-element set M, i.e., the number of families {A1, A2, . . . , As} of nonempty subsets of M such that Ai \capAj = \emptysetfor i ̸= j and %n i=1 Ai = M. Prove that nn \leqg(2n) \leq(2n)2n for every n.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX46

IMO 1983 LL LUX46 Origin: LUX Problem Let f be a real-valued function deﬁned on I = (0, +\infty) and having no zeros on I. Suppose that lim x\to+\infty f ′(x) f(x) = +\infty. For the sequence un = ln f(n+1) f(n) , prove that un \to+\infty(n \to+\infty).

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM31

IMO 1966 LL ROM31 Origin: ROM Problem Solve the equation |x2 −1| + |x2 −4| = mx as a function of the parameter m. Which pairs (x, m) of integers satisfy this equation?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET46

IMO 1969 LL NET46 Origin: NET Problem The vertices of an (n + 1)-gon are placed on the edges of a regular n-gon so that the perimeter of the n-gon is divided into equal parts. How does one choose these n + 1 points in order to obtain the (n + 1)gon with (a) maximal area; (b) minimal area?

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USA36

IMO 1976 LL USA36 Origin: USA Problem Three concentric circles with common center O are cut by a common chord in successive points A, B, C. Tangents drawn to the circles at the points A, B, C enclose a triangular region. If the distance from point O to the common chord is equal to p, prove that the area of the region enclosed by the tangents is equal to AB...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL USA65

IMO 1987 LL USA65 Origin: USA Problem The runs of a decimal number are its increasing or decreasing blocks of digits. Thus 024379 has three runs: 024, 43, and 379. Determine the average number of runs for a decimal number in the set {d1d2 . . . dn | dk ̸= dk+1, k = 1, 2, . . ., n −1}, where n \geq2.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FIN13

IMO 1987 LL FIN13 Origin: FIN Problem A be an inﬁnite set of positive integers such that every n \inA is the product of at most 1987 prime numbers. Prove that there is an inﬁnite set B \subsetA and a number p such that the greatest common divisor of any two distinct numbers in B is b.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL KOR53

IMO 1988 LL KOR53 Origin: KOR Problem Let x = p, y = q, z = r, w = s be the unique solution of the system of linear equations x + aiy + a2 i z + a3 i w = a4 i , i = 1, 2, 3, 4. Express the solution of the following system in terms of p, q, r, and s: x + a2 i...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL KOR57

IMO 1988 LL KOR57 Origin: KOR Problem Let C be a cube with edges of length 2. Construct a solid with fourteen faces by cutting oﬀall eight corners of C, keeping the new faces perpendicular to the diagonals of the cube and keeping the newly formed faces identical. If at the conclusion of this process the fourteen faces so formed have the same area, ﬁnd the area of each face...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS7

IMO 1966 LL USS7 Origin: USS Problem For which arrangements of two inﬁnite circular cylinders does their intersection lie in a plane?

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET33

IMO 1970 LL NET33 Origin: NET Problem The vertices of a given square are clockwise lettered A, B, C, D. On the side AB is situated a point E such that AE = AB/3. Starting from an arbitrarily chosen point P0 on segment AE and go- ing clockwise around the perimeter of the square, a series of points P0, P1, P2, . . . is marked on the perimeter such...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FRA14

IMO 1987 LL FRA14 Origin: FRA Problem Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that (1 −t2 n) t1 (1 −t2 1)2 + t2 (1 −t3 2)2 + \cdot \cdot \cdot + tn n (1 −tn+1 n )2 < 1.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FRA22

IMO 1979 LL FRA22 Origin: FRA Problem Consider two quadrilaterals ABCD and A′B′C′D′ in an aﬃne Euclidian plane such that AB = A′B′, BC = B′C′, CD = C′D′, and DA = D′A′. Prove that the following two statements are true: (a) If the diagonals BD and AC are mutually perpendicular, then the diagonals B′D′ and A′C′ are also mutually perpendicular. (b) If the perpendicular bisector of BD intersects AC...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL NOR62

IMO 1985 LL NOR62 Origin: NOR Problem A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed \pi 3 cm per second (relative to the disk). Describe the path of the insect (a) relative to the disk; (b) relative to the wall.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS8

IMO 1967 LL CZS8 Origin: CZS Problem ABCD is a parallelogram; AB = a, AD = 1, \alpha is the size of \angleDAB, and the three angles of the triangle ABD are acute. Prove that the four circles KA, KB, KC, KD, each of radius 1, whose centers are the vertices A, B, C, D, cover the parallelogram if and only if a \leq cos \alpha + \sqrt 3 sin...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL USS64

IMO 1969 LL USS64 Origin: USS Problem Prove that for a natural number n > 2, (n!)! > n[(n −1)!]n!.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL FRA24

IMO 1985 LL FRA24 Origin: FRA Problem Let d \geq1 be an integer that is not the square of an integer. Prove that for every integer n \geq1, (n \sqrt d + 1)| sin(n\pi \sqrt d)| \geq1.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HUN37

IMO 1988 LL HUN37 Origin: HUN Problem Let n points be given on the surface of a sphere. Show that the surface can be divided into n congruent regions such that each of them contains exactly one of the given points.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USA47

IMO 1982 LL USA47 Origin: USA Problem Evaluate sec′′ \pi 4 +sec′′ 3\pi 4 +sec′′ 5\pi 4 +sec′′ 7\pi 4 . (Here sec′′ means the second derivative of sec.)

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL BRA8

IMO 1985 LL BRA8 Origin: BRA Problem Let K be a convex set in the xy-plane, symmetric with respect to the origin and having area greater than 4. Prove that there exists a point (m, n) ̸= (0, 0) in K such that m and n are integers.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL FRG28

IMO 1985 LL FRG28 Origin: FRG Problem Let M be the set of the lengths of an octahedron whose sides are congruent quadrangles. Prove that M has at most three elements. (FRG 1a) Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL MON50

IMO 1992 LL MON50 Origin: MON Problem Let N be a point inside the triangle ABC. Through the mid- points of the segments AN, BN, and CN the lines parallel to the opposite sides of \triangleABC are constructed. Let AN, BN, and CN be the intersection points of these lines. If N is the orthocenter of the triangle ABC, prove that the nine-point circles of \triangleABC and \triangleANBNCN coincide. Remark....

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GBR25

IMO 1987 LL GBR25 Origin: GBR Problem Numbers d(n, m), with m, n integers, 0 \leqm \leqn, ae deﬁned by d(n, 0) = d(n, n) = 0 for all n \geq0 and md(n, m) = md(n −1, m) + (2n −m)d(n −1, m −1) for all 0 < m < n. Prove that all the d(n, m) are integers.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL VIE75

IMO 1979 LL VIE75 Origin: VIE Problem Given an equilateral triangle ABC, let M be an arbitrary point in space. (a) Prove that one can construct a triangle from the segments MA, MB, MC. (b) Suppose that P and Q are two points symmetric with respect to the center O of ABC. Prove that the two triangles constructed from the segments PA, PB, PC and QA, QB, QC are of...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GBR28

IMO 1989 LL GBR28 Origin: GBR Problem Let b1, b2, . . . , b1989 be positive real numbers such that the equations xr−1 −2xr + xr+1 + brxr = 0 (1 \leqr \leq1989) have a solution with x0 = x1990 = 0 but not all of x1, . . . , x1989 are equal to zero. Prove that b1 + b2 + \cdot \cdot \cdot + b1989 \geq 995.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL SPA68

IMO 1992 LL SPA68 Origin: SPA Problem Show that the numbers tan(r\pi/15), where r is a positive integer less than 15 and relatively prime to 15, satisfy x8 −92x6 + 134x4 −28x2 + 1 = 0.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA46

IMO 1989 LL INA46 Origin: INA Problem Given two distinct numbers b1 and b2, their product can be formed in two ways: b1 \times b2 and b2 \times b1. Given three distinct numbers, b1, b2, b3, their product can be formed in twelve ways: b1 \times(b2 \timesb3); (b1 \times b2) \times b3; b1 \times (b3 \times b2); (b1 \times b3) \times b2; b2 \times (b1 \times b3); (b2 \times b1)...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL FIN18

IMO 1989 LL FIN18 Origin: FIN Problem There are some boys and girls sitting in an n \times n quadratic array. We know the number of girls in every column and row and every line parallel to the diagonals of the array. For which n is this information suﬃcient to determine the exact positions of the girls in the array? For which seats can we say for sure that a...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HKG35

IMO 1989 LL HKG35 Origin: HKG Problem Find all square numbers S1 and S2 such that S1 −S2 = 1989.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRA23

IMO 1986 LL FRA23 Origin: FRA Problem Let I and J be the centers of the incircle and the excircle in the angle BAC of the triangle ABC. For any point M in the plane of the triangle, not on the line BC, denote by IM and JM the centers of the incircle and the excircle (touching BC) of the triangle BCM. Find the locus of points M for which...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GDR15

IMO 1967 LL GDR15 Origin: GDR Problem Suppose tan \alpha = p/q, where p and q are integers and q ̸= 0. Prove that the number tan \beta for which tan 2\beta = tan 3\alpha is rational only when p2 + q2 is the square of an integer. Solution Given that tan \alpha \inQ, we have that tan \beta is rational if and only if tan \gamma is rational, where...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL TUR36

IMO 1978 LL TUR36 Origin: TUR Problem The integers 1 through 1000 are located on the circumference of a circle in natural order. Starting with 1, every ﬁfteenth number (i.e., 1, 16, 31, . . .) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BUL9

IMO 1984 LL BUL9 Origin: BUL Problem The circle inscribed in the triangle A1A2A3 is tangent to its sides A1A2, A2A3, A3A1 at points T1, T2, T3, respectively. Denote by M1, M2, M3 the midpoints of the segments A2A3, A3A1, A1A2, respec- tively. Prove that the perpendiculars through the points M1, M2, M3 to the lines T2T3, T3T1, T1T2 meet at one point.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL POL32

IMO 1976 LL POL32 Origin: POL Problem We consider the inﬁnite chessboard covering the whole plane. In every ﬁeld of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent ﬁelds of the chessboard. Prove that the numbers occurring in the ﬁelds of the chessboard are all equal.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL POL65

IMO 1985 LL POL65 Origin: POL Problem Deﬁne the functions f, F : N \toN, by f(n) = 3 − \sqrt n , F(k) = min{n \inN|f k(n) > 0}, where f k = f ◦\cdot \cdot \cdot ◦f is f iterated n times. Prove that F(k + 2) = 3F(k + 1) −F(k) for all k \inN.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL CAN15

IMO 1984 LL CAN15 Origin: CAN Problem Consider all the sums of the form 1985 k=1 ekk5 = \pm15 \pm 25 \pm \cdot \cdot \cdot \pm 19855, where ek = \pm1. What is the smallest nonnegative value attained by a sum of this type?

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL SPA59

IMO 1987 LL SPA59 Origin: SPA Problem It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2 are complex numbers, and that a11a22 = a12a12 (where a12 is the conju- gate of a12). We consider the following system in x1, x2: x1(a11x1 + a12x2) = b1, x2(a12x1 + a22x2) = b2. (a) Give one condition to make the system consistent. (b) Give one condition to...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL6

IMO 1972 LL BUL6 Origin: BUL Problem Prove the inequality (n + 1) cos \pi n + 1 −n cos \pi n > 1 for all natural numbers n \geq2.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USS42

IMO 1976 LL USS42 Origin: USS Problem For a point O inside a triangle ABC, denote by A1, B1, C1 the respective intersection points of AO, BO, CO with the corresponding sides. Let n1 = AO A1O, n2 = BO B1O, n3 = CO C1O . What possible values of n1, n2, n3 can all be positive integers?

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM46

IMO 1967 LL ROM46 Origin: ROM Problem If x, y, z are real numbers satisfying the relations x+y+z = 1 and arctanx + arctan y + arctan z = \pi/4, prove that x2n+1 + y2n+1 + z2n+1 = 1 for all positive integers n. Solution Let us set arctan x = a, arctan y = b, arctanz = c. Then tan(a+b) = x+y 1−xy and tan(a + b + c)...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BEL6

IMO 1970 LL BEL6 Origin: BEL Problem Prove that the equation in x n i=1 bi x −ai = c, bi > 0, a1 < a2 < a3 < \cdot \cdot \cdot < an, has n −1 roots x1, x2, x3, . . . , xn−1 such that a1 < x1 < a2 < x2 < a3 < x3 < \cdot \cdot \cdot < xn−1 < an.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE38

IMO 1972 LL SWE38 Origin: SWE Problem Congruent rectangles with sides m (cm) and n (cm) are given (m, n positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BEL7

IMO 1983 LL BEL7 Origin: BEL Problem Find all numbers x \inZ for which the number x4 + x3 + x2 + x + 1 is a perfect square.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL FRA29

IMO 1982 LL FRA29 Origin: FRA Problem Let f : R \toR be a continuous function. Suppose that the restriction of f to the set of irrational numbers is injective. What can we say about f? Answer the analogous question if f is restricted to rationals.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ROM55

IMO 1979 LL ROM55 Origin: ROM Problem Let a, b be coprime integers. Show that the equation ax2 + by2 = z3 has an inﬁnite set of solutions (x, y, z) with x, y, z \inZ and x, y mutually coprime (in each solution).

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL YUG77

IMO 1979 LL YUG77 Origin: YUG Problem By h(n), where n is an integer greater than 1, let us denote the greatest prime divisor of the number n. Are there inﬁnitely many numbers n for which h(n) < h(n + 1) < h(n + 2) holds?

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BUL13

IMO 1983 LL BUL13 Origin: BUL Problem Let p be a prime number and a1, a2, . . . , a(p+1)/2 diﬀerent nat- ural numbers less than or equal to p. Prove that for each natural number r less than or equal to p, there exist two numbers (perhaps equal) ai and aj such that p \equivaiaj(mod r).

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL SAF65

IMO 1992 LL SAF65 Origin: SAF Problem If A, B, C, and D are four distinct points in space, prove that there is a plane P on which the orthogonal projections of A, B, C, and D form a parallelogram (possibly degenerate).

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL GDR2

IMO 1966 LL GDR2 Origin: GDR Problem Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an = 1, prove that (1 + a1)(1 + a2) \cdot \cdot \cdot (1 + an) \geq2n.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM29

IMO 1966 LL ROM29 Origin: ROM Problem (a) Find the number of ways 500 can be represented as a sum of consecutive integers. (b) Find the number of such representations for N = 2\alpha3\beta5\gamma, \alpha, \beta, \gamma \inN. Which of these representations consist only of natural numbers? (c) Determine the number of such representations for an arbitrary natural number N.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL HUN24

IMO 1977 LL HUN24 Origin: HUN Problem Determine all real functions f(x) that are deﬁned and contin- uous on the interval (−1, 1) and that satisfy the functional equation f(x + y) = f(x) + f(y) 1 −f(x)f(y) (x, y, x + y \in(−1, 1)). Solution Setting x = y = 0 gives us f(0) = 0. Let us put g(x) = arctanf(x). The given functional equation becomes tan g(x...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL TUR101

IMO 1989 LL TUR101 Origin: TUR Problem Let ABC be an equilateral triangle and \Gamma the semicircle drawn exteriorly to the triangle, having BC as diameter. Show that if a line passing through A trisects BC, it also trisects the arc \Gamma.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GBR18

IMO 1967 LL GBR18 Origin: GBR Problem If x is a positive rational number, show that x can be uniquely expressed in the form x = a1 + a2 2! + a3 3! + \cdot \cdot \cdot , where a1, a2, . . . are integers, 0 \leqan \leqn −1 for n > 1, and the series terminates. Show also that x can be expressed as the sum of reciprocals...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL KUW38

IMO 1983 LL KUW38 Origin: KUW Problem Let {un} be the sequence deﬁned by its ﬁrst two terms u0, u1 and the recursion formula un+2 = un −un+1. (a) Show that un can be written in the form un = \alphaan + \betabn, where a, b, \alpha, \beta are constants independent of n that have to be determined. (b) If Sn = u0 + u1 + \cdot \cdot \cdot +...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL USA78

IMO 1992 LL USA78 Origin: USA Problem Let Fn be the nth Fibonacci number, deﬁned by F1 = F2 = 1 and Fn = Fn−1 + Fn−2 for n > 2. Let A0, A1, A2, . . . be a sequence of points on a circle of radius 1 such that the minor arc from Ak−1 to Ak runs clockwise and such that µ(Ak−1Ak) = 4F2k+1 F 2 2k+1 +...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL GDR29

IMO 1984 LL GDR29 Origin: GDR Problem Let Sn = {1, . . . , n} and let f be a function that maps every subset of Sn into a positive real number and satisﬁes the following con- dition: For all A \subseteqSn and x, y \inSn, x ̸= y, f(A \cup{x})f(A \cup{y}) \leq f(A \cup{x, y})f(A). Prove that for all A, B \subseteqSn the following inequality holds: f(A) \cdot...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL HUN22

IMO 1971 LL HUN22 Origin: HUN Problem We are given an n \times n board, where n is an odd number. In each cell of the board either +1 or −1 is written. Let ak and bk denote the products of numbers in the kth row and in the kth column respectively. Prove that the sum a1 + a2 + \cdot \cdot \cdot + an + b1 + b2 +...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GBR23

IMO 1987 LL GBR23 Origin: GBR Problem A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance d from it....

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS57

IMO 1966 LL USS57 Origin: USS Problem Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is ﬁxed under each isometry of the cube into itself? Justify your answer.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL MON70

IMO 1989 LL MON70 Origin: MON Problem Three mutually nonparallel lines li (i = 1, 2, 3) are given in a plane. The lines li determine a triangle and reﬂections fi with axes on lines li. Prove that for every point of the plane, there exists a ﬁnite composition of the reﬂections fi that maps that point to a point interior to the triangle.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL POL41

IMO 1970 LL POL41 Origin: POL Problem Let a cube of side 1 be given. Prove that there exists a point A on the surface S of the cube such that every point of S can be joined to A by a path on S of length not exceeding 2. Also prove that there is a point of S that cannot be joined with A by a path on S...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE52

IMO 1967 LL SWE52 Origin: SWE Problem In the plane a point O and a sequence of points P1, P2, P3, . . . are given. The distances OP1, OP2, OP3, . . . are r1, r2, r3, . . . , where r1 \leq r2 \leqr3 \leq\cdot \cdot \cdot . Let \alpha satisfy 0 < \alpha < 1. Suppose that for every n the distance from the point Pn...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IRE56

IMO 1989 LL IRE56 Origin: IRE Problem Let n = 2k −1, where k \geq6 is an integer. Let T be the set of all n-tuples (x1, x2, . . . , xn) where xi is 0 or 1 (i = 1, 2, . . ., n). For x = (x1, . . . , xn) and y = (y1, . . . , yn) in T , let d(x,...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GDR29

IMO 1978 LL GDR29 Origin: GDR Problem (Variant of GDR 4) Given a nonconstant function f : R+ \toR such that f(xy) = f(x)f(y) for any x, y > 0, ﬁnd functions c, s : R+ \toR that satisfy c(x/y) = c(x)c(y)−s(x)s(y) for all x, y > 0 and c(x)+s(x) = f(x) for all x > 0.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL FRA25

IMO 1970 LL FRA25 Origin: FRA Problem Suppose that f is a real function deﬁned for 0 \leqx \leq1 having the ﬁrst derivative f ′ for 0 \leqx \leq1 and the second derivative f ′′ for 0 < x < 1. Prove that if f(0) = f ′(0) = f ′(1) = f(1) −1 = 0, there exists a number 0 < y < 1 such that |f ′′(y)| \geq4.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS56

IMO 1966 LL USS56 Origin: USS Problem Let ABCD be a tetrahedron such that AB \perpCD, AC \perpBD, and AD \perpBC. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL MON24

IMO 1972 LL MON24 Origin: MON Problem The diagonals of a convex 18-gon are colored in 5 diﬀerent colors, each color appearing on an equal number of diagonals. The diag- onals of one color are numbered 1, 2, . . .. One randomly chooses one-ﬁfth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly n pairs of diagonals of the same...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX42

IMO 1983 LL LUX42 Origin: LUX Problem Consider the square ABCD in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square ABCD.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL ROM53

IMO 1983 LL ROM53 Origin: ROM Problem Let a \inR and let z1, z2, . . . , zn be complex numbers of mod- ulus 1 satisfying the relation n k=1 z3 k = 4(a + (a −n)i) −3 n k=1 zk. Prove that a \in{0, 1, . . ., n} and zk \in{1, i} for all k.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL2

IMO 1972 LL BUL2 Origin: BUL Problem Find all real values of the parameter a for which the system of equations x4 = yz −x2 + a, y4 = zx −y2 + a, z4 = xy −z2 + a, has at most one real solution.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL NET67

IMO 1988 LL NET67 Origin: NET Problem Given a set of 1988 points in the plane, no three points of the set collinear, the points of a subset with 1788 points are colored blue, and the remaining 200 are colored red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL BUL5

IMO 1972 LL BUL5 Origin: BUL Problem Given a pyramid whose base is an n-gon inscribable in a circle, let H be the projection of the top vertex of the pyramid to its base. Prove that the projections of H to the lateral edges of the pyramid lie on a circle.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL NET29

IMO 1971 LL NET29 Origin: NET Problem A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii r1, r2, while the incircle has radius r. Given that r1 and r2 are natural numbers and that r1r2 = r, ﬁnd r1, r2, and r.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MON51

IMO 1986 LL MON51 Origin: MON Problem Let a, b, c, d be the lengths of the sides of a quadrilateral circumscribed about a circle and let S be its area. Prove that S \leq \sqrt abcd and ﬁnd conditions for equality.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL TUR80

IMO 1985 LL TUR80 Origin: TUR Problem Let E = {1, 2, . . ., 16} and let M be the collection of all 4 \times 4 matrices whose entries are distinct members of E. If a matrix A = (aij)4\times4 is chosen randomly from M, compute the probability p(k) of maxi minj aij = k for k \inE. Furthermore, determine l \inE such that p(l) = max{p(k) | k...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE37

IMO 1972 LL SWE37 Origin: SWE Problem On a chessboard (8 \times 8 squares with sides of length 1) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths 1 and 2?

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USA59

IMO 1984 LL USA59 Origin: USA Problem Determine the smallest positive integer m such that 529n +m\cdot 132n is divisible by 262417 for all odd positive integers n.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL GRE29

IMO 1989 LL GRE29 Origin: GRE Problem Let L denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points A, B, C of L there is a fourth point D, diﬀerent from A, B, C, such that the interiors of the segments AD, BD, CD contain no points of L. Is the statement true if one considers four points of...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL SWE50

IMO 1970 LL SWE50 Origin: SWE Problem The area of a triangle is S and the sum of the lengths of its sides is L. Prove that 36S \leqL2\sqrt 3 and give a necessary and suﬃcient condition for equality.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL GBR19

IMO 1977 LL GBR19 Origin: GBR Problem Given any integer m > 1 prove that there exist inﬁnitely many positive integers n such that the last m digits of 5n are a sequence am, am−1, . . . , a1 = 5 (0 \leqaj < 10) in which each digit except the last is of opposite parity to its successor (i.e., if ai is even, then ai−1 is odd, and...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL MON49

IMO 1992 LL MON49 Origin: MON Problem Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that 4x+2 i=1 (−1)i+1xixi+1 = 4m (x1 = x4k+3), prove that it is possible to choose numbers xk1, . . . , xk6 such that 6 The problem in this formulation is senseless. The correct formulation could be, “Find . . . such that \infty m=1...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL IRE45

IMO 1986 LL IRE45 Origin: IRE Problem Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, deﬁne M1 = 1 n n i=1 ai, M2 = n(n −1) 1\leqi<j\leqn aiaj, Q = M 2 1 −M2. Prove that a1 \leqM1 −Q \leqM1 + Q \leqan and that equality holds if and only if a1 = a2 = \cdot \cdot \cdot = an.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USS51

IMO 1977 LL USS51 Origin: USS Problem Several segments, which we shall call white, are given, and the sum of their lengths is 1. Several other segments, which we shall call black, are given, and the sum of their lengths is 1. Prove that every such system of segments can be distributed on the segment that is 1.51 long in the following way: Segments of the same color are disjoint,...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL USS43

IMO 1972 LL USS43 Origin: USS Problem A ﬁxed point A inside a circle is given. Consider all chords XY of the circle such that \angleXAY is a right angle, and for all such chords construct the point M symmetric to A with respect to XY . Find the locus of points M.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IRN37

IMO 1992 LL IRN37 Origin: IRN Problem Let the circles C1, C2, and C3 be orthogonal to the circle C and intersect each other inside C forming acute angles of measures A, B, and C. Show that A + B + C < \pi.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL CZS7

IMO 1976 LL CZS7 Origin: CZS Problem Let P be a ﬁxed point and T a given triangle that contains the point P. Translate the triangle T by a given vector v and denote by T ′ this new triangle. Let r, R, respectively, be the radii of the smallest disks centered at P that contain the triangles T , T ′, respectively. Prove that r + |v| \leq3R and...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ROM58

IMO 1979 LL ROM58 Origin: ROM Problem Prove that there exists a natural number k0 such that for every natural number k > k0 we may ﬁnd a ﬁnite number of lines in the plane, not all parallel to one of them, that divide the plane exactly in k regions. Find k0.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS40

IMO 1966 LL CZS40 Origin: CZS Problem For a positive real number p, ﬁnd all real solutions to the equation x2 + 2px −p2 − x2 −2px −p2 = 1.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL CUB10

IMO 1989 LL CUB10 Origin: CUB Problem Given the equation 4x3 + 4x2y −15xy2 −18y3 −12x2 + 6xy + 36y2 + 5x −10y = 0, ﬁnd all positive integer solutions.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL YUG78

IMO 1979 LL YUG78 Origin: YUG Problem By \omega(n), where n is an integer greater than 1, let us denote the number of diﬀerent prime divisors of the number n. Prove that there exist inﬁnitely many numbers n for which \omega(n) < \omega(n + 1) < \omega(n + 2) holds.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL ISR45

IMO 1979 LL ISR45 Origin: ISR Problem For any positive integer n we denote by F(n) the number of ways in which n can be expressed as the sum of three diﬀerent positive integers, without regard to order. Thus, since 10 = 7+2+1 = 6+3+1 = 5 + 4 + 1 = 5 + 3 + 2, we have F(10) = 4. Show that F(n) is even if n \equiv2...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CUB17

IMO 1985 LL CUB17 Origin: CUB Problem Set An = n k=1 k6 2k . Find limn\to\inftyAn.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POL79

IMO 1989 LL POL79 Origin: POL Problem To each pair (x, y) of distinct elements of a ﬁnite set X a number f(x, y) equal to 0 or 1 is assigned in such a way that f(x, y) ̸= f(y, x) for all x, y (x ̸= y). Prove that exactly one of the following situations occurs: (i) X is the union of two disjoint nonempty subsets U, V such...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL CZS8

IMO 1977 LL CZS8 Origin: CZS Problem A hexahedron ABCDE is made of two regular congruent tetra- hedra ABCD and ABCE. Prove that there exists only one isometry Z that maps points A, B, C, D, E onto B, C, A, E, D, respectively. Find all points X on the surface of hexahedron whose distance from Z(X) is minimal. Solution There is exactly one point satisfying the given condition on...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL SWE41

IMO 1971 LL SWE41 Origin: SWE Problem Consider the set of grid points (m, n) in the plane, m, n inte- gers. Let \sigma be a ﬁnite subset and deﬁne S(\sigma) = (m,n)\in\sigma (100 −|m| −|n|). Find the maximum of S, taken over the set of all such subsets \sigma.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL VIE51

IMO 1978 LL VIE51 Origin: VIE Problem Find the relations among the angles of the triangle ABC whose altitude AH and median AM satisfy \angleBAH = \angleCAM.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL USS66

IMO 1969 LL USS66 Origin: USS Problem (a) Prove that if 0 \leqa0 \leqa1 \leqa2, then (a0 + a1x −a2x2)2 \leq(a0 + a1 + a2)2 1 + 1 2x + 1 3x2 + 1 2x3 + x4 . (b) Formulate and prove the analogous result for polynomials of third degree.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL CZS11

IMO 1978 LL CZS11 Origin: CZS Problem Find all natural numbers n < 1978 with the following property: If m is a natural number, 1 < m < n, and (m, n) = 1 (i.e., m and n are relatively prime), then m is a prime number.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GDR20

IMO 1976 LL GDR20 Origin: GDR Problem Let (an), n = 0, 1, . . ., be a sequence of real numbers such that a0 = 0 and a3 n+1 = 1 2a2 n −1, n = 0, 1, . . .. Prove that there exists a positive number q, q < 1, such that for all n = 1, 2, . . ., |an+1 −an| \leqq|an −an−1|, and give...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA48

IMO 1989 LL INA48 Origin: INA Problem Let S be the point of intersection of the two lines l1 : 7x−5y + 8 = 0 and l2 : 3x + 4y −13 = 0. Let P = (3, 7), Q = (11, 13), and let A and B be points on the line PQ such that P is between A and Q, and B is between P and Q, and...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL VIE110

IMO 1989 LL VIE110 Origin: VIE Problem Do there exist two sequences of real numbers {ai}, {bi}, i \in N = {1, 2, 3, . . .}, satisfying the following conditions: 3\pi 2 \leqai \leqbi, cos aix + cos bix \geq−1 i for all i \inN and all x, 0 < x < 1?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL NET30

IMO 1971 LL NET30 Origin: NET Problem Prove that the system of equations 2yz + x −y −z = a, 2xz −x + y −z = a, 2xy −x −y + z = a, a being a parameter, cannot have ﬁve distinct solutions. For what values of a does this system have four distinct integer solutions?

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL BEL5

IMO 1986 LL BEL5 Origin: BEL Problem Let ABC and DEF be acute-angled triangles. Write d = EF, e = FD, f = DE. Show that there exists a point P in the interior of ABC for which the value of the expression d\cdotAP +e\cdotBP +f \cdotCP attains a minimum.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SPA51

IMO 1984 LL SPA51 Origin: SPA Problem Two cyclists leave simultaneously a point P in a circular run- way with constant velocities v1, v2 (v1 > v2) and in the same sense. A pedestrian leaves P at the same time, moving with velocity v3 = v1+v2 . If the pedestrian and the cyclists move in opposite directions, the pedes- trian meets the second cyclist 91 seconds after he meets the...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FIN19

IMO 1986 LL FIN19 Origin: FIN Problem Let f : [0, 1] \to[0, 1] satisfy f(0) = 0, f(1) = 1 and f(x + y) −f(x) = f(x) −f(x −y) for all x, y \geq0 with x −y, x + y \in[0, 1]. Prove that f(x) = x for all x \in[0, 1].

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FRA21

IMO 1979 LL FRA21 Origin: FRA Problem Let E be the set of all bijective mappings from R to R satisfying (\forallt \inR) f(t) + f −1(t) = 2t, where f −1 is the mapping inverse to f. Find all elements of E that are monotonic mappings.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM34

IMO 1972 LL ROM34 Origin: ROM Problem If p is a prime number greater than 2 and a, b, c integers not divisible by p, prove that the equation ax2 + by2 = pz + c has an integer solution.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BRA10

IMO 1983 LL BRA10 Origin: BRA Problem Which of the numbers 1, 2, . . ., 1983 has the largest number of divisors?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GBR27

IMO 1969 LL GBR27 Origin: GBR Problem The segment AB perpendicularly bisects CD at X. Show that, subject to restrictions, there is a right circular cone whose axis passes through X and on whose surface lie the points A, B, C, D. What are the restrictions?

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CAN12

IMO 1985 LL CAN12 Origin: CAN Problem Find the maximum value of sin2 \theta1 + sin2 \theta2 + \cdot \cdot \cdot + sin2 \thetan subject to the restrictions 0 \leq\thetai \leq\pi, \theta1 + \theta2 + \cdot \cdot \cdot + \thetan = \pi.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL USA65

IMO 1979 LL USA65 Origin: USA Problem Given f(x) \leqx for all real x and f(x + y) \leqf(x) + f(y) for all real x, y, prove that f(x) = x for all x.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRA9

IMO 1988 LL FRA9 Origin: FRA Problem If a0 is a positive real number, consider the sequence {an} deﬁned by an+1 = a2 n −1 n + 1 for n \geq0. Show that there exists a real number a > 0 such that: (i) for all real a0 \geqa, the sequence {an} \to+\infty(n \to\infty); (ii) for all real a0 < a, the sequence {an} \to0.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SPA52

IMO 1984 LL SPA52 Origin: SPA Problem Construct a scalene triangle such that a(tan B −tan C) = b(tan A −tan C).

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TUR73

IMO 1992 LL TUR73 Origin: TUR Problem Let {An | n = 1, 2, . . .} be a set of points in the plane such that for each n, the disk with center An and radius 2n contains no other point Aj. For any given positive real numbers a < b and R, show that there is a subset G of the plane satisfying: (i) the area of G...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SPA75

IMO 1985 LL SPA75 Origin: SPA Problem Let ABCD be a rectangle, AB = a, BC = b. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being d) that are at an the angle \varphi, 0 \leq\varphi \leq90◦, with respect to AB. Let L be the sum of the lengths of all the segments intersecting the rectangle. Find: (a) how L varies, (b)...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL CAN7

IMO 1992 LL CAN7 Origin: CAN Problem Let X be a bounded, nonempty set of points in the Cartesian plane. Let f(X) be the set of all points that are at a distance of at most 1 from some point in X. Let f n(X) = f(f(. . . (f(X)) . . . )) (n times). Show that f n(X) becomes “more circular” as n gets larger. In other words,...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM44

IMO 1967 LL ROM44 Origin: ROM Problem Suppose p and q are two diﬀerent positive integers and x is a real number. Form the product (x + p)(x + q). (a) Find the sum S(x, n) = (x + p)(x + q), where p and q take values from 1 to n. (b) Do there exist integer values of x for which S(x, n) = 0? Solution (a) S(x, n)...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL THA99

IMO 1989 LL THA99 Origin: THA Problem An arithmetic function is a real-valued function whose do- main is the set of positive integers. Deﬁne the convolution product of two arithmetic functions f and g to be the arithmetic function f ⋆g, where (f ⋆g)(n) = ij=n f(i)g(i), and f ⋆k = f ⋆f ⋆\cdot \cdot \cdot ⋆f (k times). We say that two arithmetic functions f and g are...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL POL71

IMO 1988 LL POL71 Origin: POL Problem Given integers a1, . . . , a10, prove that there exists a nonzero sequence (x1, . . . , x10) such that all xi belong to {−1, 0, 1} and the number 10 i=1 xiai is divisible by 1001.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MON53

IMO 1985 LL MON53 Origin: MON Problem For each P inside the triangle ABC, let A(P), B(P), and C(P) be the points of intersection of the lines AP, BP, and CP with the sides opposite to A, B, and C, respectively. Determine P in such a way that the area of the triangle A(P)B(P)C(P) is as large as possible.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL GBR28

IMO 1984 LL GBR28 Origin: GBR Problem A “number triangle” (tnk) (0 \leqk \leqn) is deﬁned by tn,0 = tn,n = 1 (n \geq0), tn+1,m = 2 − \sqrt m tn,m + 2 + \sqrt n−m+1 tn,m−1 (1 \leqm \leqn). Prove that all tn,m are integers.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL FRG21

IMO 1984 LL FRG21 Origin: FRG Problem (1) Start with a white balls and b black balls. (2) Draw one ball at random. (3) If the ball is white, then stop. Otherwise, add two black balls and go to step 2. Let S be the number of draws before the process terminates. For the cases a = b = 1 and a = b = 2 only, ﬁnd an =...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL COL9

IMO 1989 LL COL9 Origin: COL Problem Let m be a positive integer and deﬁne f(m) to be the number of factors of 2 in m! (that is, the greatest positive integer k such that 2k | m!). Prove that there are inﬁnitely many positive integers m such that m −f(m) = 1989.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE89

IMO 1988 LL VIE89 Origin: VIE Problem We match sets M of points in the coordinate plane to sets M∗ according to the rule that (x∗, y∗) belongs to M∗if and only if xx∗+yy∗\leq 1 whenever (x, y) \inM. Find all triangles Y such that Y∗is the reﬂection of Y at the origin.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GBR17

IMO 1976 LL GBR17 Origin: GBR Problem Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are \sqrt 3 : \sqrt 3 : 2.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GRE40

IMO 1986 LL GRE40 Origin: GRE Problem Find the maximum value that the quantity 2m + 7n can have such that there exist distinct positive integers xi (1 \leqi \leqm), yj (1 \leqj \leq n) such that the xi’s are even, the yj’s are odd, and m i=1 xi +n j=1 yj = 1986.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL NET27

IMO 1976 LL NET27 Origin: NET Problem In a plane three points P, Q, R, not on a line, are given. Let k, l, m be positive numbers. Construct a triangle ABC whose sides pass through P, Q, and R such that P divides the segment AB in the ratio 1 : k, Q divides the segment BC in the ratio 1 : l, and R divides the segment CA...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS11

IMO 1966 LL CZS11 Origin: CZS Problem Does there exist an integer z that can be written in two diﬀerent ways as z = x! + y!, where x, y are natural numbers with x \leqy?

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG33

IMO 1988 LL HKG33 Origin: HKG Problem Find a necessary and suﬃcient condition on the natural num- ber n for the equation xn + (2 + x)n + (2 −x)n = 0 to have a real root.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SPA76

IMO 1988 LL SPA76 Origin: SPA Problem The quadrilateral A1A2A3A4 is cyclic and its sides are a1 = A1A2, a2 = A2A3, a3 = A3A4, and a4 = A4A1. The respective circles with centers Ii and radii \rhoi are tangent externally to each side ai and to the sides ai+1 and ai−1 extended (a0 = a4). Show that i=1 ai \rhoi = 4(csc A1 + csc A2)2.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ROM92

IMO 1989 LL ROM92 Origin: ROM Problem Find the set of all a \inR for which there is no inﬁnite sequence (xn)n\geq0 \subsetR satisfying x0 = a, xn+1 = xn+\alpha \betaxn+1, n = 0, 1, . . . , where \alpha\beta > 0.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL USA42

IMO 1978 LL USA42 Origin: USA Problem A, B, C, D, E are points on a circle O with radius equal to r. Chords AB and DE are parallel to each other and have length equal to x. Diagonals AC, AD, BE, CE are drawn. If segment XY on O meets AC at X and EC at Y , prove that lines BX and DY meet at Z on the...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL6

IMO 1967 LL BUL6 Origin: BUL Problem Solve the system |x + y| + |1 −x| = 6, |x + y + 1| + |1 −y| = 4. Solution The given system has two solutions: (−2, −1) and (−14/3, 13/3).

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL CZS16

IMO 1979 LL CZS16 Origin: CZS Problem Let Q be a square with side length 6. Find the smallest integer n such that in Q there exists a set S of n points with the property that any square with side 1 completely contained in Q contains in its interior at least one point from S.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL VIE60

IMO 1977 LL VIE60 Origin: VIE Problem Suppose x0, x1, . . . , xn are integers and x0 > x1 > \cdot \cdot \cdot > xn. Prove that at least one of the numbers |F(x0)|, |F(x1)|, |F(x2)|, . . . , |F(xn)|, where F(x) = xn + a1xn−1 + \cdot \cdot \cdot + an, ai \inR, i = 1, . . . , n, is greater than n! 2n...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL GBR17

IMO 1972 LL GBR17 Origin: GBR Problem A solid right circular cylinder with height h and base-radius r has a solid hemisphere of radius r resting upon it. The center of the hemisphere O is on the axis of the cylinder. Let P be any point on the surface of the hemisphere and Q the point on the base circle of the cylinder that is furthest from P (measuring along...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USA39

IMO 1976 LL USA39 Origin: USA Problem In \triangleABC, the inscribed circle is tangent to side BC at X. Segment AX is drawn. Prove that the line joining the midpoint of segment AX to the midpoint of side BC passes through the center I of the inscribed circle.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL POL37

IMO 1971 LL POL37 Origin: POL Problem Let S be a circle, and \alpha = {A1, . . . , An} a family of open arcs in S. Let N(\alpha) = n denote the number of elements in \alpha. We say that \alpha is a covering of S if %n k=1 Ak \supsetS. Let \alpha = {A1, . . . , An} and \beta = {B1, . . . ,...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GDR16

IMO 1967 LL GDR16 Origin: GDR Problem Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by numbers of the form zk1,k2 = k1r1+k2r2, k1, k2 integers; i.e., for every real number x and every positive real number p two integers k1 and k2 can be found such that |x −(k1r1 + k2r2)| <...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE51

IMO 1967 LL SWE51 Origin: SWE Problem A subset S of the set of integers 0, . . . , 99 is said to have property A if it is impossible to ﬁll a crossword puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets S with property A. Solution...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FIN10

IMO 1987 LL FIN10 Origin: FIN Problem In a Cartesian coordinate system, the circle C1 has center O1(−2, 0) and radius 3. Denote the point (1, 0) by A and the origin by O. Prove that there is a constant c > 0 such that for every X that is exterior to C1, OX −1 \geqc min{AX, AX2}. Find the largest possible c.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FRA16

IMO 1987 LL FRA16 Origin: FRA Problem Let ABC be a triangle. For every point M belonging to segment BC we denote by B′ and c′ the orthogonal projections of M on the straight lines AC and BC. Find points M for which the length of segment B′C′ is a minimum.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL23

IMO 1966 LL BUL23 Origin: BUL Problem Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. (a) Prove that a necessary and suﬃcient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. (b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL SWE59

IMO 1979 LL SWE59 Origin: SWE Problem Determine the maximum value of x2y2z2w when x, y, z, w \geq0 and 2x + xy + z + yzw = 1.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL USA68

IMO 1983 LL USA68 Origin: USA Problem Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0 are tan A, tan B, and tan C, where A, B, and C are angles of a triangle. Determine the fourth root as a function only of p, q, r, and s.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL FIN23

IMO 1982 LL FIN23 Origin: FIN Problem Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL KOR55

IMO 1988 LL KOR55 Origin: KOR Problem Find all positive integers x such that the product of all digits of x is given by x2 −10x −22.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MON52

IMO 1986 LL MON52 Origin: MON Problem Solve the system of equations tan x1 + cot x1 = 3 tan x2, tan x2 + cot x2 = 3 tan x3, \cdot \cdot \cdot \cdot \cdot \cdot tan xn + cot xn = 3 tan x1.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL GDR30

IMO 1984 LL GDR30 Origin: GDR Problem Decide whether it is possible to color the 1984 natural numbers 1, 2, 3, . . ., 1984 using 15 colors so that no geometric sequence of length 3 of the same color exists.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GDR19

IMO 1976 LL GDR19 Origin: GDR Problem For a positive integer n, let 6(n) be the natural number whose decimal representation consists of n digits 6. Let us deﬁne, for all natural numbers m, k with 1 \leqk \leqm, m k = 6(m) \cdot 6(m−1) \cdot \cdot \cdot 6(m−k+1) 6(1) \cdot 6(2) \cdot \cdot \cdot 6(k) . Prove that for all m, k, m k is a natural number whose...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GRE30

IMO 1987 LL GRE30 Origin: GRE Problem Consider the regular 1987-gon A1A2 . . . A1987 with center O. Show that the sum of vectors belonging to any proper subset of M = {OAj | j = 1, 2, . . . , 1987} is nonzero.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL TUN44

IMO 1982 LL TUN44 Origin: TUN Problem Let A and B be positions of two ships M and N, respectively, at the moment when N saw M moving with constant speed v following the line Ax. In search of help, N moves with speed kv (k < 1) along the line By in order to meet M as soon as possible. Denote by C the point of meeting of the...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL FIN17

IMO 1989 LL FIN17 Origin: FIN Problem Let a, 0 < a < 1, be a real number and f a continuous function on [0, 1] satisfying f(0) = 0, f(1) = 1, and f x + y = (1 −a)f(x) + af(y) for all x, y \in[0, 1] with x \leqy. Determine f(1/7).

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SWE79

IMO 1985 LL SWE79 Origin: SWE Problem Let a, b, and c be real numbers such that bc −a2 + ca −b2 + ab −c2 = 0. Prove that a (bc −a2)2 + b (ca −b2)2 + c (ab −c2)2 = 0.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL3

IMO 1967 LL BUL3 Origin: BUL Problem Prove the trigonometric inequality cos x < 1 −x2 2 + x4 16, where x \in(0, \pi/2). Solution Consider the function f : [0, \pi/2] \toR deﬁned by f(x) = 1 −x2/2 + x4/16 −cos x. It is easy to calculate that f ′(0) = f ′′(0) = f ′′′(0) = 0 and f ′′′′(x) = 3/2 −cos x. Since f ′′′′(x) >...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL ROM46

IMO 1984 LL ROM46 Origin: ROM Problem Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers such that an+1 = nan + 1, bn+1 = nbn −1 for every n \geq1. Show that these two sequences can have only a ﬁnite number of terms in common.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL FIN10

IMO 1976 LL FIN10 Origin: FIN Problem Show that the reciprocal of any number of the form 2(m2 + m + 1), where m is a positive integer, can be represented as a sum of consecutive terms in the sequence (aj)\infty j=1, aj = j(j + 1)(j + 2).

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL3

IMO 1966 LL BUL3 Origin: BUL Problem A regular triangular prism has height h and a base of side length a. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS43

IMO 1966 LL CZS43 Origin: CZS Problem Five points in a plane are given, no three of which are collinear. Every two of them are joined by a segment, colored either red or gray, so that no three segments form a triangle colored in one color. (a) Prove that (1) every point is a vertex of exactly two red and two gray segments, and (2) the red segments form a...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL TUR83

IMO 1985 LL TUR83 Origin: TUR Problem Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an angle of measure 2\alpha such that each \Gammai is externally tangent to \Gammai+1 and ri+1 < ri. Show that the sum of the areas of the circles \Gammai is equal to the area of a circle of radius r = 1 2r0( \sqrt sin...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TUR74

IMO 1992 LL TUR74 Origin: TUR Problem Let S = \pin 1992m | n, m \inZ . Show that every real number x \geq0 is an accumulation point of S.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GBR19

IMO 1967 LL GBR19 Origin: GBR Problem The n points P1, P2, . . . , Pn are placed inside or on the bound- ary of a disk of radius 1 in such a way that the minimum distance dn between any two of these points has its largest possible value Dn. Calcu- late Dn for n = 2 to 7 and justify your answer. Solution Suppose n \leq6. Let...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL AUS3

IMO 1987 LL AUS3 Origin: AUS Problem A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop X to bus stop Y ̸= X, then we shall say Y can be reached from X. We shall use the phrase Y comes after X when...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL BEL5

IMO 1983 LL BEL5 Origin: BEL Problem Consider the set Q2 of points in R2, both of whose coordinates are rational. (a) Prove that the union of segments with vertices from Q2 is the entire set R2. (b) Is the convex hull of Q2 (i.e., the smallest convex set in R2 that con- tains Q2) equal to R2?

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL POL40

IMO 1970 LL POL40 Origin: POL Problem Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with \pi. Starting from a point P interior to the triangle, a ball reﬂects on the sides of ABC, respecting the law of reﬂection that the angle of incidence is equal to the angle of reﬂection. Prove that, supposing that the ball never reaches any of the vertices A, B, C, the...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL BUL5

IMO 1976 LL BUL5 Origin: BUL Problem Let ABCDS be a pyramid with four faces and with ABCD as a base, and let a plane \alpha through the vertex A meet its edges SB and SD at points M and N, respectively. Prove that if the intersection of the plane \alpha with the pyramid ABCDS is a parallelogram, then SM \cdot SN > BM \cdot DN.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG34

IMO 1988 LL HKG34 Origin: HKG Problem Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS12

IMO 1967 LL CZS12 Origin: CZS Problem Given a segment AB of the length 1, deﬁne the set M of points in the following way: it contains the two points A, B, and also all points obtained from A, B by iterating the following rule: (∗) for every pair of points X, Y in M, the set M also contains the point Z of the segment XY for which Y...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL3

IMO 1969 LL BEL3 Origin: BEL Problem Construct the circle that is tangent to three given circles.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM45

IMO 1967 LL ROM45 Origin: ROM Problem (a) Solve the equation sin3 x + sin3 2\pi 3 + x sin3 4\pi 3 + x 3 4 cos 2x = 0. (b) Suppose the solutions are in the form of arcs AB of the trigonometric circle (where A is the beginning of arcs of the trigonometric circle), and P is a regular n-gon inscribed in the circle with one...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL SWE62

IMO 1983 LL SWE62 Origin: SWE Problem A circle \gamma is drawn and let AB be a diameter. The point C on \gamma is the midpoint of the line segment BD. The line segments AC and DO, where O is the center of \gamma, intersect at P. Prove that there is a point E on AB such that P is on the circle with diameter AE.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA43

IMO 1988 LL INA43 Origin: INA Problem (a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find the value of k. (b) If p, q, and r are distinct roots of x3 −x2 + x −2 = 0, ﬁnd the value of p3 + q3 + r3. (c) If r is the remainder when each of the numbers 1059, 1417, and 2312 is...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USS52

IMO 1982 LL USS52 Origin: USS Problem We are given 2n natural numbers 1, 1, 2, 2, 3, 3, . . ., n −1, n −1, n, n. Find all n for which these numbers can be arranged in a row such that for each k \leqn, there are exactly k numbers between the two numbers k.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL ROM29

IMO 1974 LL ROM29 Origin: ROM Problem Let A, B, C, D be points in space. If for every point M on the segment AB the sum area(AMC)+area(CMD)+area(DMB) is constant show that the points A, B, C, D lie in the same plane.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GRE41

IMO 1986 LL GRE41 Origin: GRE Problem Let M, N, P be the midpoints of the sides BC, CA, AB of a triangle ABC. The lines AM, BN, CP intersect the circumcircle of ABC at points A′, B′, C′, respectively. Show that if A′B′C′ is an equilateral triangle, then so is ABC.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GBR27

IMO 1987 LL GBR27 Origin: GBR Problem Find, with proof, the smallest real number C with the following property: For every inﬁnite sequence {xi} of positive real numbers such that x1 + x2 + \cdot \cdot \cdot + xn \leqxn+1 for n = 1, 2, 3, . . ., we have \sqrtx1 + \sqrtx2 + \cdot \cdot \cdot + \sqrtxn \leqc\sqrtx1 + x2 + \cdot \cdot \cdot + xn for...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USS93

IMO 1985 LL USS93 Origin: USS Problem The sphere inscribed in tetrahedron ABCD touches the sides ABD and DBC at points K and M, respectively. Prove that ∡AKB = ∡DMC.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL VIE109

IMO 1989 LL VIE109 Origin: VIE Problem Let Ax, By be two noncoplanar rays with AB as a common per- pendicular, and let M, N be two mobile points on Ax and By respectively such that AM + BN = MN. First version. Prove that there exist inﬁnitely many lines coplanar with each of the lines MN. Second version. Prove that there exist inﬁnitely many rotations around a ﬁxed axis...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM31

IMO 1972 LL ROM31 Origin: ROM Problem Find values of n \inN for which the fraction 3n−2 2n−3 is reducible.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL POL49

IMO 1983 LL POL49 Origin: POL Problem Given positive integers k, m, n with km \leqn and nonnegative real numbers x1, . . . , xk, prove that n k i=1 xm i −1 \leqm k i=1 (xn i −1) .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GDR14

IMO 1967 LL GDR14 Origin: GDR Problem Which fraction p/q, where p, q are positive integers less than 100, is closest to \sqrt 2? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of \sqrt 2 (without using any tables). Solution We have that p q − \sqrt = |p −q \sqrt 2| q = |p2 −2q2|...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GDR22

IMO 1976 LL GDR22 Origin: GDR Problem A regular pentagon A1A2A3A4A5 with side length s is given. At each point Ai a sphere Ki of radius s/2 is constructed. There are two spheres K1′ and K2′ eah of radius s/2 touching all the ﬁve spheres Ki. Decide whether K1′ and K2′ intersect each other, touch each other, or have no common points.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS54

IMO 1966 LL USS54 Origin: USS Problem Find the last two digits of a sum of eighth powers of 100 consecutive integers.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL USS78

IMO 1986 LL USS78 Origin: USS Problem If T and T1 are two triangles with angles x, y, z and x1, y1, z1, respectively, prove the inequality cos x1 sin x + cos y1 sin y + cos z1 sin z \leqcot x + cot y + cot z.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL HUN40

IMO 1979 LL HUN40 Origin: HUN Problem A polynomial P(x) has degree at most 2k, where k = 0, 1, 2, . . . . Given that for an integer i, the inequality −k \leqi \leqk implies |P(i)| \leq1, prove that for all real numbers x, with −k \leqx \leqk, the following inequality holds: |P(x)| < (2k + 1) 2k k .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL GDR13

IMO 1967 LL GDR13 Origin: GDR Problem Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius r there exists one (or more) with maximal area. If so, determine their shape and area. Solution The maximum area is 3 \sqrt 3r2/4 (where r is the radius of the semicircle) and is attained in the case of a trapezoid with two vertices at the endpoints of the diameter...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IRE54

IMO 1989 LL IRE54 Origin: IRE Problem Let f be a function from the real numbers to the real numbers such that f(1) = 1, f(a+b) = f(a)+f(b) for all a, b, and f(x)f(1/x) = 1 for all x ̸= 0. Prove that f(x) = x for all real numbers x.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ROM63

IMO 1986 LL ROM63 Origin: ROM Problem Let AA′, BB′, CC′ be the bisectors of the angles of a triangle ABC (A′ \inBC, B′ \inCA, C′ \inAB). Prove that each of the lines A′B′, B′C′, C′A′ intersects the incircle in two points.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL FIN13

IMO 1978 LL FIN13 Origin: FIN Problem The satellites A and B circle the Earth in the equatorial plane at altitude h. They are separated by distance 2r, where r is the radius of the Earth. For which h can they be seen in mutually perpendicular directions from some point on the equator?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL USS51

IMO 1971 LL USS51 Origin: USS Problem Suppose that the sides AB and DC of a convex quadrilateral ABCD are not parallel. On the sides BC and AD, pairs of points (M, N) and (K, L) are chosen such that BM = MN = NC and AK = KL = LD. Prove that the areas of triangles OKM and OLN are diﬀerent, where O is the intersection point of AB...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL MON48

IMO 1992 LL MON48 Origin: MON Problem Find all the functions f : R+ \toR satisfying the identity f(x)f(y) = y\alpha \cdot f x x\beta \cdot f y , x, y \inR+, where \alpha, \beta are given real numbers.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL KOR56

IMO 1988 LL KOR56 Origin: KOR Problem The Fibonacci sequence is deﬁned by an+1 = an + an−1 (n \geq1), a0 = 0, a1 = a2 = 1. Find the greatest common divisor of the 1960th and 1988th terms of the Fibonacci sequence.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS8

IMO 1966 LL USS8 Origin: USS Problem We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL SWE66

IMO 1986 LL SWE66 Origin: SWE Problem One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL ROM55

IMO 1983 LL ROM55 Origin: ROM Problem For every a \inN denote by M(a) the number of elements of the set {b \inN | a + b is a divisor of ab}. Find maxa\leq1983 M(a).

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GBR34

IMO 1986 LL GBR34 Origin: GBR Problem For each nonnegative integer n, Fn(x) is a polynomial in x of degreee n. Prove that if the identity Fn(2x) = n r=0 (−1)n−r n r 2rFr(x) holds for each n, then Fn(tx) = n r=0 n r tr(1 −t)n−rFr(x) for each n and all t.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL IRE40

IMO 1985 LL IRE40 Origin: IRE Problem Each of the numbers x1, x2, . . . , xn equals 1 or −1 and x1x2x3x4 + x2x3x4x5 + \cdot \cdot \cdot + xn−3xn−2xn−1xn +xn−2xn−1xnx1 + xn−1xnx1x2 + xnx1x2x3 = 0. Prove that n is divisible by 4.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM47

IMO 1970 LL ROM47 Origin: ROM Problem Given a polynomial P(x) = ab(a −c)x3 + (a3 −a2c + 2ab2 −b2c + abc)x2 +(2a2b + b2c + a2c + b3 −abc)x + ab(b + c), where a, b, c ̸= 0, prove that P(x) is divisible by Q(x) = abx2 + (a2 + b2)x + ab and conclude that P(x0) is divisible by (a + b)3 for x0 = (a +...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL ROM64

IMO 1992 LL ROM64 Origin: ROM Problem For any positive integer n consider all representations n = a1 + \cdot \cdot \cdot + ak, where a1 > a2 > \cdot \cdot \cdot > ak > 0 are integers such that for all i \in{1, 2, . . ., k −1}, the number ai is divisible by ai+1. Find the longest such representation of the number 1992.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MOR55

IMO 1985 LL MOR55 Origin: MOR Problem The points A, B, C are in this order on line D, and AB = 4BC. Let M be a variable point on the perpendicular to D through C. Let MT1 and MT2 be tangents to the circle with center A and radius AB. Determine the locus of the orthocenter of the triangle MT1T2.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GBR32

IMO 1986 LL GBR32 Origin: GBR Problem Find, with proof, all solutions of the equation 1 x + 2 y −3 z = 1 in positive integers x, y, z.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL GDR34

IMO 1982 LL GDR34 Origin: GDR Problem Let M be the set of all functions f with the following proper- ties: (i) f is deﬁned for all real numbers and takes only real values. (ii) For all x, y \inR the following equality holds: f(x)f(y) = f(x + y) + f(x −y). (iii) f(0) ̸= 0. Determine all functions f \inM such that (a) f(1) = 5/2; (b) f(1) =...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL AUS3

IMO 1985 LL AUS3 Origin: AUS Problem A function f has the following property: If k > 1, j > 1, and (k, j) = m, then f(kj) = f(m) (f(k/m) + f(j/m)). What values can f(1984) and f(1985) take?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL HUN24

IMO 1971 LL HUN24 Origin: HUN Problem Let A, B, and C denote the angles of a triangle. If sin2 A + sin2 B + sin2 C = 2, prove that the triangle is right-angled.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL14

IMO 1966 LL POL14 Origin: POL Problem Compute the largest number of regions into which one can divide a disk by joining n points on its circumference.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL VIE50

IMO 1976 LL VIE50 Origin: VIE Problem Find a function f(x) deﬁned for all real values of x such that for all x, f(x + 2) −f(x) = x2 + 2x + 4, and if x \in[0, 2), then f(x) = x2.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL CUB5

IMO 1988 LL CUB5 Origin: CUB Problem Let k be a positive integer and Mk the set of all the integers that are between 2k2 + k and 2k2 + 3k, both included. Is it possible to partition Mk into two subsets A and B such that x\inA x2 = x\inB x2?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL USS68

IMO 1969 LL USS68 Origin: USS Problem Given 5 points in the plane, no three of which are collinear, prove that we can choose 4 points among them that form a convex quadrilateral.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL MON63

IMO 1988 LL MON63 Origin: MON Problem Let ABCD be a quadrilateral. Let A′BCD′ be the reﬂection of ABCD in BC, while A′′B′CD′ is the reﬂection of A′BCD′ in CD′ and A′′B′′C′D′ is the reﬂection of A′′B′CD′ in D′A′′. Show that if the lines AA′′ and BB′′ are parallel, then ABCD is a cyclic quadrilateral.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GDR25

IMO 1978 LL GDR25 Origin: GDR Problem Consider a polynomial P(x) = ax2 + bx + c with a > 0 that has two real roots x1, x2. Prove that the absolute values of both roots are less than or equal to 1 if and only if a + b + c \geq0, a −b + c \geq0, and a −c \geq0.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MOR56

IMO 1986 LL MOR56 Origin: MOR Problem Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with center O. Consider the circular arc with endpoints A1, A6 not containing

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA45

IMO 1989 LL INA45 Origin: INA Problem The expressions a + b + c, ab + ac + bc, and abc are called the elementary symmetric expressions on the three letters a, b, c; symmetric because if we interchange any two letters, say a and c, the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let Sk(n) denote the elementary...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL SWE33

IMO 1978 LL SWE33 Origin: SWE Problem A sequence (an)\infty of real numbers is called convex if 2an \leq an−1 +an+1 for all positive integers n. Let (bn)\infty 0 be a sequence of positive numbers and assume that the sequence (\alphanbn)\infty 0 is convex for any choice of \alpha > 0. Prove that the sequence (log bn)\infty 0 is convex.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USA54

IMO 1977 LL USA54 Origin: USA Problem If 0 \leqa \leqb \leqc \leqd, prove that abbccdda \geqbacbdcad. Solution We shall use the following lemma. Lemma. If a real function f is convex on the interval I and x, y, z \inI, x \leqy \leqz, then (y −z)f(x) + (z −x)f(y) + (x −y)f(z) \leq0. Proof. The inequality is obvious for x = y = z. If x < z, then...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GBR26

IMO 1969 LL GBR26 Origin: GBR Problem A smooth solid consists of a right circular cylinder of height h and base-radius r, surmounted by a hemisphere of radius r and center O. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL BUL4

IMO 1976 LL BUL4 Origin: BUL Problem Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1 is the square of some integer.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN34

IMO 1969 LL HUN34 Origin: HUN Problem Let a and b be arbitrary integers. Prove that if k is an integer not divisible by 3, then (a + b)2k + a2k + b2k is divisible by a2 + ab + b2.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GBR31

IMO 1986 LL GBR31 Origin: GBR Problem Let P and Q be distinct points in the plane of a triangle ABC such that AP : AQ = BP : BQ = CP : CQ. Prove that the line PQ passes through the circumcenter of the triangle.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ROM67

IMO 1985 LL ROM67 Origin: ROM Problem Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the property n2 | 2n1 −1, n3 | 2n2 −1, . . . , nk | 2nk−1 −1, and n1 | 2nk −1. Show that n1 = n2 = \cdot \cdot \cdot = nk = 1.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL GBR33

IMO 1982 LL GBR33 Origin: GBR Problem A sequence (un) of integers is deﬁned for n \geq0 by u0 = 0, u1 = 1, and un −2un−1 + (1 −c)un−2 = 0 (n \geq2), where c is a ﬁxed integer independent of n. Find the least value of c for which both of the following statements are true: (i) If p is a prime less than or equal to P,...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL USS88

IMO 1988 LL USS88 Origin: USS Problem There are six circles inside a ﬁxed circle, each tangent to the ﬁxed circle and tangent to the two adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, A1, A2, A3, A4, A5, and A6, prove that A1A2 \cdot A3A4 \cdot A5A6 = A2A3 \cdot A4A5 \cdot A6A1.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL MON68

IMO 1989 LL MON68 Origin: MON Problem If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n, prove that a1 a2 + \cdot \cdot \cdot + an k \cdot \cdot \cdot + an a1 + \cdot \cdot \cdot + an−1 k \geq n (n −1)k .

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GDR35

IMO 1985 LL GDR35 Origin: GDR Problem We call a coloring f of the elements in the set M = {(x, y) | x = 0, 1, . . . , kn −1; y = 0, 1, . . ., ln −1} with n colors allowable if every color appears exactly k and l times in each row and column and there are no rectangles with sides parallel to the...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN24

IMO 1967 LL HUN24 Origin: HUN Problem Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS28

IMO 1966 LL CZS28 Origin: CZS Problem Let there be given a circle with center S and radius 1 in the plane, and let ABC be an arbitrary triangle circumscribed about the circle such that SA \leqSB \leqSC. Find the loci of the vertices A, B, C.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL VIE73

IMO 1983 LL VIE73 Origin: VIE Problem Let ABC be a nonequilateral triangle. Prove that there exist two points P and Q in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of ABC, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL4

IMO 1966 LL POL4 Origin: POL Problem Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL POL57

IMO 1992 LL POL57 Origin: POL Problem For positive numbers a, b, c deﬁne A = (a + b + c)/3, G = (abc)1/3, H = 3/(a−1 + b−1 + c−1). Prove that A G 3 \geq1 4 + 3 4 \cdot A H , for every a, b, c > 0.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL YUG54

IMO 1978 LL YUG54 Origin: YUG Problem Let p, q and r be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes \alpha, \beta, and \gamma, containing p, q, and r respectively, that are perpendicular to each other (\alpha \perp\beta, \beta \perp\gamma, \gamma \perp\alpha).

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE41

IMO 1972 LL SWE41 Origin: SWE Problem The ternary expansion x = 0.10101010 . . . is given. Give the binary expansion of x. Alternatively, transform the binary expansion y = 0.110110110 . . . into a ternary expansion.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL AUS2

IMO 1983 LL AUS2 Origin: AUS Problem Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules oﬀer round-trip ﬂights. Prove that at least one of the airlines can oﬀer a round trip with an odd number of landings.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL15

IMO 1966 LL POL15 Origin: POL Problem Points A, B, C, D lie on a circle such that AB is a diameter and CD is not. If the tangents at C and D meet at P while AC and BD meet at Q, show that PQ is perpendicular to AB.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL CZS16

IMO 1966 LL CZS16 Origin: CZS Problem We are given a circle K with center S and radius 1 and a square Q with center M and side 2. Let XY be the hypotenuse of an isosceles right triangle XY Z. Describe the locus of points Z as X varies along K and Y varies along the boundary of Q.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL AUS3

IMO 1983 LL AUS3 Origin: AUS Problem (a) Given a tetrahedron ABCD and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from D passes through the orthocenter H4 of ∆ABC. Prove that this altitude DH4 intersects all the other three altitudes. (b) If we further know that a second altitude, say the one from vertex A to the face BCD,...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL BEL6

IMO 1985 LL BEL6 Origin: BEL Problem On a one-way street, an unending sequence of cars of width a, length b passes with velocity v. The cars are separated by the distance c. A pedestrian crosses the street perpendicularly with velocity w, without paying attention to the cars. (a) What is the probability that the pedestrian crosses the street unin- jured? (b) Can he improve this probability by crossing the...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON44

IMO 1969 LL MON44 Origin: MON Problem Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation x2 −ax + b = 0.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL VIE107

IMO 1989 LL VIE107 Origin: VIE Problem Let E be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let f be the function of area of a triangle. Determine the set of values f(E) of f.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON40

IMO 1969 LL MON40 Origin: MON Problem Find the number of ﬁve-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from diﬀerent pairs are diﬀerent, and the remaining digit (which does not belong to any of the pairs) is diﬀerent from the other digits.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MON54

IMO 1986 LL MON54 Origin: MON Problem Find the least integer n with the following property: For any set V of 8 points in the plane, no three lying on a line, and for any set E of n line segments with endpoints in V , one can ﬁnd a straight line intersecting at least 4 segments in E in interior points.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL NET41

IMO 1984 LL NET41 Origin: NET Problem Determine positive integers p, q, and r such that the diagonal of a block consisting of p\timesq \timesr unit cubes passes through exactly 1984 of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USA46

IMO 1982 LL USA46 Origin: USA Problem Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL VIE72

IMO 1979 LL VIE72 Origin: VIE Problem Let f(x) be a polynomial with integer coeﬃcients. Prove that if f(x) equals 1979 for four diﬀerent integer values of x, then f(x) cannot be equal to 2 \times 1979 for any integral value of x.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL AUS3

IMO 1992 LL AUS3 Origin: AUS Problem Let ABC be a triangle, O its circumcenter, S its centroid, and H its orthocenter. Denote by A1, B1, and C1 the centers of the circles circumscribed about the triangles CHB, CHA, and AHB, respectively. Prove that the triangle ABC is congruent to the triangle A1B1C1 and that the nine-point circle of \triangleABC is also the nine-point circle of \triangleA1B1C1.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL ISR51

IMO 1988 LL ISR51 Origin: ISR Problem Let A1, A2, . . . , A29 be 29 diﬀerent sequences of positive integers. For 1 \leqi < j \leq29 and any natural number x, we deﬁne Ni(x) to be the number of elements of the sequence Ai that are less than or equal to x, and Nij(x) to be the number of elements of the intersection Ai \capAj that are less...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL POL45

IMO 1984 LL POL45 Origin: POL Problem Let X be an arbitrary nonempty set contained in the plane and let sets A1, A2, . . . , Am and B1, B2, . . . , Bn be its images under parallel translations. Let us suppose that A1 \cupA2 \cup\cdot \cdot \cdot \cupAm \subsetB1 \cupB2 \cup\cdot \cdot \cdot \cupBn and that the sets A1, A2, . . . , Am are...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN22

IMO 1967 LL HUN22 Origin: HUN Problem The distance between the centers of the circles k1 and k2 with radii r is equal to r. Points A and B are on the circle k1, symmetric with respect to the line connecting the centers of the circles. Point P is an arbitrary point on k2. Prove that PA2 + PB2 \geq2r2. When does equality hold? Solution Let O1 and O2 be...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS11

IMO 1967 LL CZS11 Origin: CZS Problem Let n be a positive integer. Find the maximal number of non- congruent triangles whose side lengths are integers less than or equal to n. Solution We have to ﬁnd the number pn of triples of positive integers (a, b, c) satisfying a \leqb \leqc \leqn and a + b > c. Let us denote by pn(k) the number of such triples with...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL37

IMO 1966 LL POL37 Origin: POL Problem Prove that the perpendiculars drawn from the midpoints of the sides of a cyclic quadrilateral to the opposite sides meet at one point.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ISR43

IMO 1985 LL ISR43 Origin: ISR Problem Suppose that 1985 points are given inside a unit cube. Show that one can always choose 32 of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than 8 \sqrt 3.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ITA27

IMO 1967 LL ITA27 Origin: ITA Problem Which regular polygons can be obtained (and how) by cutting a cube with a plane? Solution Regular polygons with 3, 4, and 6 sides can be obtained by cutting a cube with a plane, as shown in the ﬁgure. A polygon with more than 6 sides cannot be obtained in such a way, for a cube has 6 faces. Also, if a pentagon...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL GBR15

IMO 1971 LL GBR15 Origin: GBR Problem Let ABCD be a convex quadrilateral whose diagonals intersect at O at an angle \theta. Let us set OA = a, OB = b, OC = c, and OD = d, c > a > 0, and d > b > 0. Show that if there exists a right circular cone with vertex V , with the properties: (1) its axis passes through...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA42

IMO 1988 LL INA42 Origin: INA Problem (a) Four balls of radius 1 are mutually tangent, three resting an the ﬂoor and the fourth resting on the others. A tetrahedron, each of whose edges has length s, is circumscribed around the balls. Find the value of s. (b) Suppose that ABCD and EFGH are opposite faces of a rectangu- lar solid, with \angleDHC = 45◦and \angleFHB = 60◦. Find the...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL NET20

IMO 1974 LL NET20 Origin: NET Problem For which natural numbers n do there exist n natural numbers ai (1 \leqi \leqn) such that n i=1 a−2 i = 1?

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL GBR21

IMO 1992 LL GBR21 Origin: GBR Problem Prove that if x, y, z > 1 and 1 x + 1 y + 1 z = 2, then \sqrtx + y + z \geq \sqrt x −1 + y −1 + \sqrt z −1.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL FRG29

IMO 1985 LL FRG29 Origin: FRG Problem Call a four-digit number (xyzt)B in the number system with base B stable if (xyzt)B = (dcba)B −(abcd)B, where a \leqb \leqc \leqd are the digits of (xyzt)B in ascending order. Determine all stable numbers in the number system with base B. (FRG 2a) The same problem with B = 1985. (FRG 2b) With assumptions as in FRG 2, determine the number of...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL SWE94

IMO 1989 LL SWE94 Origin: SWE Problem Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When does equality occur?

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON31

IMO 1967 LL MON31 Origin: MON Problem An urn contains balls of k diﬀerent colors; there are ni balls of the ith color. Balls are drawn at random from the urn, one by one, without replacement. Find the smallest number of draws necessary for getting m balls of the same color. Solution Suppose that n1 \leqn2 \leq\cdot \cdot \cdot \leqnk. If nk < m, there is no solution. Otherwise, the...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL ICE40

IMO 1988 LL ICE40 Origin: ICE Problem A sequence of numbers an, n = 1, 2, . . ., is deﬁned as follows: a1 = 1/2, and for each n \geq2, an = 2n −3 2n an−1. Prove that n k=1 ak < 1 for all n \geq1.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL6

IMO 1969 LL BEL6 Origin: BEL Problem Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two diﬀerent ways and prove that 10 − 10 1 10 = 24.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL CUB6

IMO 1978 LL CUB6 Origin: CUB Problem Prove that for all X > 1 there exists a triangle whose sides have lengths P1(X) = X4+X3+2X2+X+1, P2(X) = 2X3+X2+2X+1, and P3(X) = X4−1. Prove that all these triangles have the same greatest angle and calculate it.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TUR72

IMO 1992 LL TUR72 Origin: TUR Problem In a school six diﬀerent courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that: (i) mathematics was ranked among the most preferred courses by all stu- dents; (ii) no student ranked music among the least preferred ones; (iii) all students preferred history...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BRA9

IMO 1979 LL BRA9 Origin: BRA Problem The real numbers \alpha1, \alpha2, \alpha3, . . . , \alphan are positive. Let us denote by h = n 1/\alpha1+1/\alpha2+\cdot\cdot\cdot+1/\alphan the harmonic mean, g = n\sqrt\alpha1\alpha2 \cdot \cdot \cdot \alphan the geometric mean, a = \alpha1+\alpha2+\cdot\cdot\cdot+\alphan n the arithmetic mean. Prove that h \leq g \leqa, and that each of the equalities implies the other one.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL AUS2

IMO 1986 LL AUS2 Origin: AUS Problem Let ABCD be a convex quadrilateral. DA and CB meet at F and AB and DC meet at E. The bisectors of the angles DFC and AED are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines AC and BD.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL COL20

IMO 1983 LL COL20 Origin: COL Problem Let f and g be functions from the set A to the same set A. We deﬁne f to be a functional nth root of g (n is a positive integer) if f n(x) = g(x), where f n(x) = f n−1(f(x)). (a) Prove that the function g : R \toR, g(x) = 1/x has an inﬁnite number of nth functional roots for...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL CZS13

IMO 1972 LL CZS13 Origin: CZS Problem Given a sphere K, determine the set of all points A that are vertices of some parallelograms ABCD that satisfy AC \leqBD and whose entire diagonal BD is contained in K.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL FIN12

IMO 1976 LL FIN12 Origin: FIN Problem Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL YUG70

IMO 1969 LL YUG70 Origin: YUG Problem A park has the shape of a convex pentagon of area 5 \sqrt 3 ha (= 50000 \sqrt 3 m2). A man standing at an interior point O of the park notices that he stands at a distance of at most 200 m from each vertex of the pentagon. Prove that he stands at a distance of at least 100 m from each...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL POR58

IMO 1992 LL POR58 Origin: POR Problem Let ABC be a triangle. Denote by a, b, and c the lengths of the sides opposite to the angles A, B, and C, respectively. Prove that7 bc a + b + c = sin A + sin B + sin C cos(A/2) sin(B/2) sin(C/2).

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BEL9

IMO 1970 LL BEL9 Origin: BEL Problem If n is even, prove that 1 −1 2 + 1 3 −1 4 + \cdot \cdot \cdot −1 n = 2 n + 2 + n + 4 + n + 6 + \cdot \cdot \cdot + 1 2n .

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE40

IMO 1972 LL SWE40 Origin: SWE Problem Prove the inequalities u v \leqsin u sin v \leq\pi u v , for 0 \lequ < v \leq\pi 2 .

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL NET48

IMO 1983 LL NET48 Origin: NET Problem Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL VIE48

IMO 1974 LL VIE48 Origin: VIE Problem Let a be a number diﬀerent from zero. For all integers n deﬁne Sn = an + a−n. Prove that if for some integer k both Sk and Sk+1 are integers, then for each integer n the number Sn is an integer.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FIN16

IMO 1992 LL FIN16 Origin: FIN Problem Find all triples (x, y, z) of integers such that x2 + 2 y2 + 3 z2 = 2 3.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USS50

IMO 1977 LL USS50 Origin: USS Problem Determine all positive integers n for which there exists a poly- nomial Pn(x) of degree n with integer coeﬃcients that is equal to n at n diﬀerent integer points and that equals zero at zero. Solution Suppose that Pn(x) = n for x \in{x1, x2, . . . , xn}. Then Pn(x) = (x −x1)(x −x2) \cdot \cdot \cdot (x −xn) + n....

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL PRK59

IMO 1992 LL PRK59 Origin: PRK Problem Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle. Prove that for any two points P, Q on the arc A0A6 the following equality holds: i=0 (−1)iPAi = i=0 (−1)iQAi.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL AUT3

IMO 1970 LL AUT3 Origin: AUT Problem Prove that for a, b \inN, a!b! divides (a + b)!.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CUB6

IMO 1974 LL CUB6 Origin: CUB Problem Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USA55

IMO 1977 LL USA55 Origin: USA Problem Through a point O on the diagonal BD of a parallelogram ABCD, segments MN parallel to AB, and PQ parallel to AD, are drawn, with M on AD, and Q on AB. Prove that diagonals AO, BP, DN (ex- tended if necessary) will be concurrent. Solution The statement is true without the assumption that O \inBD. Let BP \cap DN = {K}. If...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS55

IMO 1966 LL USS55 Origin: USS Problem Given the vertex A and the centroid M of a triangle ABC, ﬁnd the locus of vertices B such that all the angles of the triangle lie in the interval [40◦, 70◦].

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL57

IMO 1969 LL POL57 Origin: POL Problem On the sides AB and AC of triangle ABC two points K and L are given such that KB AK + LC AL = 1. Prove that KL passes through the centroid of ABC.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL VIE76

IMO 1987 LL VIE76 Origin: VIE Problem Given two sequences of positive numbers {ak} and {bk} (k \inN) such that (i) ak < bk, (ii) cos akx + cos bkx \geq−1 k for all k \inN and x \inR, prove the existence of limk\to\inftyak bk and ﬁnd this limit.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL12

IMO 1966 LL BUL12 Origin: BUL Problem Find digits x, y, z such that the equality \sqrtxx \cdot \cdot \cdot x 2n −yy \cdot \cdot \cdot y n = zz \cdot \cdot \cdot z n holds for at least two values of n \inN, and in that case ﬁnd all n for which this equality is true.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL NET58

IMO 1985 LL NET58 Origin: NET Problem Prove that there are inﬁnitely many pairs (k, N) of positive integers such that 1 + 2 + \cdot \cdot \cdot + k = (k + 1) + (k + 2) + \cdot \cdot \cdot + N.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL BUL6

IMO 1971 LL BUL6 Origin: BUL Problem Let squares be constructed on the sides BC, CA, AB of a trian- gle ABC, all to the outside of the triangle, and let A1, B1, C1 be their cen- ters. Starting from the triangle A1B1C1 one analogously obtains a triangle A2B2C2. If S, S1, S2 denote the areas of triangles ABC, A1B1C1, A2B2C2, respectively, prove that S = 8S1 −4S2.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL FRA27

IMO 1985 LL FRA27 Origin: FRA Problem Let O be a point on the oriented Euclidean plane and (i, j) a directly oriented orthonormal basis. Let C be the circle of radius 1, centered at O. For every real number t and nonnegative integer n let Mn be the point on C for which ⟨i, −−−\to OMn⟩= cos 2nt (or −−−\to OMn = cos 2nti+sin 2ntj). Let k \geq2 be...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL HUN18

IMO 1966 LL HUN18 Origin: HUN Problem Solve the equation sin x + cos x = 1 p, where p is a real parameter. Discuss for which values of p the equation has at least one real solution and determine the number of solutions in [0, 2\pi) for a given p.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL SPA66

IMO 1992 LL SPA66 Origin: SPA Problem A circle of radius \rho is tangent to the sides AB and AC of the triangle ABC, and its center K is at a distance p from BC. (a) Prove that a(p −\rho) = 2s(r −\rho), where r is the inradius and 2s the perimeter of ABC. (b) Prove that if the circle intersect BC at D and E, then DE = 4...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL GBR30

IMO 1983 LL GBR30 Origin: GBR Problem Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .) of positive integers such that u2 n = n r=0 n + r r un−r for all n \geq0, where m r is the usual binomial coeﬃcient.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IND49

IMO 1989 LL IND49 Origin: IND Problem Let A, B denote two distinct ﬁxed points in space. Let X, P denote variable points (in space), while K, N, n denote positive integers. Call (X, K, N, P) admissible if (N −K)PA + K \cdot PB \geqN \cdot PX. Call (X, K, N) admissible if (X, K, N, P) is admissible for all choices of P. Call (X, N) admissible if...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL MON45

IMO 1969 LL MON45 Origin: MON Problem Given n points in the plane such that no three of them are collinear, prove that one can ﬁnd at least n−3 convex quadrilaterals with their vertices at these points.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL YUG69

IMO 1969 LL YUG69 Origin: YUG Problem Suppose that positive real numbers x1, x2, x3 satisfy x1x2x3 > 1, x1 + x2 + x3 < 1 x1 1 x2 1 x3 . Prove that: (a) None of x1, x2, x3 equals 1. (b) Exactly one of these numbers is less than 1.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FRG20

IMO 1992 LL FRG20 Origin: FRG Problem Let X and Y be two sets of points in the plane and M be a set of segments connecting points from X and Y . Let k be a natural number. Prove that the segments from M can be painted using k colors in such a way that for any point x \inX \cupY and two colors \alpha and \beta (\alpha ̸=...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL AUS2

IMO 1985 LL AUS2 Origin: AUS Problem We are given a triangle ABC and three rectangles R1, R2, R3 with sides parallel to two ﬁxed perpendicular directions and such that their union covers the sides AB, BC, and CA; i.e., each point on the perimeter of ABC is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL NET33

IMO 1971 LL NET33 Origin: NET Problem A square 2n \times 2n grid is given. Let us consider all possible paths along grid lines, going from the center of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its center at the grid center is passed through only once. (a)...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SPA74

IMO 1985 LL SPA74 Origin: SPA Problem Find the triples of positive integers x, y, z satisfying x + 1 y + 1 z = 4 5.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL40

IMO 1967 LL POL40 Origin: POL Problem Exactly one side of a tetrahedron is of length greater than Solution Suppose CD is the longest edge of the tetrahedron ABCD, AB = a, CK and DL are the altitudes of the triangles ABC and ABD respectively, and DM is the altitude of the tetrahedron ABCD. Then CK2 \leq1 −a2/4, since CK is a leg of the right triangle whose other leg...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL AUT3

IMO 1971 LL AUT3 Origin: AUT Problem Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that for any positive real numbers x, y, z the following inequality holds: (ax + by + cz) x a + y b + z c \leq(x + y + z)2 (a + c)2 4ac .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ITA28

IMO 1967 LL ITA28 Origin: ITA Problem Find values of the parameter u for which the expression y = tan(x −u) + tan x + tan(x + u) tan(x −u) tan x tan(x + u) does not depend on x. Solution The given expression can be transformed into y = 4 cos 2u + 2 cos 2u −cos 2x −3. It does not depend on x if and only if...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN37

IMO 1969 LL HUN37 Origin: HUN Problem If a1, a2, . . . , an are real constants, and if y = cos(a1 + x) + 2 cos(a2 + x) + \cdot \cdot \cdot + n cos(an + x) has two zeros x1 and x2 whose diﬀerence is not a multiple of \pi, prove that y \equiv0.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GBR21

IMO 1978 LL GBR21 Origin: GBR Problem A circle touches the sides AB, BC, CD, DA of a square at points K, L, M, N respectively, and BU, KV are parallel lines such that U is on DM and V on DN. Prove that UV touches the circle.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BUL11

IMO 1969 LL BUL11 Origin: BUL Problem Let Z be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of Z. Let us call such a pair of points unjoinable. Prove that for each real r > 0 there exists an unjoinable pair of points separated by distance r.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CZS21

IMO 1985 LL CZS21 Origin: CZS Problem Let A be a set of positive integers such that for any two elements x, y of A, |x−y| \geqxy 25 . Prove that A contains at most nine elements. Give an example of such a set of nine elements.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL POL38

IMO 1971 LL POL38 Origin: POL Problem Let A, B, C be three points with integer coordinates in the plane and K a circle with radius R passing through A, B, C. Show that AB\cdotBC\cdotCA \geq2R, and if the center of K is in the origin of the coordinates, show that AB \cdot BC \cdot CA \geq4R.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BUL16

IMO 1970 LL BUL16 Origin: BUL Problem Show that the equation 2 −x2 + 3 −x3 = 0 has no real roots.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR85

IMO 1989 LL POR85 Origin: POR Problem Let P(x) be a polynomial with integer coeﬃcients such that P(m1) = P(m2) = P(m3) = P(m4) = 7 for given distinct integers m1, m2, m3, and m4. Show that there is no integer m such that P(m) = 14.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL POL31

IMO 1977 LL POL31 Origin: POL Problem Let f be a function deﬁned on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that f satisﬁes the following conditions: (1) f(ab, c) = f(a, c)f(b, c), f(c, ab) = f(c, a)f(c, b); (2) f(a, 1 −a) = 1. Prove that f(a, a) = f(a, −a) = 1, f(a, b)f(b, a) = 1. Solution We...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MON35

IMO 1984 LL MON35 Origin: MON Problem Prove that there exist distinct natural numbers m1, m2, . . . , mk satisfying the conditions \pi−1984 < 25 − 1 m1 1 m2 \cdot \cdot \cdot + 1 mk < \pi−1960 where \pi is the ratio between circle and its diameter.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL POL50

IMO 1987 LL POL50 Origin: POL Problem Let P, Q, R be polynomials with real coeﬃcients, satisfying P 4+Q4 = R2. Prove that there exist real numbers p, q, r and a polynomial S such that P = pS, Q = qS and R = rS2. Variants: (1) P 4 + Q4 = R4; (2) gcd(P, Q) = 1; (3) \pmP 4 + Q4 = R2 or R4.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL FRG26

IMO 1983 LL FRG26 Origin: FRG Problem Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) =

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE93

IMO 1988 LL VIE93 Origin: VIE Problem Given a natural number n, ﬁnd all polynomials P(x) of degree less than n satisfying the following condition: n i=0 P(i)(−1)i n i = 0.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE42

IMO 1972 LL SWE42 Origin: SWE Problem The decimal number 13101 is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL SPA58

IMO 1987 LL SPA58 Origin: SPA Problem Find, with argument, the integer solutions of the equation 3z2 = 2x3 + 385x2 + 256x −58195.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL BUL2

IMO 1988 LL BUL2 Origin: BUL Problem Let an = (n + 1)2 + n2 , n = 1, 2, . . . , where [x] denotes the integer part of x. Prove that (a) there are inﬁnitely many positive integers m such that am+1−am > 1; (b) there are inﬁnitely many positive integers m such that am+1−am = 1.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL NET23

IMO 1976 LL NET23 Origin: NET Problem Prove that in a Euclidean plane there are inﬁnitely many concentric circles C such that all triangles inscribed in C have at least one irrational side.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL YUG55

IMO 1971 LL YUG55 Origin: YUG Problem Prove that the polynomial x4 + \lambdax3 + µx2 + \nux + 1 has no real roots if \lambda, µ, \nu are real numbers satisfying |\lambda| + |µ| + |\nu| \leq \sqrt 2.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL FRA23

IMO 1970 LL FRA23 Origin: FRA Problem Let E be a ﬁnite set, PE the family of its subsets, and f a mapping from PE to the set of nonnegative real numbers such that for any two disjoint subsets A, B of E, f(A \cupB) = f(A) + f(B). Prove that there exists a subset F of E such that if with each A \subsetE we associate a subset A′...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL CZS10

IMO 1978 LL CZS10 Origin: CZS Problem Show that for any natural number n there exist two prime numbers p and q, p ̸= q, such that n divides their diﬀerence.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL TUR73

IMO 1986 LL TUR73 Origin: TUR Problem Let (ai)i\inN be a strictly increasing sequence of positive real numbers such that limi\to\inftyai = +\inftyand ai+1/ai \leq10 for each i. Prove that for every positive integer k there are inﬁnitely many pairs (i, j) with 10k \leqai/aj \leq10k+1.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL ITA40

IMO 1992 LL ITA40 Origin: ITA Problem The colonizers of a spherical planet have decided to build N towns, each having area 1/1000 of the total area of the planet. They also decided that any two points belonging to diﬀerent towns will have diﬀerent latitude and diﬀerent longitude. What is the maximal value of N?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HKG33

IMO 1989 LL HKG33 Origin: HKG Problem Let n be a positive integer. Show that ( \sqrt 2 + 1)n = \sqrtm + \sqrtm −1 for some positive integer m.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL33

IMO 1966 LL BUL33 Origin: BUL Problem Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle one draws segments tangent to the smaller circle. Prove that the length of one of these segments equals the sum of the lengths of the other two.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL SWE42

IMO 1971 LL SWE42 Origin: SWE Problem Let Li, i = 1, 2, 3, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths li, i = 1, 2, 3. By L∗ i we denote the segment of length li with its midpoint on the midpoint of the corresponding side of the triangle. Let M(L) be the set of points in the plane...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BEL2

IMO 1979 LL BEL2 Origin: BEL Problem For a ﬁnite set E of cardinality n \geq3, let f(n) denote the maximum number of 3-element subsets of E, any two of them having exactly one common element. Calculate f(n).

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET32

IMO 1970 LL NET32 Origin: NET Problem Let there be given an acute angle \angleAOB = 3\alpha, where OA = OB. The point A is the center of a circle with radius OA. A line s parallel to OA passes through B. Inside the given angle a variable line t is drawn through O. It meets the circle in O and C and the given line s in D, where...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG35

IMO 1988 LL HKG35 Origin: HKG Problem In the triangle ABC, let D, E, and F be the midpoints of the three sides, X, Y , and Z the feet of the three altitudes, H the orthocenter, and P, Q, and R the midpoints of the line segments joining H to the three vertices. Show that the nine points D, E, F, P, Q, R, X, Y, Z lie on...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TWN77

IMO 1992 LL TWN77 Origin: TWN Problem Show that if 994 integers are chosen from 1, 2, . . . , 1992 and one of the chosen integers is less than 64, then there exist two among the chosen integers such that one of them is a factor of the other.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL NET36

IMO 1970 LL NET36 Origin: NET Problem Let x, y, z be nonnegative real numbers satisfying x2 + y2 + z2 = 5 and yz + zx + xy = 2. Which values can the greatest of the numbers x2 −yz, y2 −xz, z2 −xy have?

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL YUG51

IMO 1976 LL YUG51 Origin: YUG Problem Four swallows are catching a ﬂy. At ﬁrst, the swallows are at the four vertices of a tetrahedron, and the ﬂy is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the ﬂy.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL GBR32

IMO 1983 LL GBR32 Origin: GBR Problem Let a, b, c be positive real numbers and let [x] denote the greatest integer that does not exceed the real number x. Suppose that f is a function deﬁned on the set of nonnegative integers n and taking real values such that f(0) = 0 and f(n) \leqan + f([bn]) + f([cn]), for all n \geq1. Prove that if b + c...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL GDR20

IMO 1971 LL GDR20 Origin: GDR Problem Let M be the circumcenter of a triangle ABC. The line through M perpendicular to CM meets the lines CA and CB at Q and P respec- tively. Prove that CP CM CQ CM AB PQ = 2.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL FRG16

IMO 1988 LL FRG16 Origin: FRG Problem Show that if n runs through all positive integers, f(n) = n + n/3 + 1/2 runs through all positive integers skipping the terms of the sequence an = 3n2 −2n.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM36

IMO 1972 LL ROM36 Origin: ROM Problem A ﬁnite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE53

IMO 1967 LL SWE53 Origin: SWE Problem In making Euclidean constructions in geometry it is permit- ted to use a straightedge and compass. In the constructions considered in this question, no compasses are permitted, but the straightedge is as- sumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the ruler....

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SWE78

IMO 1988 LL SWE78 Origin: SWE Problem A two-person game is played with nine boxes arranged in a 3 \times 3 square, initially empty, and with white and black stones. At each move a player puts three stones, not necessarily of the same color, in three boxes in either a horizontal or a vertical row. No box can contain stones of diﬀerent colors: If, for instance, a player puts a...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON30

IMO 1967 LL MON30 Origin: MON Problem Given m+n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . ., n), determine the number of pairs (ai, bj) for which |i −j| \geqk, where k is a nonnegative integer. Solution We assume w.l.o.g. that m \leqn. Let r and s be the numbers of pairs for which i −j \geqk and of...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL USS79

IMO 1986 LL USS79 Origin: USS Problem Let AA1, BB1, CC1 be the altitudes in an acute-angled triangle ABC. K and M are points on the line segments A1C1 and B1C1 respec- tively. Prove that if the angles MAK and CAA1 are equal, then the angle C1KM is bisected by AK.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL GRE35

IMO 1979 LL GRE35 Origin: GRE Problem Given a sequence (an), with a1 = 4 and an+1 = a2 n−2 (\foralln \inN), prove that there is a triangle with side lengths an −1, an, an + 1, and that its area is equal to an integer.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL CZS20

IMO 1982 LL CZS20 Origin: CZS Problem Consider a cube C and two planes \sigma, \tau, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MON42

IMO 1987 LL MON42 Origin: MON Problem Find the integer solutions of the equation \sqrt 2 m (2 + \sqrt 2)n .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL1

IMO 1967 LL BUL1 Origin: BUL Problem Prove that all numbers in the sequence 107811 , 110778111 , 111077781111 , . . . are perfect cubes. Solution Let us denote the nth term of the given sequence by an. Then an = 1 103n+3 −102n+3 7102n+2 −10n+1 10n+2 −1 = 1 27(103n+3 −3 \cdot 102n+2 + 3 \cdot 10n+1 −1) = 10n+1 −1 3 .

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MOR57

IMO 1986 LL MOR57 Origin: MOR Problem In a triangle ABC, the incircle touches the sides BC, CA, AB in the points A′, B′, C′, respectively; the excircle in the angle A touches the lines containing these sides in A1, B1, C1, and similarly, the excircles in the angles B and C touch these lines in A2, B2, C2 and A3, B3, C3. Prove that the triangle ABC is right-angled...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BUL11

IMO 1979 LL BUL11 Origin: BUL Problem Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges and equal space angles between adjacent lateral walls is regular. Variant. Prove that a pyramid A1 . . . A2k+1S with equal space angles between adjacent lateral walls is regular if there exists a sphere tangent to all its edges.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USS44

IMO 1976 LL USS44 Origin: USS Problem A circle of radius 1 rolls around a circle of radius \sqrt

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL IRE48

IMO 1988 LL IRE48 Origin: IRE Problem Find all plane triangles whose sides have integer length and whose incircles have unit radius.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL GBR27

IMO 1984 LL GBR27 Origin: GBR Problem The function f(n) is deﬁned on the nonnegative integers n by: f(0) = 0, f(1) = 1, f(n) = f n −1 2m(m −1) −f 1 2m(m + 1) −n , for 1 2m(m −1) < n \leq1 2m(m + 1), m \geq2. Find the smallest integer n for which f(n) = 5.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN39

IMO 1969 LL HUN39 Origin: HUN Problem Find the positions of three points A, B, C on the boundary of a unit cube such that min{AB, AC, BC} is the greatest possible.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS59

IMO 1967 LL USS59 Origin: USS Problem On the circle with center O and radius 1 the point A0 is ﬁxed and points A1, A2, . . . , A999, A1000 are distributed in such a way that \angleA0OAk = k (in radians). Cut the circle at points A0, A1, . . . , A1000. How many arcs with diﬀerent lengths are obtained? Solution By the arc AB we shall...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL NET28

IMO 1972 LL NET28 Origin: NET Problem The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MOR44

IMO 1987 LL MOR44 Origin: MOR Problem Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot + sin \thetan = 0. Prove that | sin \theta1 + 2 sin \theta2 + \cdot \cdot \cdot + n sin \thetan| \leq n2 .

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USS43

IMO 1974 LL USS43 Origin: USS Problem An (n2 +n+1)\times(n2 +n+1) matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed (n + 1)(n2 + n + 1).

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL SWE32

IMO 1974 LL SWE32 Origin: SWE Problem Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb for k = 1, 2, . . ., n. If m1 = 1 n(a1 + a2 + \cdot \cdot \cdot + an) and m2 = 1 n(a2 1 + a2 2 + \cdot \cdot \cdot + a2 n), prove that m2 \leq(a+b)2 4ab m2...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL GDR29

IMO 1970 LL GDR29 Origin: GDR Problem Prove that the equation 4x+6x = 9x has no rational solutions.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USS65

IMO 1984 LL USS65 Origin: USS Problem A tetrahedron is inscribed in a sphere of radius 1 such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL GDR21

IMO 1976 LL GDR21 Origin: GDR Problem Find the largest positive real number p (if it exists) such that the inequality x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \geqp(x1x2 + x2x3 + \cdot \cdot \cdot + xn−1xn) (1) is satisﬁed for all real numbers xi, and (a) n = 2; (b) n = 5. Find the largest positive real number p (if it exists) such...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL YUG77

IMO 1987 LL YUG77 Origin: YUG Problem Find the least natural number k such that for any n \in[0, 1] and any natural number n, ak(1 −a)n < (n + 1)3 .

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS9

IMO 1967 LL CZS9 Origin: CZS Problem The circle k and its diameter AB are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on AB and two other vertices on k. Solution The incenter of any such triangle lies inside the circle k. We shall show that every point S interior to the circle S is the incenter of one such triangle....

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL BUL2

IMO 1978 LL BUL2 Origin: BUL Problem If f(x) = (x + 2x2 + \cdot \cdot \cdot + nxn)2 = a2x2 + a3x3 + \cdot \cdot \cdot + a2nx2n, prove that an+1 + an+2 + \cdot \cdot \cdot + a2n = n + 1 5n2 + 5n + 2 .

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL CUB11

IMO 1971 LL CUB11 Origin: CUB Problem Prove that n! cannot be the square of any natural number.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL BUL4

IMO 1971 LL BUL4 Origin: BUL Problem Let xn = 22n + 1 and let m be the least common multiple of x2, x3, . . . , x1971. Find the last digit of m.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN20

IMO 1967 LL HUN20 Origin: HUN Problem In space, n points (n \geq3) are given. Every pair of points determines some distance. Suppose all distances are diﬀerent. Connect every point with the nearest point. Prove that it is impossible to obtain a polygonal line in such a way. 1 Solution The statement so formulated is false. It would be true under the additional assumption that the polygonal line is closed....

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL CZS15

IMO 1989 LL CZS15 Origin: CZS Problem A sequence a1, a2, a3, . . . is deﬁned recursively by a1 = 1 and a2k+j = −aj (j = 1, 2, . . . , 2k). Prove that this sequence is not periodic.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL MON53

IMO 1986 LL MON53 Origin: MON Problem For given positive integers r, v, n let S(r, v, n) denote the num- ber of n-tuples of nonnegative integers (x1, . . . , xn) satisfying the equation x1 + \cdot \cdot \cdot + xn = r and such that xi \leqv for i = 1, . . . , n. Prove that S(r, v, n) = m k=0 (−1)k n...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL NET57

IMO 1985 LL NET57 Origin: NET Problem The solid S is deﬁned as the intersection of the six spheres with the six edges of a regular tetrahedron T , with edge length 1, as diameters. Prove that S contains two points at a distance \sqrt 6. (NET 1a) Using the same assumptions, prove that no pair of points in S has a distance larger than \sqrt 6.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL USA82

IMO 1988 LL USA82 Origin: USA Problem The triangle ABC has a right angle at C. The point P is located on segment AC such that triangles PBA and PBC have congruent inscribed circles. Express the length x = PC in terms of a = BC, b = CA, and c = AB.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL FRA22

IMO 1989 LL FRA22 Origin: FRA Problem Let ABC be an equilateral triangle with side length equal to a natural number N. Consider the set S of all points M inside the triangle ABC such that −−\to AM = N (n−−\to AB + m−\to AC), where m, n are integers and 0 \leqm, n, m+n \leqN. Every point of S is colored in one of the three colors blue, white,...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL GBR20

IMO 1988 LL GBR20 Origin: GBR Problem It is proposed to partition the set of positive integers into two disjoint subsets A and B subject to the following conditions: (i) 1 is in A; (ii) no two distinct members of A have a sum of the form 2k + 2 (k = 0, 1, 2, . . .); and (iii) no two distinct members of B have a sum of...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET47

IMO 1969 LL NET47 Origin: NET Problem Let A and B be points on the circle \gamma. A point C, diﬀerent from A and B, is on the circle \gamma. Let D be the projection of the point C onto the line AB. Consider three other circles \gamma1, \gamma2, and \gamma3 with the common tangent AB: \gamma1 inscribed in the triangle ABC, and \gamma2 and \gamma3 tangent to both (the...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL52

IMO 1969 LL POL52 Origin: POL Problem Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL CAN17

IMO 1983 LL CAN17 Origin: CAN Problem In how many ways can 1, 2, . . . , 2n be arranged in a 2 \times n rectangular array a1 a2 \cdot \cdot \cdot an b1 b2 \cdot \cdot \cdot bn for which: (i) a1 < a2 < \cdot \cdot \cdot < an, (ii) b1 < b2 < \cdot \cdot \cdot < bn, (iii) a1 < b1, a2 <...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL ROM35

IMO 1972 LL ROM35 Origin: ROM Problem (a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b = −cm + d = m, (i) \sqrt a2 + b2 + \sqrt c2 + d2 + (a −c)2 + (b −d)2 \geq 4m2 1+m2 , and (ii) 2 \leq 4m2 1+m2 < 4. (b) Express a, b, c, d as functions of m so that there...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL GBR21

IMO 1977 LL GBR21 Origin: GBR Problem Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0, prove that x2 x2 1 + x2 2 + x2 + y2 y2 1 + y2 2 + y2 = 2 3. Solution Let us consider the vectors v1 = (x1, x2, x3), v2 = (y1, y2, y3), v3 = (1, 1, 1) in space. The given equalities express the condition that these three...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA44

IMO 1989 LL INA44 Origin: INA Problem Let A and B be ﬁxed distinct points on the X axis, none of which coincides with the origin O(0, 0), and let C be a point on the Y axis of an orthogonal Cartesian coordinate system. Let g be a line through the origin O(0, 0) and perpendicular to the line AC. Find the locus of the point of intersection of the...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL ROM37

IMO 1977 LL ROM37 Origin: ROM Problem Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1) = 1 for every i = 1, 2, . . ., n. Show that the equation xA1 xA2 \cdot \cdot \cdot + xAn n = xAn+1 n+1 has an inﬁnite set of solutions (x1, x2, . . . , xn+1) in positive integers. Solution We look for a...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GBR35

IMO 1986 LL GBR35 Origin: GBR Problem Establish the maximum and minimum values that the sum |a| + |b| + |c| can have if a, b, c are real numbers such that the maximum value of |ax2 + bx + c| is 1 for −1 \leqx \leq1.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL BEL3

IMO 1979 LL BEL3 Origin: BEL Problem Is it possible to partition 3-dimensional Euclidean space into 1979 mutually isometric subsets?

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL FRA15

IMO 1987 LL FRA15 Origin: FRA Problem Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real numbers. We set S1 = a1b2c3, S2 = a2b3c1, S3 = a3b1c2; T1 = a1b3c2, T2 = a2b1c3, T3 = a3b2c1. Suppose that the set {S1, S2, S3, T1, T2, T3} has at most two elements. Prove that S1 + S2 + S3 = T1 + T2 +...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL38

IMO 1967 LL POL38 Origin: POL Problem Does there exist an integer such that its cube is equal to 3n2 + 3n + 7, where n is integer? Solution Suppose that there exist integers n and m such that m3 = 3n2 + 3n + 7. Then from m3 \equiv1 (mod 3) it follows that m = 3k + 1 for some k \inZ. Substituting into the initial equation we...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM38

IMO 1966 LL ROM38 Origin: ROM Problem Two concentric circles have radii R and r respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between 2 \cdot \sqrt R+\sqrtr \sqrt R−\sqrtr −1 and 63 20 \cdot R+r R−r.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRA21

IMO 1986 LL FRA21 Origin: FRA Problem Let AB be a segment of unit length and let C, D be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set {A, B, C, D}.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL HUN21

IMO 1967 LL HUN21 Origin: HUN Problem Without using any tables, ﬁnd the exact value of the product P = cos \pi 15 cos 2\pi 15 cos 3\pi 15 cos 4\pi 15 cos 5\pi 15 cos 6\pi 15 cos 7\pi 15 . Solution Using the formula cos x cos 2x cos4x \cdot \cdot \cdot cos 2n−1x = sin 2nx 2n sin x, which is shown by simple induction, we obtain...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL USA104

IMO 1989 LL USA104 Origin: USA Problem For each nonzero complex number z, let arg z be the unique real number t such that −\pi < t \leq\pi and z = |z|(cos t + ı sin t). Given a real number c > 0 and a complex number z ̸= 0 with arg z ̸= \pi, deﬁne B(c, z) = {b \inR | |w −z| < b ⇒| arg w...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL USA75

IMO 1986 LL USA75 Origin: USA Problem The incenter of a triangle is the midpoint of the line seg- ment of length 4 joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL GDR28

IMO 1970 LL GDR28 Origin: GDR Problem A set G with elements u, v, w, . . . is a group if the following conditions are fulﬁlled: (1) There is a binary algebraic operation ◦deﬁned on G such that for all u, v \inG there is a w \inG with u ◦v = w. (2) This operation is associative; i.e., for all u, v, w \inG, (u ◦v) ◦w =...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL YUG13

IMO 1966 LL YUG13 Origin: YUG Problem Let a1, a2, . . . , an be positive real numbers. Prove the inequality n i<j aiaj \geq4 ⎛ ⎝ i<j ai + aj ⎞ ⎠ and ﬁnd the conditions on the numbers ai for equality to hold.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IRE32

IMO 1992 LL IRE32 Origin: IRE Problem Let Sn = {1, 2, . . ., n} and fn : Sn \toSn be deﬁned inductively as follows: f1(1) = 1, fn(2j) = j (j = 1, 2, . . ., [n/2]) and (i) if n = 2k (k \geq1), then fn(2j −1) = fk(j) + k (j = 1, 2, . . ., k); (ii) if n = 2k + 1...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL ROM54

IMO 1987 LL ROM54 Origin: ROM Problem Let n be a natural number. Solve in integers the equation xn + yn = (x −y)n+1.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL ROM35

IMO 1977 LL ROM35 Origin: ROM Problem Find all numbers N = a1a2 . . . an for which 9 \times a1a2 . . . an = an . . . a2a1 such that at most one of the digits a1, a2, . . . , an is zero. Solution The solutions are 0 and Nk = 10 99 . . .9 k 89, where k =...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MON36

IMO 1984 LL MON36 Origin: MON Problem The set {1, 2, . . ., 49} is divided into three subsets. Prove that at least one of these subsets contains three diﬀerent numbers a, b, c such that a + b = c.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL MOR72

IMO 1989 LL MOR72 Origin: MOR Problem Let ABCD be a quadrilateral inscribed in a circle with diam- eter AB such that BC = a, CD = 2a, DA = 3 \sqrt 5−1 a. For each point M on the semicircle AB not containing C and D, denote by h1, h2, h3 the distances from M to the sides BC, CD, and DA. Find the maximum of h1 + h2...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL VIE47

IMO 1978 LL VIE47 Origin: VIE Problem Given the expression Pn(x) = 1 2n x + x2 −1 n + x − x2 −1 n , prove: (a) Pn(x) satisﬁes the identity Pn(x) −xPn−1(x) + 1 4Pn−2(x) \equiv0. (b) Pn(x) is a polynomial in x of degree n.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL USA43

IMO 1978 LL USA43 Origin: USA Problem If p is a prime greater than 3, show that at least one of the numbers p2 , 4 p2 , . . . , p−2 p2 is expressible in the form 1 x + 1 y, where x and y are positive integers.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BEL5

IMO 1984 LL BEL5 Origin: BEL Problem For a real number x, let [x] denote the greatest integer not exceeding x. If m \geq3, prove that m(m + 1) 2(2m −1) m + 1 .

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FIN20

IMO 1986 LL FIN20 Origin: FIN Problem For any angle \alpha with 0 < \alpha < 180◦, we call a closed convex planar set an \alpha-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is \alpha. Given a (closed) triangle T , ﬁnd the greatest \alpha such that any two points in T are contained in an \alpha-set S...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL AUS2

IMO 1984 LL AUS2 Origin: AUS Problem Given a regular convex 2m-sided polygon P, show that there is a 2m-sided polygon \pi with the same vertices as P (but in diﬀerent order) such that \pi has exactly one pair of parallel sides.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HKG32

IMO 1989 LL HKG32 Origin: HKG Problem Let ABC be an equilateral triangle. Let D, E, F, M, N, and P bee the mid-points of BC, CA, AB, FD, FB, and DC respectively. (a) Show that the line segments AM, EN, and FP are concurrent. (b) Let O be the point of intersection of AM, EN, and FP. Find OM : OF : ON : OE : OP : OA.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL POL32

IMO 1977 LL POL32 Origin: POL Problem In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted. Solution It is a known result that among six persons there are 3 mutually ac- quainted or 3 mutually unacquainted. By the condition of the problem the last case is excluded. If there is a man in...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL GDR15

IMO 1977 LL GDR15 Origin: GDR Problem Let n be an integer greater than 1. In the Cartesian coordinate system we consider all squares with integer vertices (x, y) such that 1 \leq x, y \leqn. Denote by pk (k = 0, 1, 2, . . . ) the number of pairs of points that are vertices of exactly k such squares. Prove that k(k −1)pk = 0. Solution...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USA86

IMO 1985 LL USA86 Origin: USA Problem Let l denote the length of the smallest diagonal of all rectangles inscribed in a triangle T . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of T .) Determine the maximum value of l2 S(T ) taken over all triangles (S(T ) denotes the area of triangle T ).

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BEL4

IMO 1984 LL BEL4 Origin: BEL Problem Given a triangle ABC, three equilateral triangles AEB, BFC, and CGA are constructed in the exterior of ABC. Prove that: (a) CE = AF = BG; (b) CE, AF, and BG have a common point.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USS91

IMO 1985 LL USS91 Origin: USS Problem Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USA40

IMO 1976 LL USA40 Origin: USA Problem Let g(x) be a ﬁxed polynomial and deﬁne f(x) by f(x) = x2 + xg(x3). Show that f(x) is not divisible by x2 −x + 1.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL HUN26

IMO 1971 LL HUN26 Origin: HUN Problem An inﬁnite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates (0, 0), (p, 0), (p, q), (0, q) for some positive integers p, q. Show that there must exist two among them one of which is entirely contained in the other.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE94

IMO 1988 LL VIE94 Origin: VIE Problem Let n + 1 (n \geq1) positive integers be given such that for each integer, the set of all prime numbers dividing this integer is a subset of the set of n given prime numbers. Prove that among these n + 1 integers one can ﬁnd numbers (possibly one number) whose product is a perfect square.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL CUB12

IMO 1971 LL CUB12 Origin: CUB Problem A system of n numbers x1, x2, . . . , xn is given such that x1 = logxn−1 xn, x2 = logxn x1, . . . , xn = logxn−2 xn−1. Prove that $n k=1 xk = 1.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS12

IMO 1969 LL CZS12 Origin: CZS Problem Given a unit cube, ﬁnd the locus of the centroids of all tetra- hedra whose vertices lie on the sides of the cube.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL NET47

IMO 1983 LL NET47 Origin: NET Problem In a plane, three pairwise intersecting circles C1, C2, C3 with centers M1, M2, M3 are given. For i = 1, 2, 3, let Ai be one of the points of intersection of Cj and Ck ({i, j, k} = {1, 2, 3}). Prove that if \angleM3A1M2 = \angleM1A2M3 = \angleM2A3M1 = \pi/3 (directed angles), then M1A1, M2A2, and M3A3 are concurrent.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS57

IMO 1967 LL USS57 Origin: USS Problem Consider the sequence (cn): c1 = a1 + a2 + \cdot \cdot \cdot + a8, c2 = a2 1 + a2 2 + \cdot \cdot \cdot + a2 8, . . . . . . . . . . . . cn = an 1 + an 2 + \cdot \cdot \cdot + an 8, . . . . . . . ....

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL GBR20

IMO 1978 LL GBR20 Origin: GBR Problem Let O be the center of a circle. Let OU, OV be perpendicular radii of the circle. The chord PQ passes through the midpoint M of UV . Let W be a point such that PM = PW, where U, V, M, W are collinear. Let R be a point such that PR = MQ, where R lies on the line PW. Prove...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL FRA18

IMO 1978 LL FRA18 Origin: FRA Problem Given a natural number n, prove that the number M(n) of points with integer coordinates inside the circle (O(0, 0), \sqrtn) satisﬁes \pin −5\sqrtn + 1 < M(n) < \pin + 4\sqrtn + 1.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USA38

IMO 1976 LL USA38 Origin: USA Problem Let x = \sqrta + \sqrt b, where a and b are natural numbers, x is not an integer, and x < 1976. Prove that the fractional part of x exceeds 10−19.76.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL AUT2

IMO 1970 LL AUT2 Origin: AUT Problem Prove that the two last digits of 999 and 9999 in decimal representation are equal.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL AUS1

IMO 1987 LL AUS1 Origin: AUS Problem Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q are positive integers. For i = 1, 2, . . ., n, put Si = xi + xi+1 + \cdot \cdot \cdot + xi+p−1 and Ti = xi+p + xi+p+1 + \cdot \cdot \cdot + xi+n−1 (it is assumed that xi+n =...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USS48

IMO 1977 LL USS48 Origin: USS Problem The intersection of a plane with a regular tetrahedron with edge a is a quadrilateral with perimeter P. Prove that 2a \leqP \leq3a. Solution Let a plane cut the edges AB, BC, CD, DA at points K, L, M, N respec- tively. Let D′, A′, B′ be distinct points in the plane ABC such that the triangles BCD′, CD′A′, D′A′B′ are equilateral, and...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL THA100

IMO 1989 LL THA100 Origin: THA Problem Let A be an n\timesn matrix whose elements are nonnegative real numbers. Assume that A is a nonsingular matrix and all elements of A−1 are nonnegative real numbers. Prove that every row and every column of A has exactly one nonzero element.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL USS67

IMO 1987 LL USS67 Origin: USS Problem If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1, ﬁnd the maximum of the expression (a + b)4 + (a + c)4 + (a + d)4 + (b + c)4 + (b + d)4 + (c + d)4.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRG30

IMO 1986 LL FRG30 Origin: FRG Problem Prove that a convex polyhedron all of whose faces are equilat- eral triangles has at most 30 edges.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL ROM17

IMO 1966 LL ROM17 Origin: ROM Problem Suppose ABCD and A′B′C′D′ are two parallelograms arbi- trarily arranged in space, and let points M, N, P, Q divide the segments AA′, BB′, CC′, DD′ respectively in equal ratios. (a) Show that MNPQ is a parallelogram; (b) Find the locus of MNPQ as M varies along the segment AA′.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL SWE68

IMO 1986 LL SWE68 Origin: SWE Problem Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real coeﬃcients a, b. Determine the number of distinct real roots and their multiplicities for various values of a and b. Display your result graphically in the (a, b) plane.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL VIE50

IMO 1978 LL VIE50 Origin: VIE Problem A variable tetrahedron ABCD has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal (BC = DA, CA = DB, AB = DC); and the vertices A, B, C lie respectively on three ﬁxed spheres with the same center P and radii 3, 4, 12. What is the maximal length of PD?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL IRE53

IMO 1989 LL IRE53 Origin: IRE Problem Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3 and a1, a2, . . . , an are distinct integers. Suppose that f(x) = g(x)h(x), where g(x), h(x) are both nonconstant polynomials with integer coeﬃ- cients. Prove that n = 3.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL TWN75

IMO 1992 LL TWN75 Origin: TWN Problem A sequence {an} of positive integers is deﬁned by an = n + \sqrtn + 1 , n \inN. Determine the positive integers that occur in the sequence.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL AUT1

IMO 1971 LL AUT1 Origin: AUT Problem The points S(i, j) with integer Cartesian coordinates 0 < i \leqn, 0 < j \leqm, m \leqn, form a lattice. Find the number of: (a) rectangles with vertices on the lattice and sides parallel to the coor- dinate axes; (b) squares with vertices on the lattice and sides parallel to the coordinate axes; (c) squares in total, with vertices on the lattice.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL CZS11

IMO 1974 LL CZS11 Origin: CZS Problem Given a line p and a triangle \trianglein the plane, construct an equilateral triangle one of whose vertices lies on the line p, while the other two halve the perimeter of \triangle.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL NET32

IMO 1971 LL NET32 Origin: NET Problem Two half-lines a and b, with the common endpoint O, make an acute angle \alpha. Let A on a and B on b be points such that OA = OB, and let b′ be the line through A parallel to b. Let \beta be the circle with center B and radius BO. We construct a sequence of half-lines c1, c2, c3, . ....

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL USS87

IMO 1988 LL USS87 Origin: USS Problem All the irreducible positive rational numbers such that the prod- uct of the numerator and the denominator is less than 1988 are written in increasing order. Prove that any two adjacent fractions a/b and c/d, a/b < c/d, satisfy the equation bc −ad = 1.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL NET60

IMO 1986 LL NET60 Origin: NET Problem Prove the inequality (−a+b+c)2(a−b+c)2(a+b−c)2 \geq(−a2+b2+c2)(a2−b2+c2)(a2+b2−c2) for all real numbers a, b, c.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL GDR18

IMO 1977 LL GDR18 Origin: GDR Problem Given an isosceles triangle ABC with a right angle at C, construct the center M and radius r of a circle cutting on segments AB, BC, CA the segments DE, FG, and HK, respectively, such that \angleDME + \angleFMG + \angleHMK = 180◦and DE : FG : HK = AB : BC : CA. Solution Let U be the midpoint of the segment...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR83

IMO 1989 LL POR83 Origin: POR Problem Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth n bourbakis and each silver coin is worth m bourbakis (n and m are positive integers). Using gold and solver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ROM71

IMO 1985 LL ROM71 Origin: ROM Problem For every integer r > 1 ﬁnd the smallest integer h(r) > 1 having the following property: For any partition of the set {1, 2, . . ., h(r)} into r classes, there exist integers a \geq0, 1 \leqx \leqy such that the numbers a + x, a + y, a + x + y are contained in the same class of the...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CZS23

IMO 1985 LL CZS23 Origin: CZS Problem Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s = [nx + y], n \inN}. Prove that if r > 1 is a rational number, there exist real numbers u and v such that S(r, 0) \capS(u, v) = \emptyset, S(r, 0) \cupS(u, v) = N.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ICE41

IMO 1989 LL ICE41 Origin: ICE Problem Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM48

IMO 1970 LL ROM48 Origin: ROM Problem Let a polynomial p(x) with integer coeﬃcients take the value 5 for ﬁve diﬀerent integer values of x. Prove that p(x) does not take the value 8 for any integer x.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL INA45

IMO 1988 LL INA45 Origin: INA Problem (a) Consider a circle K with diameter AB, a circle L tangent to AB and to K, and a circle M tangent to circle K, circle L, and AB. Calculate the ratio of the area of circle K to the area of circle M. (b) In triangle ABC, AB = AC and ∡CAB = 80◦. If points D, E, and F lie on...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL FRA25

IMO 1985 LL FRA25 Origin: FRA Problem Find eight positive integers n1, n2, . . . , n8 with the follow- ing property: For every integer k, −1985 \leqk \leq1985, there are eight integers \alpha1, \alpha2, . . . , \alpha8, each belonging to the set {−1, 0, 1}, such that k = 8 i=1 \alphaini.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL CUB5

IMO 1978 LL CUB5 Origin: CUB Problem Prove that for any triangle ABC there exists a point P in the plane of the triangle and three points A′, B′, and C′ on the lines BC, AC, and AB respectively such that AB \cdot PC′ = AC \cdot PB′ = BC \cdot PA′ = 0.3M 2, where M = max{AB, AC, BC}.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CAN10

IMO 1986 LL CAN10 Origin: CAN Problem A set of n standard dice are shaken and randomly placed in a straight line. If n < 2r and r < s, then the probability that there will be a string of at least r, but not more than s, consecutive 1’s can be written as P/6s+2. Find an explicit expression for P.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CHN12

IMO 1986 LL CHN12 Origin: CHN Problem Let O be an interior point of a tetrahedron A1A2A3A4. Let S1, S2, S3, S4 be spheres with centers A1, A2, A3, A4, respectively, and let U, V be spheres with centers at O. Suppose that for i, j = 1, 2, 3, 4, i ̸= j, the spheres Si and Sj are tangent to each other at a point Bij lying on...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS51

IMO 1966 LL USS51 Origin: USS Problem In a school, n children numbered 1 to n are initially arranged in the order 1, 2, . . . , n. At a command, every child can either exchange its position with any other child or not move. Can they rearrange into the order n, 1, 2, . . . , n −1 after two commands?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS16

IMO 1969 LL CZS16 Origin: CZS Problem A convex quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d and angles \alpha = \angleDAB, \beta = \angleABC, \gamma = \angleBCD, and \delta = \angleCDA is given. Let s = (a + b + c + d)/2 and P be the area of the quadrilateral. Prove that P 2 = (s −a)(s −b)(s −c)(s −d)...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL BEL6

IMO 1986 LL BEL6 Origin: BEL Problem In an urn there are one ball marked 1, two balls marked 2, and so on, up to n balls marked n. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL CUB13

IMO 1989 LL CUB13 Origin: CUB Problem Let n be a natural number not greater than 44. Prove that for any function f deﬁned over N2 whose images are in the set {1, 2, . . ., n}, there are four ordered pairs (i, j), (i, k), (l, j), and (l, k) such that f(i, j) = f(i, k) = f(l, j) = f(l, k), where i, j, k, l...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL BUL4

IMO 1974 LL BUL4 Origin: BUL Problem Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a triangle ABC, touching the interiors of the sides BC, CA, AB at points Ta, Tb, Tc respectively. Prove that the lines OaTa, ObTb, OcTc are concurrent in a point P for which POa = POb = POc = 2R holds, where R denotes the circumradius of ABC. Also prove...

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL GBR18

IMO 1972 LL GBR18 Origin: GBR Problem We have p players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers s1 \leqs2 \leqs3 \leq\cdot \cdot \cdot \leqsp is given. Show that it is possible for this sequence to be a set of ﬁnal scores of the players in the...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL BEL7

IMO 1987 LL BEL7 Origin: BEL Problem Let f : (0, +\infty) \toR be a function having the property that f(x) = f(1/x) for all x > 0. Prove that there exists a function u : [1, +\infty) \toR satisfying u x+1/x = f(x) for all x > 0.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL MOR45

IMO 1987 LL MOR45 Origin: MOR Problem Let us consider a variable polygon with 2n sides (n \inN) in a ﬁxed circle such that 2n −1 of its sides pass through 2n −1 ﬁxed points lying on a straight line ∆. Prove that the last side also passes through a ﬁxed point lying on ∆.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL CUB10

IMO 1971 LL CUB10 Origin: CUB Problem In how many diﬀerent ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USA41

IMO 1974 LL USA41 Origin: USA Problem Through the circumcenter O of an arbitrary acute-angled trian- gle, chords A1A2, B1B2, C1C2 are drawn parallel to the sides BC, CA, AB of the triangle respectively. If R is the radius of the circumcircle, prove that A1O \cdot OA2 + B1O \cdot OB2 + C1O \cdot OC2 = R2.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL VIE49

IMO 1974 LL VIE49 Origin: VIE Problem Determine an equation of third degree with integral coeﬃcients having roots sin \pi 14, sin 5\pi 14 and sin −3\pi 14 .

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HUN38

IMO 1988 LL HUN38 Origin: HUN Problem In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any 3 of them there was a question that the three students answered diﬀerently. What is the maximal possible number of students tested?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL CUB13

IMO 1971 LL CUB13 Origin: CUB Problem One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: Martian : I have spent 1/12 of my life on Pluton. Human : I also have. Venusian : Me too. Martian : But Venusian and I have spend much more time here than you, Human. Human : That is true. However, Venusian and I are of...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL NET47

IMO 1987 LL NET47 Origin: NET Problem Through a point P within a triangle ABC the lines l, m, and n perpendicular respectively to AP, BP, CP are drawn. Prove that if l intersects the line BC in Q, m intersects AC in R, and n intersects AB in S, then the points Q, R, and S are collinear.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL SWE49

IMO 1970 LL SWE49 Origin: SWE Problem For n \inN, let f(n) be the number of positive integers k \leqn that do not contain the digit 9. Does there exist a positive real number p such that f(n) n \geqp for all positive integers n?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL NET48

IMO 1979 LL NET48 Origin: NET Problem In the plane a circle C of unit radius is given. For any line l a number s(l) is deﬁned in the following way: If l and C intersect in two points, s(l) is their distance; otherwise, s(l) = 0. Let P be a point at distance r from the center of C. One deﬁnes M(r) to be the maximum value of the...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL YUG71

IMO 1969 LL YUG71 Origin: YUG Problem Let four points Ai (i = 1, 2, 3, 4) in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by S. What is the exact placement of the points Ai if S = 180◦?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL SWE61

IMO 1979 LL SWE61 Origin: SWE Problem Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two sequences such that m k=1 ak \geqm k=1 bk for all m \leqn with equality for m = n. Let f be a convex function deﬁned on the real numbers. Prove that n k=1 f(ak) \leq n k=1 f(bk).

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL ROM48

IMO 1984 LL ROM48 Origin: ROM Problem Let ABC be a triangle with interior angle bisectors AA1, BB1, CC1 and incenter I. If \sigma[IA1B] + \sigma[IB1C] + \sigma[IC1A] = 1 2\sigma[ABC], where \sigma[ABC] denotes the area of ABC, show that ABC is isosceles.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MOR56

IMO 1985 LL MOR56 Origin: MOR Problem Let ABCD be a rhombus with angle \angleA = 60◦. Let E be a point, diﬀerent from D, on the line AD. The lines CE and AB intersect at F. The lines DF and BE intersect at M. Determine the angle ∡BMD as a function of the position of E on AD.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL USS48

IMO 1966 LL USS48 Origin: USS Problem Find all positive numbers p for which the equation x2+px+3p = 0 has integral roots.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL GDR34

IMO 1983 LL GDR34 Origin: GDR Problem In a plane are given n points Pi (i = 1, 2, . . . , n) and two angles \alpha and \beta. Over each of the segments PiPi=1 (Pn+1 = P1) a point Qi is constructed such that for all i: (i) upon moving from Pi to Pi+1, Qi is seen on the same side of PiPi+1, (ii) \anglePi+1PiQi = \alpha, (iii)...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL POL49

IMO 1987 LL POL49 Origin: POL Problem In the coordinate system in the plane we consider a convex polygon W and lines given by equations x = k, y = m, where k and m are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of W intersects a square if the boundary contains an interior point of the square. Prove that...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL USA67

IMO 1983 LL USA67 Origin: USA Problem The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BUL12

IMO 1970 LL BUL12 Origin: BUL Problem Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7. Prove that at least one of the expressions of the form \pmx1 \pm x2 \pm x3 \pm x4 \pm x5 \pm x6 is divisible by 7, where the signs are selected in all possible ways. (Gener- alize the statement to every prime number!)

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL POL36

IMO 1966 LL POL36 Origin: POL Problem Let ABCD be a cyclic quadrilateral. Show that the centroids of the triangles ABC, CDA, BCD, DAB lie on a circle.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SPA73

IMO 1985 LL SPA73 Origin: SPA Problem Let A1A2, B1B2, C1C2 be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines B2C1, C2A1, A2B1, the segments B2C1, C2A1, A2B1 are proportional to the sides in which they are contained.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL ISR59

IMO 1989 LL ISR59 Origin: ISR Problem Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1 for 1 \leqr \leq1989. Show that it is possible to ﬁnd ϵr (1 \leqr \leq1989), each equal to \pm1, such that 1989 r=1 ϵrvr \leq \sqrt 3.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL USA48

IMO 1982 LL USA48 Origin: USA Problem Given a ﬁnite sequence of complex numbers c1, c2, . . . , cn, show that there exists an integer k (1 \leqk \leqn) such that for every ﬁnite sequence a1, a2, . . . , an of real numbers with 1 \geqa1 \geqa2 \geq\cdot \cdot \cdot \geqan \geq0, the following inequality holds: n m=1 amcmn \leq n m=1 cm .

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL NET32

IMO 1978 LL NET32 Origin: NET Problem Let C be the circumcircle of the square with vertices (0, 0), (0, 1978), (1978, 0), (1978, 1978) in the Cartesian plane. Prove that C con- tains no other point for which both coordinates are integers.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA21

IMO 1969 LL FRA21 Origin: FRA Problem A right-angled triangle OAB has its right angle at the point B. An arbitrary circle with center on the line OB is tangent to the line OA. Let AT be the tangent to the circle diﬀerent from OA (T is the point of tangency). Prove that the median from B of the triangle OAB intersects AT at a point M such that MB...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL AUS1

IMO 1986 LL AUS1 Origin: AUS Problem Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove the inequality 1 + bk + b2k + \cdot \cdot \cdot + bnk \geq(1 + bn)k for all real b \geq0.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SWE77

IMO 1985 LL SWE77 Origin: SWE Problem Two equilateral triangles are inscribed in a circle with radius r. Let A be the area of the set consisting of all points interior to both triangles. Prove that 2A \geqr2\sqrt 3.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL POL39

IMO 1982 LL POL39 Origin: POL Problem Let S be the unit circle with center O and let P1, P2, . . . , Pn be points of S such that the sum of vectors vi = −−\to OPi is the zero vector. Prove that the inequality n i=1 XPi \geqn holds for every point X.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL ISR36

IMO 1983 LL ISR36 Origin: ISR Problem The set X has 1983 members. There exists a family of subsets {S1, S2, . . . , Sk} such that: (i) the union of any three of these subsets is the entire set X, while (ii) the union of any two of them contains at most 1979 members. What is the largest possible value of k?

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL FIN24

IMO 1983 LL FIN24 Origin: FIN Problem Every x, 0 \leqx \leq1, admits a unique representation x = \infty j=0 aj2−j, where all the aj belong to {0, 1} and inﬁnitely many of them are 0. If b(0) = 1+c 2+c, b(1) = 2+c, c > 0, and f(x) = a0 + \infty j=0 b(a0) \cdot \cdot \cdot b(aj)aj+1, show that 0 < f(x) −x < c for every...

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ITA48

IMO 1985 LL ITA48 Origin: ITA Problem In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, A, B, and C. Person A says, “If C is a knight, B is a knave.” Person C says, “A and I are diﬀerent; one is a knight and the other is a knave.” Who are the knights, and who are...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL TUR72

IMO 1986 LL TUR72 Origin: TUR Problem A one-person game with two possible outcomes is played as follows: After each play, the player receives either a or b points, where a and b are integers with 0 < b < a < 1986. The game is played as many times as one wishes and the total score of the game is deﬁned as the sum of points received after successive...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM46

IMO 1970 LL ROM46 Origin: ROM Problem Given a triangle ABC and a plane \pi having no common points with the triangle, ﬁnd a point M such that the triangle determined by the points of intersection of the lines MA, MB, MC with \pi is congruent to the triangle ABC.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL FIN12

IMO 1978 LL FIN12 Origin: FIN Problem The equation x3 + ax2 + bx + c = 0 has three (not necessarily distinct) real roots t, u, v. For which a, b, c do the numbers t3, u3, v3 satisfy the equation x3 + a3x2 + b3x + c3 = 0?

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL CAN4

IMO 1992 LL CAN4 Origin: CAN Problem Let p, q, and r be the angles of a triangle, and let a = sin 2p, b = sin 2q, and c = sin 2r. If s = (a + b + c)/2, show that s(s −a)(s −b)(s −c) \geq0. When does equality hold?

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL HUN42

IMO 1986 LL HUN42 Origin: HUN Problem The integers 1, 2, . . ., n2 are placed on the ﬁelds of an n \times n chessboard (n > 2) in such a way that any two ﬁelds that have a common edge or a vertex are assigned numbers diﬀering by at most n + 1. What is the total number of such placements?

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USA52

IMO 1977 LL USA52 Origin: USA Problem Two perpendicular chords are drawn through a given interior point P of a circle with radius R. Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from P to the center of the circle is kR. Solution The maximum and minimum are 2R \sqrt 4 −2k2 and 2R 1 +...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL VIE82

IMO 1992 LL VIE82 Origin: VIE Problem Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) = xn + b1xn−1 + \cdot \cdot \cdot + bn−1 + bn be two polynomials with real coeﬃcients such that for each real number x, f(x) is the square of an integer if and only if so is g(x). Prove that if n + m > 0,...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL USS72

IMO 1983 LL USS72 Origin: USS Problem Prove that for all x1, x2, . . . , xn \inR the following inequality holds: n\geqi>j\geq1 cos2(xi −xj) \geqn(n −2) .

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL MON51

IMO 1985 LL MON51 Origin: MON Problem Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers. From f1 one constructs a sequence fk of sequences as follows: if fk = (c1, c2, . . . , cn), then fk+1 = (ci1, ci2, ci3 + 1, ci4 + 1, . . . , cin + 1), where (ci1, ci2, . . ....

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL BUL3

IMO 1974 LL BUL3 Origin: BUL Problem Let ABCD be an arbitrary quadrilateral. Let squares ABB1A2, BCC1B2, CDD1C2, DAA1D2 be constructed in the exterior of the quadrilateral. Furthermore, let AA1PA2 and CC1QC2 be parallelograms. For any arbitrary point P in the interior of ABCD, parallelograms RASC and RPTQ are constructed. Prove that these two parallelograms have two vertices in common.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS58

IMO 1967 LL USS58 Origin: USS Problem A linear binomial l(z) = Az + B with complex coeﬃcients A and B is given. It is known that the maximal value of |l(z)| on the segment −1 \leqx \leq1 (y = 0) of the real line in the complex plane (z = x + iy) is equal to M. Prove that for every z |l(z)| \leqM\rho, where \rho is the sum...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL MEX61

IMO 1988 LL MEX61 Origin: MEX Problem Prove that the numbers A, B, and C are equal, where we deﬁne A as the number of ways that we can cover a 2 \times n rectangle with 2 \times 1 rectangles, B as the number of sequences of ones and twos that add up to n, and C as . m + m+1 \cdot \cdot \cdot + 2m 2m...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL AUS4

IMO 1987 LL AUS4 Origin: AUS Problem Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that (a1b2 + a2b1 + a1b3 + a3b1 + a2b3 + a3b2)2 \geq4(a1a2 + a2a3 + a3a1)(b1b2 + b2b3 + b3b1) and show that the two sides of the inequality are equal if and only if a1/b1 = a2/b2 = a3/b3.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL CZS14

IMO 1969 LL CZS14 Origin: CZS Problem Let a and b be two positive real numbers. If x is a real solution of the equation x2 + px + q = 0 with real coeﬃcients p and q such that |p| \leqa, |q| \leqb, prove that |x| \leq1 a + a2 + 4b . (1) Conversely, if x satisﬁes (1), prove that there exist real numbers p and q...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL USS55

IMO 1970 LL USS55 Origin: USS Problem A turtle runs away from an UFO with a speed of 0.2 m/s. The UFO ﬂies 5 meters above the ground, with a speed of 20 m/s. The UFO’s path is a broken line, where after ﬂying in a straight path of length ℓ(in meters) it may turn through for any acute angle \alpha such that tan \alpha < ℓ 1000. When the...

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USA56

IMO 1977 LL USA56 Origin: USA Problem The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles. Solution See the solution to (LL67-38).

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL GBR14

IMO 1971 LL GBR14 Origin: GBR Problem Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that if the diﬀerence between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL BUL11

IMO 1985 LL BUL11 Origin: BUL Problem Let a and b be integers and n a positive integer. Prove that bn−1a(a + b)(a + 2b) \cdot \cdot \cdot (a + (n −1)b) n! is an integer.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL ROM43

IMO 1970 LL ROM43 Origin: ROM Problem Prove that the equation x3 −3 tan \pi 12x2 −3x + tan \pi 12 = 0 has one root x1 = tan \pi 36, and ﬁnd the other roots.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL LUX65

IMO 1989 LL LUX65 Origin: LUX Problem A regular n-gon A1A2A3 . . . Ak . . . An inscribed in a circle of radius R is given. If S is a point on the circle, calculate T = SA2 1 +SA2 2 + \cdot \cdot \cdot + SA2 n.

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL NET25

IMO 1972 LL NET25 Origin: NET Problem We consider n real variables xi (1 \leqi \leqn), where n is an integer and n \geq2. The product of these variables will be denoted by p, their sum by s, and the sum of their squares by S. Furthermore, let \alpha be a positive constant. We now study the inequality ps \leqS\alpha. Prove that it holds for every n-tuple (xi) if and...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL VIE111

IMO 1989 LL VIE111 Origin: VIE Problem Find the greatest number c such that for all natural numbers n, {n \sqrt 2} \geqc n (where {n \sqrt 2} = n \sqrt 2 −[n \sqrt 2]; [x] is the integer part of x). For this number c, ﬁnd all natural numbers n for which {n \sqrt 2} = c n.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL YUG50

IMO 1974 LL YUG50 Origin: YUG Problem Let m and n be natural numbers with m > n. Prove that 2(m −n)2(m2 −n2 + 1) \geq2m2 −2mn + 1.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL GBR31

IMO 1985 LL GBR31 Origin: GBR Problem Let E1, E2, and E3 be three mutually intersecting ellipses, all in the same plane. Their foci are respectively F2, F3; F3, F1; and F1, F2. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IRE33

IMO 1992 LL IRE33 Origin: IRE Problem Let a, b, c be positive real numbers and p, q, r complex numbers. Let S be the set of all solutions (x, y, z) in C of the system of simultaneous equations ax + by + cz = p, ax2 + by2 + cz2 = q, ax3 + bx3 + cx3 = r. Prove that S has at most six elements.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE62

IMO 1969 LL SWE62 Origin: SWE Problem Which natural numbers can be expressed as the diﬀerence of squares of two integers?

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL SWE39

IMO 1972 LL SWE39 Origin: SWE Problem How many tangents to the curve y = x3 −3x (y = x3 + px) can be drawn from diﬀerent points in the plane?

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ITA29

IMO 1967 LL ITA29 Origin: ITA Problem The triangles A0B0C0 and A′B′C′ have all their angles acute. Describe how to construct one of the triangles ABC, similar to A′B′C′ and circumscribing A0B0C0 (so that A, B, C correspond to A′, B′, C′, and AB passes through C0, BC through A0, and CA through B0). Among these triangles ABC, describe, and prove, how to construct the triangle with the maximum area....

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GDR30

IMO 1969 LL GDR30 Origin: GDR Problem Prove that there exist inﬁnitely many natural numbers a with the following property: The number z = n4 + a is not prime for any natural number n.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USA61

IMO 1984 LL USA61 Origin: USA Problem A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL BUL2

IMO 1976 LL BUL2 Origin: BUL Problem Let P be a set of n points and S a set of l segments. It is known that: (i) No four points of P are coplanar. (ii) Any segment from S has its endpoints at P. (iii) There is a point, say g, in P that is the endpoint of a maximal number of segments from S and that is not a...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL USS67

IMO 1969 LL USS67 Origin: USS Problem Under the conditions x1, x2 > 0, x1y1 > z2 1, and x2y2 > z2 2, prove the inequality (x1 + x2)(y1 + y2) −(z1 + z2)2 \leq x1y1 −z2 + x2y2 −z2 .

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL POL30

IMO 1976 LL POL30 Origin: POL Problem Prove that if P(x) = (x−a)kQ(x), where k is a positive integer, a is a nonzero real number, Q(x) is a nonzero polynomial, then P(x) has at least k + 1 nonzero coeﬃcients.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL AUS2

IMO 1987 LL AUS2 Origin: AUS Problem Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A perfect shuﬄe of these cards changes the order to n+1, 1, n+2, 2, . . ., n− 1, 2n, n; i.e., the cards originally in the ﬁrst n positions have been moved to the places 2, 4, . . . , 2n, while the...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BRA11

IMO 1982 LL BRA11 Origin: BRA Problem A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers a and b. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between a and b be for this to happen?

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL USA62

IMO 1984 LL USA62 Origin: USA Problem From a point P exterior to a circle K, two rays are drawn intersecting K in the respective pairs of points A, A′ and B, B′. For any other pair of points C, C′ on K, let D be the point of intersection of the circumcircles of triangles PAC and PB′C′ other than point P. Similarly, let D′ be the point of intersection...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL NET68

IMO 1988 LL NET68 Origin: NET Problem Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}. The distance between two elements {ai} and {bi} of S is deﬁned as 7 i=1 |ai −bi|. Let T be a subset of S in which any two elements have a distance apart greater than or equal to 3. Prove that T contains at most...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL VIE73

IMO 1987 LL VIE73 Origin: VIE Problem Let f(x) be a periodic function of period T > 0 deﬁned over R. Its ﬁrst derivative is continuous on R. Prove that there exist x, y \in[0, T ) such that x ̸= y and f(x)f ′(y) = f(y)f ′(x).

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CAN9

IMO 1986 LL CAN9 Origin: CAN Problem In a triangle ABC, \angleBAC = 100◦, AB = AC. A point D is chosen on the side AC such that \angleABD = \angleCBD. Prove that AD + DB = BC.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL BUL6

IMO 1989 LL BUL6 Origin: BUL Problem The circles c1 and c2 are tangent at the point A. A straight line l through A intersects c1 and c2 at points C1 and C2 respectively. A circle c, which contains C1 and C2, meets c1 and c2 at points B1 and B2 respectively. Let \kappa be the circle circumscribed around triangle AB1B2. The circle k tangent to \kappa at the point...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GBR28

IMO 1969 LL GBR28 Origin: GBR Problem Let us deﬁne u0 = 0, u1 = 1 and for n \geq0, un+2 = aun+1+bun, a and b being positive integers. Express un as a polynomial in a and b. Prove the result. Given that b is prime, prove that b divides a(ub −1).

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BRA10

IMO 1982 LL BRA10 Origin: BRA Problem Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that S1 r2 S2 r2 \cdot \cdot \cdot + Sn r2n = 4\pi.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL GBR29

IMO 1983 LL GBR29 Origin: GBR Problem Let O be a point outside a given circle. Two lines OAB, OCD through O meet the circle at A, B, C, D, where A, C are the midpoints of OB, OD, respectively. Additionally, the acute angle \theta between the lines is equal to the acute angle at which each line cuts the circle. Find cos \theta and show that the tangents at...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL BUL2

IMO 1967 LL BUL2 Origin: BUL Problem Prove that 1 3n2 + 1 2n + 1 6 \geq(n!)2/n (n is a positive integer) and that equality is possible only in the case n = 1. Solution (n!)2/n = ((1 \cdot 2 \cdot \cdot \cdot n)1/n)2 \leq 1+2+\cdot\cdot\cdot+n n 2 = n+1 2 \leq1 3n2 + 1 2n + 1 6.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL BUL32

IMO 1966 LL BUL32 Origin: BUL Problem The sides a, b, c of a triangle ABC form an arithmetic progression; the sides of another triangle A1B1C1 also form an arithmetic progression. Suppose that \angleA = \angleA1. Prove that the triangles ABC and A1B1C1 are similar.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL TUR61

IMO 1987 LL TUR61 Origin: TUR Problem Let PQ be a line segment of constant length \lambda taken on the side BC of a triangle ABC with the order B, P, Q, C, and let the lines through P and Q parallel to the lateral sides meet AC at P1 and Q1 and AB at P2 and Q2 respectively. Prove that the sum of the areas of the trapezoids PQQ1P1...

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL NET49

IMO 1979 LL NET49 Origin: NET Problem Let there be given two sequences of integers fi(1), fi(2), . . . (i = 1, 2) satisfying: (i) fi(nm) = fi(n)fi(m) if gcd(n, m) = 1; (ii) for every prime P and all k = 2, 3, 4, . . ., fi(P k) = fi(P)fi(P k−1) − P 2f(P k−2). Moreover, for every prime P: (iii) f1(P) = 2P, (iv) f2(P) <...

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL YUG44

IMO 1966 LL YUG44 Origin: YUG Problem What is the greatest number of balls of radius 1/2 that can be placed within a rectangular box of size 10 \times 10 \times 1?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FRA24

IMO 1979 LL FRA24 Origin: FRA Problem Let a and b be coprime integers, greater than or equal to 1. Prove that all integers n greater than or equal to (a −1)(b −1) can be written in the form: n = ua + vb, with (u, v) \inN \times N.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL53

IMO 1969 LL POL53 Origin: POL Problem Given two segments AB and CD not in the same plane, ﬁnd the locus of points M such that MA2 + MB2 = MC2 + MD2.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL POL54

IMO 1992 LL POL54 Origin: POL Problem Suppose that n > m \geq1 are integers such that the string of digits 143 occurs somewhere in the decimal representation of the fraction m/n. Prove that n > 125

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL ICE24

IMO 1992 LL ICE24 Origin: ICE Problem Let Q+ denote the set of nonnegative rational numbers. Show that there exists exactly one function f : Q+ \toQ+ satisfying the follow- ing conditions: (i) if 0 < q < 1 2, then f(q) = 1 + f q 1−2q ; (ii) if 1 < q \leq2, then f(q) = 1 + f(q + 1); (iii) f(q)f(1/q) = 1 for all...

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL GBR26

IMO 1984 LL GBR26 Origin: GBR Problem A cylindrical container has height 6 cm and radius 4 cm. It rests on a circular hoop, also of radius 4 cm, ﬁxed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BEL4

IMO 1969 LL BEL4 Origin: BEL Problem Let O be a point on a nondegenerate conic. A right angle with vertex O intersects the conic at points A and B. Prove that the line AB passes through a ﬁxed point located on the normal to the conic through the point O.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ROM47

IMO 1967 LL ROM47 Origin: ROM Problem Prove the inequality x1x2 \cdot \cdot \cdot xk xn−1 xn−1 \cdot \cdot \cdot + xn−1 k \leqxn+k−1 +xn+k−1 +\cdot \cdot \cdot+xn+k−1 k , where xi > 0 (i = 1, 2, . . ., k), k \inN, n \inN. Solution Using the A–G mean inequality we get (n + k −1)xn 1x2 \cdot \cdot \cdot xk \leqnxn+k−1 xn+k−1 \cdot \cdot \cdot...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL HKG36

IMO 1989 LL HKG36 Origin: HKG Problem Prove the identity 1+ 1 2 −2 3 + 1 4 + 1 5 −2 6 +\cdot \cdot \cdot+ 1 478 + 1 479 −2 480 = 2 k=0 (161 + k)(480 −k).

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX45

IMO 1983 LL LUX45 Origin: LUX Problem Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coﬀee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass 2, and then take the same spoon of the new mixture from glass 2 and pour it back into the ﬁrst glass. What happens...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL MON33

IMO 1967 LL MON33 Origin: MON Problem In what case does the system x + y + mz = a, x + my + z = b, mx + y + z = c, have a solution? Find the conditions under which the unique solution of the above system is an arithmetic progression. Solution If m ̸\in{−2, 1}, the system has the unique solution x = b + a −(1...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL NET49

IMO 1969 LL NET49 Origin: NET Problem A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number...

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL YUG54

IMO 1971 LL YUG54 Origin: YUG Problem A set M is formed of 2n n men, n = 1, 2, . . .. Prove that we can choose a subset P of the set M consisting of n + 1 men such that one of the following conditions is satisﬁed: (1) every member of the set P knows every other member of the set P; (2) no member of...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL GDR31

IMO 1970 LL GDR31 Origin: GDR Problem Prove that for any triangle with sides a, b, c and area P the following inequality holds: P \leq \sqrt 4 (abc)2/3. Find all triangles for which equality holds.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA22

IMO 1969 LL FRA22 Origin: FRA Problem Let \alpha(n) be the number of pairs (x, y) of integers such that x + y = n, 0 \leqy \leqx, and let \beta(n) be the number of triples (x, y, z) such that x + y + z = n and 0 \leqz \leqy \leqx. Find a simple relation between \alpha(n) and the integer part of the number n+2 and the relation...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL SWE48

IMO 1967 LL SWE48 Origin: SWE Problem Determine all positive roots of the equation xx = 1/ \sqrt 2. Solution Put f(x) = x ln x. The given equation is equivalent to f(x) = f(1/2), which has the solutions x1 = 1/2 and x2 = 1/4. Since the function f is decreasing on (0, 1/e), and increasing on (1/e, +\infty), this equation has no other solutions.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL BEL8

IMO 1987 LL BEL8 Origin: BEL Problem Determine the least possible value of the natural number n such that n! ends in exactly 1987 zeros.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL YUG52

IMO 1974 LL YUG52 Origin: YUG Problem A fox stands in the center of the ﬁeld which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole ﬁeld, while the rabbit can move only along the border of the ﬁeld. The maximal speeds of the fox and rabbit are equal to u and v, respectively. Prove that:...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL ITA26

IMO 1967 LL ITA26 Origin: ITA Problem Let ABCD be a regular tetrahedron. To an arbitrary point M on one edge, say CD, corresponds the point P = P(M), which is the intersection of two lines AH and BK, drawn from A orthogonally to BM and from B orthogonally to AM. What is the locus of P as M varies? Solution Let L be the midpoint of the edge AB....

Practice › Mathematics › IMO Longlist › IMO 1972 Longlist ›

IMO 1972 LL NET29

IMO 1972 LL NET29 Origin: NET Problem Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a triangle A1B1C1 such that A1A, B1B, C1C are the bisectors of angles of the triangle. We have that AC = BC and A1C1 ̸= B1C1. (a) Prove that C1 lies on the circumcircle of the triangle ABC. (b) Suppose that ∡BAC1 = \pi/6; ﬁnd the form of triangle ABC.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX40

IMO 1983 LL LUX40 Origin: LUX Problem Four faces of tetrahedron ABCD are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are a < b < c, determine the radius of the sphere circumscribed about the tetrahedron as a function on a, b, and c. What is the ratio c/a if R = a?

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL SWE43

IMO 1971 LL SWE43 Origin: SWE Problem Show that for nonnegative real numbers a, b and integers n \geq2, an + bn \geq a + b n . When does equality hold?

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL CYP18

IMO 1985 LL CYP18 Origin: CYP Problem The circles (R, r) and (P, \rho), where r > \rho, touch externally at A. Their direct common tangent touches (R, r) at B and (P, \rho) at C. The line RP meets the circle (P, \rho) again at D and the line BC at E. If |BC| = 6|DE|, prove that: (a) the lengths of the sides of the triangle RBE are...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL POL39

IMO 1967 LL POL39 Origin: POL Problem Show that the triangle whose angles satisfy the equality sin2 A + sin2 B + sin2 C cos2 A + cos2 B + cos2 C = 2 is a right-angled triangle. Solution Since sin2 A + sin2 B + sin2 C + cos2 A + cos2 B + cos2 C = 3, the given equality is equivalent to cos2 A+cos2 B+cos2 C =...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BEL6

IMO 1982 LL BEL6 Origin: BEL Problem On the three distinct lines a, b, and c three points A, B, and C are given, respectively. Construct three collinear points X, Y, Z on lines a, b, c, respectively, such that BY AX = 2 and CZ AX = 3.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USS90

IMO 1985 LL USS90 Origin: USS Problem Decompose the number 51985−1 into a product of three integers, each of which is larger than 5100.

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL CZS10

IMO 1967 LL CZS10 Origin: CZS Problem The square ABCD is to be decomposed into n triangles (nonoverlapping) all of whose angles are acute. Find the smallest inte- ger n for which there exists a solution to this problem and construct at least one decomposition for this n. Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL BEL7

IMO 1970 LL BEL7 Origin: BEL Problem Let ABCD be any quadrilateral. A square is constructed on each side of the quadrilateral, all in the same manner (i.e., outward or inward). Denote the centers of the squares by M1, M2, M3, and M4. Prove: (a) M1M3 = M2M4; (b) M1M3 is perpendicular to M2M4.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL FRA23

IMO 1979 LL FRA23 Origin: FRA Problem Consider the set E consisting of pairs of integers (a, b), with a \geq 1 and b \geq1, that satisfy in the decimal system the following properties: (i) b is written with three digits, as \alpha2\alpha1\alpha0, \alpha2 ̸= 0; (ii) a is written as \betap . . . \beta1\beta0 for some p; (iii) (a + b)2 is written as \betap . . ....

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ITA46

IMO 1985 LL ITA46 Origin: ITA Problem Let C be the curve determined by the equation y = x3 in the rectangular coordinate system. Let t be the tangent to C at a point P of C; t intersects C at another point Q. Find the equation of the set L of the midpoints M of PQ as P describes C. Is the correspondence associating P and M a bijection...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL USA64

IMO 1983 LL USA64 Origin: USA Problem The sum of all the face angles about all of the vertices except one of a given polyhedron is 5160. Find the sum of all of the face angles of the polyhedron.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POL80

IMO 1989 LL POL80 Origin: POL Problem We are given a ﬁnite collection of segments in the plane, of total length 1. Prove that there exists a line ℓsuch that the sum of the lengths of the projections of the given segments to the line ℓis less than 2/\pi.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL AUT1

IMO 1970 LL AUT1 Origin: AUT Problem Prove that bc b + c + ca c + a + ab a + b \leq1 2(a + b + c) (a, b, c > 0).

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL SWE61

IMO 1983 LL SWE61 Origin: SWE Problem Let a and b be integers. Is it possible to ﬁnd integers p and q such that the integers p + na and q + nb have no common prime factor no matter how the integer n is chosen.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL GRE31

IMO 1987 LL GRE31 Origin: GRE Problem Construct a triangle ABC given its side a = BC, its circum- radius R (2R \geqa), and the diﬀerence 1/k = 1/c−1/b, where c = AB and b = AC.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL FRA24

IMO 1986 LL FRA24 Origin: FRA Problem Two families of parallel lines are given in the plane, consisting of 15 and 11 lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the ﬁrst family are perpendicular to the lines of the second family. Let V be the set of 165 intersection points of the lines under consideration. Show that...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL USS86

IMO 1988 LL USS86 Origin: USS Problem Let a, b, c be integers diﬀerent from zero. It is known that the equation ax2 + by2 + cz2 = 0 has a solution (x, y, z) in integers diﬀerent from the solution x = y = z = 0. Prove that the equation ax2+by2+cz2 = 1 has a solution in rational numbers.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL GDR32

IMO 1979 LL GDR32 Origin: GDR Problem Let n, k \geq1 be natural numbers. Find the number A(n, k) of solutions in integers of the equation |x1| + |x2| + \cdot \cdot \cdot + |xk| = n.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL HUN23

IMO 1971 LL HUN23 Origin: HUN Problem Find all integer solutions of the equation x2 + y2 = (x −y)3.

Practice › Mathematics › IMO Longlist › IMO 1971 Longlist ›

IMO 1971 LL GDR18

IMO 1971 LL GDR18 Origin: GDR Problem Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n their geometric mean, and ma = (a1 + a2 + \cdot \cdot \cdot + an)/n their arithmetic mean. Prove that (1 + mg)n \leq(1 + a1) \cdot \cdot \cdot (1 + an) \leq(1 + ma)n.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL AUS5

IMO 1987 LL AUS5 Origin: AUS Problem Let there be given three circles K1, K2, K3 with centers O1, O2, O3 respectively, which meet at a common point P. Also, let K1 \capK2 = {P, A}, K2 \capK3 = {P, B}, K3 \capK1 = {P, C}. Given an arbitrary point X on K1, join X to A to meet K2 again in Y , and join X to C to...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL AUT4

IMO 1970 LL AUT4 Origin: AUT Problem Solve the system of equations x2 + xy = a2 + ab y2 + xy = a2 −ab, a, b real, a ̸= 0.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL USA81

IMO 1992 LL USA81 Origin: USA Problem Suppose that points X, Y, Z are located on sides BC, CA, and AB, respectively, of \triangleABC in such a way that \triangleXY Z is similar to \triangleABC. Prove that the orthocenter of \triangleXY Z is the circumcenter of \triangleABC.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL USA106

IMO 1989 LL USA106 Origin: USA Problem Let n > 1 be a ﬁxed integer. Deﬁne functions f0(x) = 0, f1(x) = 1 −cos x, and for k > 0, fk+1(x) = 2fk(x) cos x −fk−1(x). If F(x) = f1(x) + f2(x) + \cdot \cdot \cdot + fn(x), prove that (a) 0 < F(x) < 1 for 0 < x < \pi n+1, and (b) F(x) > 1 for \pi...

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL JAP41

IMO 1992 LL JAP41 Origin: JAP Problem Let S be a set of positive integers n1, n2, . . . , n6 and let n(f) denote the number n1nf(1) + n2nf(2) + \cdot \cdot \cdot + n6nf(6), where f is a permu- tation of {1, 2, . . ., 6}. Let Ω= {n(f) | f is a permutation of {1, 2, . . ., 6}}. Give an example of positive...

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GBR24

IMO 1969 LL GBR24 Origin: GBR Problem The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where a0, . . . , ak are integers, is said to be divisible by an integer m if P(x) is a multiple of m for every integral value of x. Show that if P(x) is divisible by m, then a0 \cdotk! is a multiple of m. Also prove that...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL VIE72

IMO 1987 LL VIE72 Origin: VIE Problem Is it possible to cover a rectangle of dimensions m \times n with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter L) if: (a) m \times n = 1985 \times 1987; (b) m \times n = 1987 \times 1989?

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BEL6

IMO 1984 LL BEL6 Origin: BEL Problem Let P, Q, R be the polynomials with real or complex coeﬃcients such that at least one of them is not constant. If P n +Qn +Rn = 0, prove that n < 3.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL GRE39

IMO 1986 LL GRE39 Origin: GRE Problem Let S be a k-element set. (a) Find the number of mappings f : S \toS such that (i) f(x) ̸= x for x \inS, (ii) f(f(x)) = x for x \inS. (b) The same with the condition (i) left out.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL AUS1

IMO 1992 LL AUS1 Origin: AUS Problem Points D and E are chosen on the sides AB and AC of the triangle ABC in such a way that if F is the intersection point of BE and CD, then AE + EF = AD + DF. Prove that AC + CF = AB + BF.

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL BEL5

IMO 1982 LL BEL5 Origin: BEL Problem Among all triangles with a given perimeter, ﬁnd the one with the maximal radius of its incircle.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL FRG11

IMO 1977 LL FRG11 Origin: FRG Problem Let n and z be integers greater than 1 and (n, z) = 1. Prove: (a) At least one of the numbers zi = 1+z+z2+\cdot \cdot \cdot+zi, i = 0, 1, . . . , n−1, is divisible by n. (b) If (z−1, n) = 1, then at least one of the numbers zi, i = 0, 1, . . ., n−2, is...

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL FRA19

IMO 1989 LL FRA19 Origin: FRA Problem Let a1, . . . , an be distinct positive integers that do not contain a 9 in their decimal representations. Prove that a1 \cdot \cdot \cdot + 1 an \leq30.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL ITA47

IMO 1985 LL ITA47 Origin: ITA Problem Let F be the correspondence associating with every point P = (x, y) the point P ′ = (x′, y′) such that x′ = ax + b, y′ = ay + 2b. (1) Show that if a ̸= 1, all lines PP ′ are concurrent. Find the equation of the set of points corresponding to P = (1, 1) for b = a2....

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SPA55

IMO 1984 LL SPA55 Origin: SPA Problem Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30), p > q being two given positive integers. (a) Prove that k = a3 + b3 + c3 is not a prime number. (b) Prove that if a \cdot b \cdot c is maximum, then 1984 divides k.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL SPA72

IMO 1985 LL SPA72 Origin: SPA Problem Construct a triangle ABC given the side AB and the distance OH from the circumcenter O to the orthocenter H, assuming that OH and AB are parallel.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL VIE47

IMO 1974 LL VIE47 Origin: VIE Problem Given two points A, B outside of a given plane P, ﬁnd the positions of points M in the plane P for which the ratio MA MB takes a minimum or maximum.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL SIN73

IMO 1988 LL SIN73 Origin: SIN Problem In a group of n people each one knows exactly three others. They are seated around a table. We say that the seating is perfect if everyone knows the two sitting by their sides. Show that if there is a perfect seating S for the group, then there is always another perfect seating that cannot be obtained from S by rotation or reﬂection.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL THA70

IMO 1992 LL THA70 Origin: THA Problem Let two circles A and B with unequal radii r and R, respec- tively, be tangent internally at the point A0. If there exists a sequence of distinct circles (Cn) such that each circle is tangent to both A and B, and each circle Cn+1 touches circle Cn at the point An, prove that \infty n=1 |An+1An| < 4\piRr R + r...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL TUR38

IMO 1978 LL TUR38 Origin: TUR Problem Given a circle, construct a chord that is trisected by two given noncollinear radii.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL CAN18

IMO 1983 LL CAN18 Origin: CAN Problem Let b \geq2 be a positive integer. (a) Show that for an integer N, written in base b, to be equal to the sum of the squares of its digits, it is necessary either that N = 1 or that N have only two digits. (b) Give a complete list of all integers not exceeding 50 that, relative to some base b, are...

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL SWE34

IMO 1976 LL SWE34 Origin: SWE Problem Let {an}\infty and {bn}\infty be two sequences determined by the recursion formulas an+1 = an + bn, bn+1 = 3an + bn, n = 0, 1, 2, . . ., and the initial values a0 = b0 = 1. Prove that there exists a uniquely determined constant c such that n|can−bn| < 2 for all nonnegative integers n.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL VIE74

IMO 1979 LL VIE74 Origin: VIE Problem Given an equilateral triangle ABC of side a in a plane, let M be a point on the circumcircle of the triangle. Prove that the sum s = MA4 + MB4 + MC4 is independent of the position of the point M on the circle, and determine that constant value as a function of a.

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL VIE90

IMO 1988 LL VIE90 Origin: VIE Problem Does there exist a number \alpha (0 < \alpha < 1) such that there is an inﬁnite sequence {an} of positive numbers satisfying 1 + an+1 \leqan + \alpha nan, n = 1, 2, . . .?

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL USS70

IMO 1979 LL USS70 Origin: USS Problem There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of triangle Tk is equal to 1/k, k = 1, 2, . . . , 1979. At what values of a number a can one place all these triangles into the equilateral triangle with side length a so that they don’t intersect (points of contact are allowed)?

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL HUN35

IMO 1969 LL HUN35 Origin: HUN Problem Prove that 1 + 1 23 + 1 33 + \cdot \cdot \cdot + 1 n3 < 5 4.

Practice › Mathematics › IMO Longlist › IMO 1979 Longlist ›

IMO 1979 LL SWE62

IMO 1979 LL SWE62 Origin: SWE Problem T is a given triangle with vertices P1, P2, P3. Consider an arbi- trary subdivision of T into ﬁnitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex V of the subtriangles there is assigned a number n(V ) according to the following rules: (i) If V = Pi, then n(V )...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL SWE33

IMO 1974 LL SWE33 Origin: SWE Problem Let a be a real number such that 0 < a < 1, and let n be a positive integer. Deﬁne the sequence a0, a1, a2, . . . , an recursively by a0 = a; ak+1 = ak + 1 na2 k for k = 0, 1, . . ., n −1. Prove that there exists a real number A, depending on...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USS46

IMO 1974 LL USS46 Origin: USS Problem Outside an arbitrary triangle ABC, triangles ADB and BCE are constructed such that \angleADB = \angleBEC = 90◦and \angleDAB = \angleEBC = 30◦. On the segment AC the point F with AF = 3FC is chosen. Prove that \angleDFE = 90◦ and \angleFDE = 30◦.

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL SWE35

IMO 1974 LL SWE35 Origin: SWE Problem If p and q are distinct prime numbers, then there are integers x0 and y0 such that 1 = px0 + qy0. Determine the maximum value of b −a, where a and b are positive integers with the following property: If a \leqt \leqb, and t is an integer, then there are integers x and y with 0 \leqx \leqq −1 and 0...

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL FRG25

IMO 1983 LL FRG25 Origin: FRG Problem How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange ai and ai+1 whenever ai > ai+1 for i running from 1 up to n −1?

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL FRG23

IMO 1984 LL FRG23 Origin: FRG Problem A 2 \times 2 \times 12 box ﬁxed in space is to be ﬁlled with twenty-four 1 \times 1 \times 2 bricks. In how many ways can this be done?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL MON67

IMO 1989 LL MON67 Origin: MON Problem A family of sets A1, A2, . . . , An has the following properties: (i) Each Ai contains 30 elements. (ii) Ai \capAj contains exactly one element for all i, j, 1 \leqi < j \leq30. Find the largest possible n if the intersection of all these sets is empty.

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL AUS3

IMO 1986 LL AUS3 Origin: AUS Problem A line parallel to the side BC of a triangle ABC meets AB in F and AC in E. Prove that the circles on BE and CF as diameters intersect in a point lying on the altitude of the triangle ABC dropped from A to BC.

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL IND30

IMO 1992 LL IND30 Origin: IND Problem Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive integer n. Determine, with proof, the least positive integer m, if it exists, for which Pm is divisible by 3333.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL SPA54

IMO 1984 LL SPA54 Origin: SPA Problem Let P be a convex planar polygon with equal angles. Let l1, . . . , ln be its sides. Show that a necessary and suﬃcient condition for P to be regular is that the sum of the ratios li li+1 (i = 1, . . . , n; ln+1 = l1) equals the number of sides.

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL VIE49

IMO 1978 LL VIE49 Origin: VIE Problem Let A, B, C, D be four arbitrary distinct points in space. (a) Prove that using the segments AB + CD, AC + BD and AD + BC it is always possible to construct a triangle T that is nondegenerate and has no obtuse angle. (b) What should these four points satisfy in order for the triangle T to be right-angled?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL INA43

IMO 1989 LL INA43 Origin: INA Problem Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num- bers. Find all values of a, b, and c such that the following three conditions are satisﬁed simultaneously: (i) f(x) = 381 if sin x = 1/2. (ii) The absolute maximum of f(x) is 444. (iii) The absolute minimum of f(x) is 364.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL FRA18

IMO 1984 LL FRA18 Origin: FRA Problem Let c be the inscribed circle of the triangle ABC, d a line tan- gent to c which does not pass through the vertices of triangle ABC. Prove the existence of points A1, B1, C1, respectively, on the lines BC, CA, AB satisfying the following two properties: (i) Lines AA1, BB1, and CC1 are parallel. (ii) Lines AA1, BB1, and CC1 meet d...

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL USS57

IMO 1970 LL USS57 Origin: USS Problem Let the numbers 1, 2, . . . , n2 be written in the cells of an n \times n square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the kth row? (k a positive integer, 1 \leqk \leqn.)

Practice › Mathematics › IMO Longlist › IMO 1992 Longlist ›

IMO 1992 LL FRG19

IMO 1992 LL FRG19 Origin: FRG Problem Denote by an the greatest number that is not divisible by 3 and that divides n. Consider the sequence s0 = 0, sn = a1 + a2 + \cdot \cdot \cdot+ an, n \inN. Denote by A(n) the number of all sums sk (0 \leqk \leq3n, k \inN0) that are divisible by 3. Prove the formula A(n) = 3n−1 + 2 \cdot 3(n/2)−1...

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL LUX39

IMO 1987 LL LUX39 Origin: LUX Problem Let A be a set of polynomials with real coeﬃcients and let them satisfy the following conditions: (i) if f \inA and deg f \leq1, then f(x) = x −1; (ii) if f \inA and deg f \geq2, then either there exists g \inA such that f(x) = x2+deg g + xg(x) −1 or there exist g, h \inA such that f(x) =...

Practice › Mathematics › IMO Longlist › IMO 1982 Longlist ›

IMO 1982 LL CZS21

IMO 1982 LL CZS21 Origin: CZS Problem All edges and all diagonals of regular hexagon A1A2A3A4A5A6 are colored blue or red such that each triangle AjAkAm, 1 \leqj < k < m \leq6 has at least one red edge. Let Rk be the number of red segments AkAj, (j ̸= k). Prove the inequality k=1 (2Rk −7)2 \leq54.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL FRA20

IMO 1969 LL FRA20 Origin: FRA Problem A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is S. If I is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, ﬁnd the number T = 2S −B −2I +...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL CZS16

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE63

IMO 1969 LL SWE63 Origin: SWE Problem Prove that there are inﬁnitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL LUX31

IMO 1984 LL LUX31 Origin: LUX Problem Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three positive roots \alpha > \beta > \gamma > 0. From the equation f1(x) = 0 one constructs the equation f2(x) = x3 + a2x2 + b2x + c2 = x(x + b1)2 −(a1x + c1)2 = 0. Continuing this process, we get equations f3, . . . , fn. Prove...

Practice › Mathematics › IMO Longlist › IMO 1967 Longlist ›

IMO 1967 LL USS56

IMO 1967 LL USS56 Origin: USS Problem In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi. Solution We shall prove by induction on n the following statement: If in some group of...

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL GBR16

IMO 1974 LL GBR16 Origin: GBR Problem A pack of 2n cards contains n diﬀerent pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players A and B. A third person called the dealer shuﬄes the pack and deals the cards one by one face upward onto the table. One of the players,...

Practice › Mathematics › IMO Longlist › IMO 1988 Longlist ›

IMO 1988 LL HKG32

IMO 1988 LL HKG32 Origin: HKG Problem Assuming that the roots of x3+px2+qx+r = 0 are all real and positive, ﬁnd a relation between p, q, and r that gives a necessary condition for the roots to be exactly the cosines of three angles of a triangle.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL FRA19

IMO 1984 LL FRA19 Origin: FRA Problem Let ABC be an isosceles triangle with right angle at point A. Find the minimum of the function F given by F(M) = BM + CM − \sqrt 3AM.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL GDR33

IMO 1969 LL GDR33 Origin: GDR Problem Given a ring G in the plane bounded by two concentric circles with radii R and R/2, prove that we can cover this region with 8 disks of radius 2R/5. (A region is covered if each of its points is inside or on the border of some disk.)

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL POL54

IMO 1969 LL POL54 Origin: POL Problem Given a polynomial f(x) with integer coeﬃcients whose value is divisible by 3 for three integers k, k + 1, and k + 2, prove that f(m) is divisible by 3 for all integers m.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL USA53

IMO 1977 LL USA53 Origin: USA Problem Find all pairs of integers a and b for which 7a + 14b = 5a2 + 5ab + 5b2. Solution The discriminant of the given equation considered as a quadratic equation in b is 196−75a2. Thus 75a2 \leq196 and hence −1 \leqa \leq1. Now the integer solutions of the given equation are easily found: (−1, 3), (0, 0), (1, 2).

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL ROM65

IMO 1986 LL ROM65 Origin: ROM Problem Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show that there is a point M on C such that MA1 −MA2 + MA3 −MA4 = 0.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BUL7

IMO 1984 LL BUL7 Origin: BUL Problem Prove that for any natural number n, the number 2n n divides the least common multiple of the numbers 1, 2, . . . , 2n −1, 2n.

Practice › Mathematics › IMO Longlist › IMO 1987 Longlist ›

IMO 1987 LL ROM55

IMO 1987 LL ROM55 Origin: ROM Problem Two moving bodies M1, M2 are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines M1M2.

Practice › Mathematics › IMO Longlist › IMO 1985 Longlist ›

IMO 1985 LL USA85

IMO 1985 LL USA85 Origin: USA Problem Let CD be a diameter of circle K. Let AB be a chord that is parallel to CD. The line segment AE, with E on K, is parallel to CB; F is the point of intersection of line segments AB and DE. The line segment FG, with G on DC, extended is parallel to CB. Is GA tangent to K at point A?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL VIE108

IMO 1989 LL VIE108 Origin: VIE Problem For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n} arranged in any order, denote by f(s) the sum of absolute values of the diﬀerences between two consecutive members of s. Find the maximum value of f(s) (where s runs through the set of all such sequences).

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL SWE60

IMO 1969 LL SWE60 Origin: SWE Problem Find the natural number n with the following properties: (1) Let S = {p1, p2, . . . } be an arbitrary ﬁnite set of points in the plane, and rj the distance from Pj to the origin O. We assign to each Pj the closed disk Dj with center Pj and radius rj. Then some n of these disks contain all points...

Practice › Mathematics › IMO Longlist › IMO 1986 Longlist ›

IMO 1986 LL BEL4

IMO 1986 LL BEL4 Origin: BEL Problem Find the last eight digits of the binary development of 271986.

Practice › Mathematics › IMO Longlist › IMO 1969 Longlist ›

IMO 1969 LL BUL10

IMO 1969 LL BUL10 Origin: BUL Problem Let M be the point inside the right-angled triangle ABC (\angleC = 90◦) such that \angleMAB = \angleMBC = \angleMCA = ϕ. Let \psi be the acute angle between the medians of AC and BC. Prove that sin(ϕ+\psi) sin(ϕ−\psi) = 5.

Practice › Mathematics › IMO Longlist › IMO 1977 Longlist ›

IMO 1977 LL FIN43

IMO 1977 LL FIN43 Origin: FIN Problem Evaluate S = n k=1 k(k + 1) \cdot \cdot \cdot (k + p), where n and p are positive integers. Solution Since k(k + 1) \cdot \cdot \cdot (k + p) = (p + 1)! k+p p+1 = (p + 1)! k+p+1 p+2 − k+p p+2 , it follows that n k=1 k(k + 1) \cdot \cdot...

Practice › Mathematics › IMO Longlist › IMO 1978 Longlist ›

IMO 1978 LL USA45

IMO 1978 LL USA45 Origin: USA Problem If r > s > 0 and a > b > c, prove that arbs + brcs + cras \geqasbr + bscr + csar.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL BUL8

IMO 1984 LL BUL8 Origin: BUL Problem In the plane of a given triangle A1A2A3 determine (with proof) a straight line l such that the sum of the distances from A1, A2, and A3 to l is the least possible.

Practice › Mathematics › IMO Longlist › IMO 1970 Longlist ›

IMO 1970 LL FRA27

IMO 1970 LL FRA27 Origin: FRA Problem Find a natural number n such that for all prime numbers p, n is divisible by p if and only if n is divisible by p −1.

Practice › Mathematics › IMO Longlist › IMO 1966 Longlist ›

IMO 1966 LL YUG45

IMO 1966 LL YUG45 Origin: YUG Problem An alphabet consists of n letters. What is the maximal length of a word, if (i) two neighboring letters in a word are always diﬀerent, and (ii) no word abab (a ̸= b) can be obtained by omitting letters from the given word?

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR82

IMO 1989 LL POR82 Origin: POR Problem Solve in the set of real numbers the equation 3x3 −[x] = 3, where [x] denotes the integer part of x.

Practice › Mathematics › IMO Longlist › IMO 1976 Longlist ›

IMO 1976 LL USS43

IMO 1976 LL USS43 Origin: USS Problem Prove that if for a polynomial P(x, y) we have P(x −1, y −2x + 1) = P(x, y), then there exists a polynomial \Phi(x) with P(x, y) = \Phi(y −x2).

Practice › Mathematics › IMO Longlist › IMO 1974 Longlist ›

IMO 1974 LL USA39

IMO 1974 LL USA39 Origin: USA Problem Let n be a positive integer, n \geq2, and consider the polynomial equation xn −xn−2 −x + 2 = 0. For each n, determine all complex numbers x that satisfy the equation and have modulus |x| = 1.

Practice › Mathematics › IMO Longlist › IMO 1989 Longlist ›

IMO 1989 LL POR84

IMO 1989 LL POR84 Origin: POR Problem Let a, b, c, r, and s be real numbers. Show that if r is a root of ax2 +bx+c = 0 and s is a root of −ax2 +bx+c = 0, then a 2x2 +bx+c = 0 has a root between r and s.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL LUX44

IMO 1983 LL LUX44 Origin: LUX Problem We are given twelve coins, one of which is a fake with a diﬀerent mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.

Practice › Mathematics › IMO Longlist › IMO 1983 Longlist ›

IMO 1983 LL ROM56

IMO 1983 LL ROM56 Origin: ROM Problem Consider the expansion (1 + x + x2 + x3 + x4)496 = a0 + a1x + \cdot \cdot \cdot + a1984x1984. (a) Determine the greatest common divisor of the coeﬃcients a3, a8, a13, . . . , a1983. (b) Prove that 10340 < a992 < 10347.

Practice › Mathematics › IMO Longlist › IMO 1984 Longlist ›

IMO 1984 LL MON34

IMO 1984 LL MON34 Origin: MON Problem One country has n cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of ﬂights that the worker will...

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 3

IMO 1973 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 11m52s Problem Let $a$ and $b$ be real numbers for which the equation $$ x^4 + ax^3 + bx^2 + ax + 1 = 0 $$ has at least one real solution. For all such pairs $(a, b)$, find the minimum value of $a^2 + b^2$. Exploration The polynomial $$P(x)=x^4+ax^3+bx^2+ax+1$$ is palindromic. Such equations often...

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 2

IMO 1973 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 14m44s Problem Determine whether or not there exists a finite set $M$ of points in space not lying in the same plane such that, for any two points $A$ and $B$ of $M$, one can select two other points $C$ and $D$ of $M$ so that lines $AB$ and $CD$ are parallel and not coincident....

Practice › Mathematics › IMO › IMO 1973 ›

IMO 1973 Problem 1

IMO 1973 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 22m51s Problem Point $O$ lies on line $g$; $\overrightarrow{OP_1}, \overrightarrow{OP_2}, \ldots, \overrightarrow{OP_n}$ are unit vectors such that points $P_1, P_2, \ldots, P_n$ all lie in a plane containing $g$ and on one side of $g$. Prove that if $n$ is odd, $$ \left\vert \overrightarrow{OP_1} + \overrightarrow{OP_2} + \ldots + \overrightarrow{OP_n} \right\vert \geq 1 $$ Here...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 6

IMO 1972 Problem 6 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 7m52s Problem Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane. Exploration The problem asks for the existence of a regular tetrahedron with one vertex on each of four given distinct parallel planes in $\mathbb{R}^3$. An initial approach using a tilted face fails because computing vertical...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 5

IMO 1972 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m24s Problem Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $$ f(x + y) + f(x - y) = 2f(x)g(y) $$ for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)|...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 6

IMO 1976 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 44m41s Problem For all positive integral $n$, $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$, $u_0=2$, and $u_1=2\frac12$. Prove that $$ 3\log_2[u_n]=2^n-(-1)^n, $$ where $[x]$ is the integral part of $x$. Exploration The recurrence is $$u_{n+1}=u_n(u_{n-1}^2-2)-u_1,$$ with $$u_0=2,\qquad u_1=2\frac12=\frac52.$$ The target identity is $$3\lfloor u_n\rfloor =2^n-(-1)^n.$$ The first task is to compute several terms exactly. For $n=1$, $$u_2=u_1(u_0^2-2)-u_1 =\frac52(4-2)-\frac52 =\frac52.$$ Thus $$3\lfloor...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 4

IMO 1972 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m59s Problem Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities $$ (x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \ ,(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \ ,(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \ ,(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \ ,(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 3

IMO 1972 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m11s Problem Let $m$ and $n$ be arbitrary non-negative integers. Prove that $$ \frac{(2m)!(2n)!}{m!n!(m+n)!} $$ is an integer. ($0! = 1$.) Exploration Consider small values of $m$ and $n$ to examine the expression $$\frac{(2m)!(2n)!}{m! n! (m+n)!}.$$ For $m = n = 0$, the expression equals $\frac{0! \cdot 0!}{0! \cdot 0! \cdot 0!} = 1$. For...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 2

IMO 1972 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 21m31s Problem Prove that if $n \geq 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle. Problem Understanding A cyclic quadrilateral is given. For every integer $n\ge 4$, we must construct a dissection of the quadrilateral into exactly $n$ quadrilaterals,...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 1

IMO 1972 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 20m29s Problem Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum. Exploration Consider a set of ten distinct two-digit numbers, $S = {a_1, a_2, \dots, a_{10}}$, and examine the sums of all its non-empty subsets. There...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 5

IMO 1976 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 45m37s Problem Let a set of $p$ equations be given, $$ \begin{array}{ccccccc} a_{11}x_1&+&\cdots&+&a_{1q}x_q&=&0,\ a_{21}x_1&+&\cdots&+&a_{2q}x_q&=&0,\ &&&\vdots&&&\ a_{p1}x_1&+&\cdots&+&a_{pq}x_q&=&0,\ \end{array} $$ with coefficients $a_{ij}$ satisfying $a_{ij}=-1$, $0$, or $+1$ for all $i=1,\dots, p$, and $j=1,\dots, q$. Prove that if $q=2p$, there exists a solution $x_1, \dots, x_q$ of this system such that all $x_j$ ($j=1,\dots, q$) are integers...

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 6

IMO 1971 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 24m01s Problem Let $A = (a_{ij})(i, j = 1, 2, \cdots, n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij} = 0$, the sum of the elements in the $i$th row and the $j$th column is $\geq n$. Prove that the sum of all the elements of the...

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 5

IMO 1971 Problem 5 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 7m23s Problem Prove that for every natural number $m$, there exists a finite set $S$ of points in a plane with the following property: For every point $A$ in $S$, there are exactly $m$ points in $S$ which are at unit distance from $A$. Exploration For small values of $m$, explicit examples suggest a graph-theoretic...

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 4

IMO 1971 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 12m37s Problem All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y, Z, T$ are interior points of edges $BC, CD, DA$, respectively. Prove: (a) If $\angle DAB +...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 5

IMO 1959 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 18m27s Problem An arbitrary point $M$ is selected in the interior of the segment $AB$. The squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with the segments $AM$ and $MB$ as their respective bases. The circles about these squares, with respective centers $P$ and $Q$, intersect at $M$ and also at...

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 3

IMO 1971 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 3m43s Problem Prove that the set of integers of the form $2^k - 3 (k = 2, 3, \cdots)$ contains an infinite subset in which every two members are relatively prime. Exploration The sequence begins $$2^2-3=1,\qquad 2^3-3=5,\qquad 2^4-3=13,\qquad 2^5-3=29,$$ $$2^6-3=61,\qquad 2^7-3=125,\qquad 2^8-3=253.$$ Among these numbers, $$\gcd(125,5)=5,\qquad \gcd(253,11)=11,$$ since $$2^7-3=5^3,\qquad 2^8-3=11\cdot 23.$$ Thus the whole sequence...

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 2

IMO 1971 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m05s Problem Consider a convex polyhedron $P_1$ with nine vertices $A_1, A_2, \cdots, A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i (i = 2, 3, \cdots, 9)$. Prove that at least two of the polyhedra $P_1, P_2, \cdots, P_9$ have an interior point in common....

Practice › Mathematics › IMO › IMO 1971 ›

IMO 1971 Problem 1

IMO 1971 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 43m36s Problem Prove that the following assertion is true for $n = 3$ and $n = 5$, and that it is false for every other natural number $n > 2$: If $a_1, a_2, \cdots, a_n$ are arbitrary real numbers, then $$ (a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) + (a_2 - a_1)(a_2 -...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 2

IMO 1975 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 35m15s Problem Let $a_1, a_2, a_3, \cdots$ be an infinite increasing sequence of positive integers. Prove that for every $p \geq 1$ there are infinitely many $a_m$ which can be written in the form $$ a_m = xa_p + ya_q $$ with $x, y$ positive integers and $q > p$. Exploration The statement concerns an...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 6

IMO 1970 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 19m45s Problem In a plane there are $100$ points, no three of which are collinear. Consider all possible triangles having these point as vertices. Prove that no more than $70 %$ of these triangles are acute-angled. Exploration The statement concerns all $\binom{100}{3}$ triangles determined by a set of points in general position. The target proportion...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 1

IMO 1975 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 15m43s Problem Let $x_i, y_i~(i = 1, 2, \ldots, n)$ be real numbers such that $$ x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n. $$ Prove that, if $z_1, z_2, \cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n$, then $$ \sum_{i=1}^n (x_i - y_i)^2...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 6

IMO 1959 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m48s Problem Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is in the plane $P$, and the point ${C}$ is in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB$ parallel to $DC$) in which a circle can be...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 6

IMO 1974 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m06s Problem Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k))^2 = 1$, prove that $n(P) - \deg(P) \leq 2$, where $\deg(P)$ denotes the degree of the polynomial $P$. Exploration The problem concerns the integers $k$ for which $P(k)=\pm1$. Since $P$ has integer...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 4

IMO 1959 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m05s Problem Construct a right triangle with a given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. Problem Understanding We are given a positive length $c$, and we must construct a right triangle whose hypotenuse has length $c$. If the legs are...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 4

IMO 1974 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 10m53s Problem Consider decompositions of an $8 \times 8$ chessboard into p non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1 < a_2 < \cdots < a_p$. Find the maximum value...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 3

IMO 1959 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 12m31s Problem Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ : $a\cos ^{2}x + b\cos{x} + c = 0.$ Using the numbers $a,b,c$, form a quadratic equation in $\cos{2x}$, whose roots are the same as those of the original equation. Compare the equations in $\cos{x}$ and $\cos{2x}$ for $a=4, b=2, c=-1$. Exploration...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 2

IMO 1959 Problem 2 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 15m36s Problem For what real values of $x$ is $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A,$ given (a) $A = \sqrt{2}$, (b) $A=1$, (c) $A=2$, where only non-negative real numbers are admitted for square roots? Problem Understanding We are asked to determine all real numbers $x$ such that $$\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A$$ for three specific values of $A$: $\sqrt{2}$, $1$,...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 1

IMO 1959 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m04s Problem Prove that $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$. Exploration The fraction $$\frac{21n+4}{14n+3}$$ is irreducible if and only if $$\gcd(21n+4,14n+3)=1.$$ The natural first step is to compute this greatest common divisor through the Euclidean algorithm. Try small cases. For $n=1$, $$\frac{25}{17},$$ and $\gcd(25,17)=1$. For $n=2$, $$\frac{46}{31},$$ and $\gcd(46,31)=1$. For $n=3$, $$\frac{67}{45},$$ and...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 1

IMO 1974 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 10m39s Problem Three players $A, B$ and $C$ play the following game: On each of three cards an integer is written. These three numbers $p, q, r$ satisfy $0 < p < q < r$. The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 3

IMO 1964 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 3m04s Problem A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 5

IMO 1963 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 5m45s Problem Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$. Problem Understanding The problem asks to prove the exact trigonometric identity $\cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7} = \frac12.$ No approximations are allowed. The angles are rational multiples of $\pi$, so all values are algebraic numbers. The task is to express the left-hand side exactly, using algebraic or trigonometric identities, and...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 3

IMO 1962 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 2m18s Problem A cube $ABCDA'B'C'D'$ is given. Point $X$ moves at constant speed along the square $ABCD$ in the direction $A \to B$; point $Y$ moves at the same speed along the square $BCC'B'$ in the direction $B' \to C'$. Initially $X = A$ and $Y = B'$. Find the locus of all midpoints of...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 6

IMO 1961 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 3m04s Problem Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 6

IMO 1960 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 3m00s Problem Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a)...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 5

IMO 1960 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m30s Problem Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 6

IMO 2024 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 17m40s Problem Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called $\emph{aquaesulian}$ if the following property holds: for every $x,y \in \mathbb{Q}$, $$ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). $$Show that there exists an integer $c$ such that for any aquaesulian...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 5

IMO 2024 Problem 5 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 11m42s Problem Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 4

IMO 2024 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 1h06m Problem Let $ABC$ be a triangle with $AB < AC < BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 3

IMO 2024 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m59s Problem Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$. Prove that at least one of the sequence $a_1, a_3,...

Practice › Mathematics › IMO › IMO 2024 ›

IMO 2024 Problem 1

IMO 2024 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m00s Problem Determine all real numbers $\alpha$ such that, for every positive integer $n$, the integer $$ \lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor $$ is a multiple of $n$. (Note that $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$. For example, $\lfloor -\pi \rfloor...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist C4

IMO 2013 Shortlist C4 Category: Combinatorics Problem Let n be a positive integer, and let A be a subset of t1,...,nu. An A-partition of n into k parts is a representation of n as a sum n “ a1 ¨¨¨ ak, where the parts a1,...,ak belong to A and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist G4

IMO 2013 Shortlist G4 Category: Geometry Problem Let ABC be a triangle with =B ą =C. Let P and Q be two different points on line AC such that =PBA “ =QBA “ =ACB and A is located between P and C. Suppose that there exists an interior point D of segment BQ for which PD “ PB. Let the ray AD intersect the circle ABC at R ‰ A....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist G3

IMO 2024 Shortlist G3 Category: Geometry Problem Let ABCDE be a convex pentagon and let M be the midpoint of AB. Suppose that segment AB is tangent to the circumcircle of triangle CME at M and that D lies on the circumcircles of triangles AME and BMC. Lines AD and ME intersect at K, and lines BD and MC intersect at L. Points P and Q lie on line EC...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist G8

IMO 2015 Shortlist G8 Category: Geometry Problem A triangulation of a convex polygon Π is a partitioning of Π into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thaiangulation if all triangles in it have the same area. Prove that any two different Thaiangulations of a convex polygon Π differ by exactly two triangles. (In other words,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist A4

IMO 2019 Shortlist A4 Category: Algebra Problem Let n ě 2 be a positive integer and a1, a2, ..., an be real numbers such that a1 a2 ¨¨¨ ` an “ 0. Deﬁne the set A by A “ pi,jq ˇ ˇ 1 ď i ă j ď n,|ai ´ aj| ě 1 ( . Prove that, if A is not empty, then ÿ pi,jqPA aiaj ă 0. (China)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N8

IMO 2015 Shortlist N8 Category: Number Theory Problem For every positive integer n with prime factorization n “ śk i“1 pαi i , define ℧pnq “ ÿ i: pią10100 αi. That is, ℧pnq is the number of prime factors of n greater than 10100 , counted with multiplicity. Find all strictly increasing functions f : Z Ñ Z such that ℧ ` fpaq ´ fpbq ˘ ď ℧pa ´ bq...

Practice › Mathematics › IMO › IMO Shortlist ›

IMO 2007 Shortlist

IMO 2007 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist C5

IMO 2023 Shortlist C5 Category: Combinatorics Problem Elisa has 2023 treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: • if more than one chests are unlocked, it locks one of them, or • if there is only one unlocked chest, it...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist A6

IMO 2017 Shortlist A6 Category: Algebra Problem Find all fun tions f : R Ñ R su h that fpfpxqfpyqq fpx yq “ fpxyq for all x,y P R. (Albania)

Practice › Mathematics › IMO ›

IMO 2000

IMO 2000 Official IMO 2000 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, so that $M$ lies closer to $AB$ than $N$. Let $CD$ be...

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IMO 2014 Shortlist N8

IMO 2014 Shortlist N8 Category: Number Theory Problem For every real number x, let }x} denote the distance between x and the nearest integer. Prove that for every pair pa,bq of positive integers there exist an odd prime p and a positive integer k satisfying › › › › a pk › › › › › › › › b pk › › › › › › › › a...

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IMO 2017 Shortlist G3

IMO 2017 Shortlist G3 Category: Geometry Problem Let O be the ir um enter of an a ute s alene triangle ABC. Line OA interse ts the altitudes of ABC through B and C at P and Q, respe tively. The altitudes meet at H. Prove that the ir um enter of triangle PQH lies on a median of triangle ABC. (Ukraine)

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IMO 2020 Shortlist C8

IMO 2020 Shortlist C8 Category: Combinatorics Problem Players A and B play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player A erases two numbers x and y from the blackboard, and then player B writes one of the numbers x ` y and |x ´ y| on the blackboard. The game terminates as soon as, at the end of some...

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IMO 2015 Shortlist N7

IMO 2015 Shortlist N7 Category: Number Theory Problem Let Zą0 denote the set of positive integers. For any positive integer k, a function f : Zą0 Ñ Zą0 is called k-good if gcd fpmq n,fpnq ` m ˘ ď k for all m ‰ n. Find all k such that there exists a k-good function. (Canada)

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IMO 2021

IMO 2021 Official IMO 2021 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $n \geq 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of...

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IMO Shortlist

IMO Shortlist Problems from the IMO Shortlist — the ~30–35 candidate problems considered each year before 6 are selected for the official competition. Total: 575 problems across 19 years. Year A C G N Other Total 2024 8 8 8 7 - 31 2023 7 7 8 8 - 30 2022 8 9 8 8 - 33 2021 8 8 8 8 - 32 2020 8 8 9 7 -...

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IMO 2014 Shortlist C4

IMO 2014 Shortlist C4 Category: Combinatorics Problem Construct a tetromino by attaching two 2 ˆ 1 dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them S- and Z-tetrominoes, respectively. S-tetrominoes Z-tetrominoes Assume that a lattice polygon P can be tiled with S-tetrominoes....

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IMO 2022 Shortlist C7

IMO 2022 Shortlist C7 Category: Combinatorics Problem Lucy starts by writing s integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples v “ pv1,...,v2022q and w “ pw1,...,w2022q that she has already written, and apply one of the following operations to obtain a new tuple: v w “ pv1 w1,...,v2022 ` w2022q v _ w “ pmaxpv1,w1q,...,maxpv2022,w2022qq and then write this tuple...

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IMO 2013 Shortlist G3

IMO 2013 Shortlist G3 Category: Geometry Problem In a triangle ABC, let D and E be the feet of the angle bisectors of angles A and B, respectively. A rhombus is inscribed into the quadrilateral AEDB (all vertices of the rhombus lie on different sides of AEDB). Let ϕ be the non-obtuse angle of the rhombus. Prove that ϕ ď maxt=BAC,=ABCu. (Serbia)

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IMO 2023 Shortlist C4

IMO 2023 Shortlist C4 Category: Combinatorics Problem Let n ě 2 be a positive integer. Paul has a 1 ˆ n2 rectangular strip consisting of n2 unit squares, where the ith square is labelled with i for all 1 ď i ď n2 . He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or...

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IMO 2021 Shortlist G8

IMO 2021 Shortlist G8 Category: Geometry Problem Letω bethecircumcircleofatriangleABC,andletΩA beitsexcirclewhichistangent to the segment BC. Let X and Y be the intersection points of ω and ΩA. Let P and Q be the projections of A onto the tangent lines to ΩA at X and Y , respectively. The tangent line at P to the circumcircle of the triangle APX intersects the tangent line at Q to the circumcircle of the...

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IMO 2023 Shortlist N5

IMO 2023 Shortlist N5 Category: Number Theory Problem Let a1 ă a2 ă a3 ă ¨¨¨ be positive integers such that ak 1 divides 2pa1 a2 ¨¨¨ akq for every k ě 1. Suppose that for infinitely many primes p, there exists k such that p divides ak. Prove that for every positive integer n, there exists k such that n divides ak. (Netherlands)

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IMO 2020 Shortlist C1

IMO 2020 Shortlist C1 Category: Combinatorics Problem Let n be a positive integer. Find the number of permutations a1,a2,...,an of the sequence 1,2,...,n satisfying a1 ď 2a2 ď 3a3 ď ... ď nan. (United Kingdom)

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IMO 2009 Shortlist C6

IMO 2009 Shortlist C6 Category: Combinatorics Problem On a 999×999 board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A non- intersecting route of the limp rook consists of a...

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IMO 2017 Shortlist N4

IMO 2017 Shortlist N4 Category: Number Theory Problem Call a rational number short if it has nitely many digits in its de imal expansion. For a positive integer m, we say that a positive integer t is m-tasti if there exists a number c P t1,2,3,...,2017u su h that 10t ´ 1 c ¨ m is short, and su h that 10k ´ 1 c ¨ m is not short...

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IMO 2018 Shortlist A1

IMO 2018 Shortlist A1 Category: Algebra Problem Let Qą0 denote the set of all positive rational numbers. Determine all fun tions f : Qą0 Ñ Qą0 satisfying f ` x2 fpyq2 ˘ “ fpxq2 fpyq for all x,y P Qą0. (Switzerland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist G2

IMO 2021 Shortlist G2 Category: Geometry Problem Let ABCD be a convex quadrilateral circumscribed around a circle with centre I. Let ω be the circumcircle of the triangle ACI. The extensions of BA and BC beyond A and C meet ω at X and Z, respectively. The extensions of AD and CD beyond D meet ω at Y and T, respectively. Prove that the perimeters of the (possibly self-intersecting) quadrilaterals...

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IMO 2009 Shortlist A7

IMO 2009 Shortlist A7 Category: Algebra Problem Find all functions f from the set of real numbers into the set of real numbers which satisfy for all real x,y the identity f(xf(x + y)) = f(yf(x)) + x2 .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G3

IMO 2007 Shortlist G3 Category: Geometry Problem The diagonals of a trapezoid ABCD intersect at point P. Point Q lies between the parallel lines BC and AD such that ∠AQD = ∠CQB, and line CD separates points P and Q. Prove that ∠BQP = ∠DAQ. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N5

IMO 2012 Shortlist N5 Category: Number Theory Problem For a nonnegative integer n define rad(n) = 1 if n = 0 or n = 1, and rad(n) = p1p2 ···pk where p1 < p2 < ··· < pk are all prime factors of n. Find all polynomials f(x) with nonnegative integer coefficients such that rad(f(n)) divides rad(f(nrad(n) )) for every nonnegative integer n. Solution 1. We are going to prove...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist A7

IMO 2020 Shortlist A7 Category: Algebra Problem Let n and k be positive integers. Prove that for a1,...,an P r1,2k s one has n ÿ i“1 ai a a2 1 ... a2 i ď 4 ? kn. (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C3

IMO 2006 Shortlist C3 Category: Combinatorics Problem Let S be a finite set of points in the plane such that no three of them are on a line. For each convex polygon P whose vertices are in S, let a(P) be the number of vertices of P, and let b(P) be the number of points of S which are outside P. Prove that for every real number x X P...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist G7

IMO 2017 Shortlist G7 Category: Geometry Problem A onvex quadrilateral ABCD has an ins ribed ir le with enter I. Let Ia, Ib, Ic, and Id be the in enters of the triangles DAB, ABC, BCD, and CDA, respe tively. Suppose that the ommon external tangents of the ir les AIbId and CIbId meet at X, and the ommon external tangents of the ir les BIaIc and DIaIc meet at...

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IMO 2016

IMO 2016 Official IMO 2016 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector...

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IMO 2019 Shortlist N3

IMO 2019 Shortlist N3 Category: Number Theory Problem We say that a set S of integers is rootiful if, for any positive integer n and any a0,a1,...,an P S, all integer roots of the polynomial a0 a1x ¨¨¨ ` anxn are also in S. Find all rootiful sets of integers that contain all numbers of the form 2a ´ 2b for positive integers a and b. (Czech Republic)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist A3

IMO 2022 Shortlist A3 Category: Algebra Problem Let Rą0 be the set of positive real numbers. Find all functions f : Rą0 Ñ Rą0 such that, for every x P Rą0, there exists a unique y P Rą0 satisfying xfpyq ` yfpxq ď 2. (Netherlands)

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IMO 2022 Shortlist N4

IMO 2022 Shortlist N4 Category: Number Theory Problem Find all triples of positive integers pa,b,pq with p prime and ap “ b! ` p. (Belgium)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A4

IMO 2011 Shortlist A4 Category: Algebra Problem Determine all pairs (f,g) of functions from the set of positive integers to itself that satisfy fg(n)+1 (n) + gf(n) (n) = f(n + 1) − g(n + 1) + 1 for every positive integer n. Here, fk (n) means f(f(...f | {z } k (n)...)).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C2

IMO 2011 Shortlist C2 Category: Combinatorics Problem Suppose that 1000 students are standing in a circle. Prove that there exists an integer k with 100 ≤ k ≤ 300 such that in this circle there exists a contiguous group of 2k students, for which the first half contains the same number of girls as the second half.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N1

IMO 2013 Shortlist N1 Category: Number Theory Problem Let Zą0 be the set of positive integers. Find all functions f : Zą0 Ñ Zą0 such that m2 fpnq | mfpmq n for all positive integers m and n. (Malaysia)

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IMO 2005

IMO 2005 Official IMO 2005 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1, A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal...

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IMO 2013 Shortlist A3

IMO 2013 Shortlist A3 Category: Algebra Problem Let Qą0 be the set of positive rational numbers. Let f : Qą0 Ñ R be a function satisfying the conditions fpxqfpyq ě fpxyq and fpx yq ě fpxq fpyq for all x,y P Qą0. Given that fpaq “ a for some rational a ą 1, prove that fpxq “ x for all x P Qą0. (Bulgaria)

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IMO 2013 Shortlist C5

IMO 2013 Shortlist C5 Category: Combinatorics Problem Let r be a positive integer, and let a0,a1,... be an infinite sequence of real numbers. Assume that for all nonnegative integers m and s there exists a positive integer n P rm 1,m rs such that am am 1 ¨¨¨ am s “ an an 1 ¨¨¨ an s. Prove that the sequence is periodic, i.e. there exists some p ě 1...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist C5

IMO 2024 Shortlist C5 Category: Combinatorics Problem Let N be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers 1, 2, ..., N on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers pk,nq, where k ě 0 and n is one of the integers on the board, and...

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IMO 2012 Shortlist G4

IMO 2012 Shortlist G4 Category: Geometry Problem Let ABC be a triangle with AB 6= AC and circumcenter O. The bisector of ∠BAC intersects BC at D. Let E be the reflection of D with respect to the midpoint of BC. The lines through D and E perpendicular to BC intersect the lines AO and AD at X and Y respectively. Prove that the quadrilateral BXCY is cyclic. Solution. The...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N4

IMO 2012 Shortlist N4 Category: Number Theory Problem An integer a is called friendly if the equation (m2 n)(n2 m) = a(m − n)3 has a solution over the positive integers. a) Prove that there are at least 500 friendly integers in the set {1,2,...,2012}. b) Decide whether a = 2 is friendly. Solution. a) Every a of the form a = 4k − 3 with k ≥ 2 is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N2

IMO 2009 Shortlist N2 Category: Number Theory Problem A positive integer N is called balanced, if N = 1 or if N can be written as a product of an even number of not necessarily distinct primes. Given positive integers a and b, consider the polynomial P defined by P(x) = (x + a)(x + b). (a) Prove that there exist distinct positive integers a and b such that all...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C2

IMO 2007 Shortlist C2 Category: Combinatorics Problem A unit square is dissected into n > 1 rectangles such that their sides are parallel to the sides of the square. Any line, parallel to a side of the square and intersecting its interior, also intersects the interior of some rectangle. Prove that in this dissection, there exists a rectangle having no point on the boundary of the square. (Japan)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist G6

IMO 2023 Shortlist G6 Category: Geometry Problem Let ABC be an acute-angled triangle with circumcircle ω. A circle Γ is internally tangent to ω at A and also tangent to BC at D. Let AB and AC intersect Γ at P and Q respectively. Let M and N be points on line BC such that B is the midpoint of DM and C is the midpoint of DN. Lines MP...

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IMO 1961

IMO 1961 Official IMO 1961 problems · 6/6 solved, 5 verified. # Status Time 1 ✓ verified 4m48s 2 ✓ verified 11m24s 3 solved 39m28s 4 ✓ verified 19m03s 5 ✓ verified 39m11s 6 ✓ verified 3m04s Problem 1 ✓ verified · 4m48s · Solution → ( Hungary ) Solve the system of equations: $\begin{matrix} \quad x + y + z !!! &= a ; , \ x^2 +y^2+z^2 !!!...

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IMO 2020 Shortlist A8

IMO 2020 Shortlist A8 Category: Algebra Problem Let R be the set of positive real numbers. Determine all functions f : R Ñ R such that, for all positive real numbers x and y, f x fpxyq ˘ y “ fpxqfpyq ` 1. (Ukraine) 6 Saint-Petersburg — Russia, 18th–28th September 2020

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist G1

IMO 2012 Shortlist G1 Category: Geometry Problem In the triangle ABC the point J is the center of the excircle opposite to A. This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L respectively. The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of intersection of...

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IMO 2006 Shortlist N4

IMO 2006 Shortlist N4 Category: Number Theory Problem Let P be a polynomial of degree n > 1 with integer coefficients and let k be any positive integer. Consider the polynomial Q(x) = P(P(...P(P(x))...)), with k pairs of parentheses. Prove that Q has no more than n integer fixed points, i.e. integers satisfying the equation Q(x) = x. (Romania)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist C3

IMO 2012 Shortlist C3 Category: Combinatorics Problem In a 999 × 999 square table some cells are white and the remaining ones are red. Let T be the number of triples (C1,C2,C3) of cells, the first two in the same row and the last two in the same column, with C1 and C3 white and C2 red. Find the maximum value T can attain. Solution. We prove that in an...

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IMO 2017 Shortlist G2

IMO 2017 Shortlist G2 Category: Geometry Problem Let R and S be distin t points on ir le Ω, and let t denote the tangent line to Ω at R. Point R1 is the ree tion of R with respe t to S. A point I is hosen on the smaller ar RS of Ω so that the ir um ir le Γ of triangle ISR1 interse ts t at...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C8

IMO 2007 Shortlist C8 Category: Combinatorics Problem Given a convex n-gon P in the plane. For every three vertices of P, consider the triangle determined by them. Call such a triangle good if all its sides are of unit length. Prove that there are not more than 2 n good triangles. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G2

IMO 2009 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB, respectively. The circle k passes through the midpoints of the segments BP, CQ, and PQ. Prove that if the line PQ is tangent to circle k then OP = OQ.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist G4

IMO 2019 Shortlist G4 Category: Geometry Problem Let P be a point inside triangle ABC. Let AP meet BC at A1, let BP meet CA at B1, and let CP meet AB at C1. Let A2 be the point such that A1 is the midpoint of PA2, let B2 be the point such that B1 is the midpoint of PB2, and let C2 be the point such that C1 is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist G1

IMO 2022 Shortlist G1 Category: Geometry Problem Let ABCDE be a convex pentagon such that BC “ DE. Assume there is a point T inside ABCDE with TB “ TD, TC “ TE and =TBA “ =AET. Let lines CD and CT intersect line AB at points P and Q, respectively, and let lines CD and DT intersect line AE at points R and S, respectively. Assume that points P,B,A,Q...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G2

IMO 2007 Shortlist G2 Category: Geometry Problem Given an isosceles triangle ABC with AB = AC. The midpoint of side BC is denoted by M. Let X be a variable point on the shorter arc MA of the circumcircle of triangle ABM. Let T be the point in the angle domain BMA, for which ∠TMX = 90◦ and TX = BX. Prove that ∠MTB − ∠CTM does not depend on...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C1

IMO 2008 Shortlist C1 Category: Combinatorics Problem In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest n for which there exist n boxes B1,...,Bn such that Bi and Bj intersect if and only if...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A3

IMO 2007 Shortlist A3 Category: Algebra Problem Let n be a positive integer, and let x and y be positive real numbers such that xn +yn = 1. Prove that n X k=1 1 + x2k 1 + x4k ! n X k=1 1 + y2k 1 + y4k ! < (1 − x)(1 − y) . (Estonia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist A1

IMO 2023 Shortlist A1 Category: Algebra Problem Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may...

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IMO 1969

IMO 1969 Official IMO 1969 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Prove that there are infinitely many natural numbers $a$ with the following property: the number $z = n^4 + a$ is not prime for any natural number $n$. Problem 2 Let $a_1, a_2, \cdots, a_n$ be real constants,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist C6

IMO 2016 Shortlist C6 Category: Combinatorics Problem There are n ⩾ 3 islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands X...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A5

IMO 2007 Shortlist A5 Category: Algebra Problem Let c > 2, and let a(1),a(2),... be a sequence of nonnegative real numbers such that a(m + n) ≤ 2a(m) + 2a(n) for all m,n ≥ 1, (1) and a(2k ) ≤ (k + 1)c for all k ≥ 0. (2) Prove that the sequence a(n) is bounded. (Croatia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N8

IMO 2021 Shortlist N8 Category: Number Theory Problem For a polynomial Ppxq with integer coefficients let P1 pxq “ Ppxq and Pk`1 pxq “ PpPk pxqq for k ě1. Find all positive integers n for which there exists a polynomial Ppxq with integer coefficients such that for every integer m ě 1, the numbers Pm p1q,...,Pm pnq leave exactly rn{2m s distinct remainders when divided by n. Shortlisted problems –...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N4

IMO 2009 Shortlist N4 Category: Number Theory Problem Find all positive integers n such that there exists a sequence of positive integers a1,a2,...,an satisfying ak+1 = a2 k + 1 ak−1 + 1 − 1 for every k with 2 ≤ k ≤ n − 1.

Practice › Mathematics › IMO ›

IMO 2018

IMO 2018 Official IMO 2018 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$...

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IMO 2011 Shortlist G1

IMO 2011 Shortlist G1 Category: Geometry Problem Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC. Suppose that ω is tangent to AB at B′ and to AC at C′ . Suppose also that the circumcenter O of the triangle ABC lies on the shorter arc B′ C′ of ω. Prove that the circumcircle of ABC and ω meet at...

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IMO 2013 Shortlist G5

IMO 2013 Shortlist G5 Category: Geometry Problem Let ABCDEF be a convex hexagon with AB “ DE, BC “ EF, CD “ FA, and =A ´ =D “ =C ´ =F “ =E ´ =B. Prove that the diagonals AD,BE, and CF are concurrent. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C3

IMO 2021 Shortlist C3 Category: Combinatorics Problem A thimblerigger has 2021 thimbles numbered from 1 through 2021. The thimbles are arranged in a circle in arbitrary order. The thimblerigger performs a sequence of 2021 moves; in the kth move, he swaps the positions of the two thimbles adjacent to thimble k. Prove that there exists a value of k such that, in the kth move, the thimblerigger swaps some thimbles...

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IMO 2020 Shortlist C4

IMO 2020 Shortlist C4 Category: Combinatorics Problem The Fibonacci numbers F0,F1,F2,... are defined inductively by F0 “ 0, F1 “ 1, and Fn 1 “ Fn Fn´1 for n ě 1. Given an integer n ě 2, determine the smallest size of a set S of integers such that for every k “ 2,3,...,n there exist some x,y P S such that x ´ y “ Fk. (Croatia)

Practice › Mathematics › IMO ›

IMO 1967

IMO 1967 Official IMO 1967 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius $1$ with centers $A$, $B$, $C$,...

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IMO 2013 Shortlist C3

IMO 2013 Shortlist C3 Category: Combinatorics Problem A crazy physicist discovered a new kind of particle which he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a...

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IMO 2021 Shortlist N1

IMO 2021 Shortlist N1 Category: Number Theory Problem Determine all integers n ě 1 for which there exists a pair of positive integers pa,bq such that no cube of a prime divides a2 b 3 and ab 3b 8 a2 b 3 “n.

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IMO 2014 Shortlist C1

IMO 2014 Shortlist C1 Category: Combinatorics Problem Let n points be given inside a rectangle R such that no two of them lie on a line parallel to one of the sides of R. The rectangle R is to be dissected into smaller rectangles with sides parallel to the sides of R in such a way that none of these rectangles contains any of the given points in its interior....

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IMO 2009

IMO 2009 Official IMO 2009 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $n$ be a positive integer and let $a_1,\ldots,a_k (k\ge2)$ be distinct integers in the set ${1,\ldots,n}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,\ldots,k-1$. Prove that $n$ doesn't divide $a_k(a_1-1)$. Author: Ross Atkins, Australia Problem 2 Let $ABC$...

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IMO 2006 Shortlist N6

IMO 2006 Shortlist N6 Category: Number Theory Problem Let a > b > 1 be relatively prime positive integers. Define the weight of an integer c, denoted by w(c), to be the minimal possible value of |x| + |y| taken over all pairs of integers x and y such that ax + by = c. An integer c is called a local champion if w(c) ≥ w(c ± a) and...

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IMO 2023 Shortlist A6

IMO 2023 Shortlist A6 Category: Algebra Problem Let k ě 2 be an integer. Determine all sequences of positive integers a1,a2,... for which there exists a monic polynomial P of degree k with non-negative integer coefficients such that Ppanq “ an 1an 2 ¨¨¨an`k for every integer n ě 1. (Malaysia)

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IMO 2006 Shortlist G5

IMO 2006 Shortlist G5 Category: Geometry Problem In triangle ABC, let J be the centre of the excircle tangent to side BC at A1 and to the extensions of sides AC and AB at B1 and C1, respectively. Suppose that the lines A1B1 and AB are perpendicular and intersect at D. Let E be the foot of the perpendicular from C1 to line DJ. Determine the angles ∠BEA1 and ∠AEB1....

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IMO 2011 Shortlist G3

IMO 2011 Shortlist G3 Category: Geometry Problem Let ABCD be a convex quadrilateral whose sides AD and BC are not parallel. Suppose that the circles with diameters AB and CD meet at points E and F inside the quadrilateral. Let ωE be the circle through the feet of the perpendiculars from E to the lines AB, BC, and CD. Let ωF be the circle through the feet of the perpendiculars...

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IMO 2012 Shortlist A4

IMO 2012 Shortlist A4 Category: Algebra Problem Let f and g be two nonzero polynomials with integer coefficients and degf > degg. Suppose that for infinitely many primes p the polynomial pf + g has a rational root. Prove that f has a rational root. Solution 1. Since degf > degg, we have |g(x)/f(x)| < 1 for sufficiently large x; more precisely, there is a real number R such that...

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IMO 2010 Shortlist C5

IMO 2010 Shortlist C5 Category: Combinatorics Problem n ©4 players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players bad if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in...

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IMO 2013 Shortlist C1

IMO 2013 Shortlist C1 Category: Combinatorics Problem Let n be a positive integer. Find the smallest integer k with the following property: Given any real numbers a1,...,ad such that a1 a2 ¨¨¨ ` ad “ n and 0 ď ai ď 1 for i “ 1,2,...,d, it is possible to partition these numbers into k groups (some of which may be empty) such that the sum of the numbers in...

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IMO 2006 Shortlist G6

IMO 2006 Shortlist G6 Category: Geometry Problem Circles ω1 and ω2 with centres O1 and O2 are externally tangent at point D and internally tangent to a circle ω at points E and F, respectively. Line t is the common tangent of ω1 and ω2 at D. Let AB be the diameter of ω perpendicular to t, so that A, E and O1 are on the same side of t....

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IMO 2014 Shortlist N1

IMO 2014 Shortlist N1 Category: Number Theory Problem Let n ě 2 be an integer, and let An be the set An “ t2n ´ 2k |k P Z, 0 ď k ă nu. Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of An. (Serbia)

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IMO 1994

IMO 1994 Official IMO 1994 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $m$ and $n$ be two positive integers. Let $a_1$, $a_2$, $\ldots$, $a_m$ be $m$ different numbers from the set ${1, 2,\ldots, n}$ such that for any two indices $i$ and $j$ with $1\leq i \leq j \leq...

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IMO 2009 Shortlist

IMO 2009 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory...

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IMO 2011 Shortlist A6

IMO 2011 Shortlist A6 Category: Algebra Problem Let f be a function from the set of real numbers to itself that satisfies f(x + y) ≤ yf(x) + f(f(x)) for all real numbers x and y. Prove that f(x) = 0 for all x ≤ 0.

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IMO 2018 Shortlist C5

IMO 2018 Shortlist C5 Category: Combinatorics Problem Let k be a positive integer. The organising ommittee of a tennis tournament is to s hedule the mat hes for 2k players so that every two players play on e, ea h day exa tly one mat h is played, and ea h player arrives to the tournament site the day of his rst mat h, and departs the day of his...

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IMO 2022 Shortlist G5

IMO 2022 Shortlist G5 Category: Geometry Problem Let ABC be a triangle, and let 1 and 2 be two parallel lines. For i “ 1,2, let `i meet the lines BC, CA, and AB at Xi, Yi, and Zi, respectively. Suppose that the line through Xi perpendicular to BC, the line through Yi perpendicular to CA, and finally the line through Zi perpendicular to AB, determine a non-degenerate triangle ∆i....

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IMO 2011 Shortlist C5

IMO 2011 Shortlist C5 Category: Combinatorics Problem Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn 90◦ clockwise and continue moving...

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IMO 2013 Shortlist N2

IMO 2013 Shortlist N2 Category: Number Theory Problem Prove that for any pair of positive integers k and n there exist k positive integers m1,m2,...,mk such that 1 2k ´ 1 n “ ˆ 1 m1 ˙ˆ 1 m2 ˙ ¨¨¨ ˆ 1 mk ˙ . (Japan)

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IMO 2020 Shortlist C3

IMO 2020 Shortlist C3 Category: Combinatorics Problem Let n be an integer with n ě 2. On a slope of a mountain, n2 checkpoints are marked, numbered from 1 to n2 from the bottom to the top. Each of two cable car companies, A and B, operates k cable cars numbered from 1 to k; each cable car provides a transfer from some checkpoint to a higher one. For each...

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IMO 2024 Shortlist C4

IMO 2024 Shortlist C4 Category: Combinatorics Problem On a board with 2024 rows and 2023 columns, Turbo the snail tries to move from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then moves one step at a time to an adjacent cell sharing a common side. He wins if he reaches any cell in the last row....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N1

IMO 2016 Shortlist N1 Category: Number Theory Problem For any positive integer k, denote the sum of digits of k in its decimal representation by S(k). Find all polynomials P(x) with integer coefficients such that for any positive integer n ⩾ 2016, the integer P(n) is positive and S(P(n)) = P(S(n)).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist C3

IMO 2009 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. Given a sequence ε1,...,εn−1 with εi = 0 or εi = 1 for each i = 1,...,n − 1, the sequences a0,...,an and b0,...,bn are constructed by the following rules: a0 = b0 = 1, a1 = b1 = 7, ai+1 = ( 2ai−1 + 3ai, if εi = 0, 3ai−1 + ai, if εi = 1,...

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IMO 2019 Shortlist N2

IMO 2019 Shortlist N2 Category: Number Theory Problem Find all triples pa,b,cq of positive integers such that a3 b3 c3 “ pabcq2 . (Nigeria)

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IMO 2016 Shortlist N7

IMO 2016 Shortlist N7 Category: Number Theory Problem Let n be an odd positive integer. In the Cartesian plane, a cyclic polygon P with area S is chosen. All its vertices have integral coordinates, and all squares of its side lengths are divisible by n. Prove that 2S is an integer divisible by n.

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IMO 2006 Shortlist N1

IMO 2006 Shortlist N1 Category: Number Theory Problem Determine all pairs (x,y) of integers satisfying the equation 1 + 2x 22x+1 = y2 . (USA)

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IMO 2020 Shortlist A4

IMO 2020 Shortlist A4 Category: Algebra Problem Let a,b,c,d be four real numbers such that a ě b ě c ě d ą 0 and a b c d “ 1. Prove that pa 2b 3c 4dqaa bb cc dd ă 1. (Belgium)

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IMO 2011 Shortlist G5

IMO 2011 Shortlist G5 Category: Geometry Problem Let ABC be a triangle with incenter I and circumcircle ω. Let D and E be the second intersection points of ω with the lines AI and BI, respectively. The chord DE meets AC at a point F, and BC at a point G. Let P be the intersection point of the line through F parallel to AD and the line through G...

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IMO 2014 Shortlist A1

IMO 2014 Shortlist A1 Category: Algebra Problem Let z0 ă z1 ă z2 ă ¨¨¨ be an infinite sequence of positive integers. Prove that there exists a unique integer n ě 1 such that zn ă z0 z1 ¨¨¨ zn n ď zn 1. (Austria)

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IMO 2012 Shortlist

IMO 2012 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number...

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IMO 2014 Shortlist C8

IMO 2014 Shortlist C8 Category: Combinatorics Problem A card deck consists of 1024 cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of...

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IMO 2014 Shortlist

IMO 2014 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics C9 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number...

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IMO 2018 Shortlist G6

IMO 2018 Shortlist G6 Category: Geometry Problem A onvex quadrilateral ABCD satises AB ¨ CD “ BC ¨ DA. A point X is hosen inside the quadrilateral so that =XAB “ =XCD and =XBC “ =XDA. Prove that =AXB` =CXD “ 180˝ . (Poland)

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IMO 2020 Shortlist N6

IMO 2020 Shortlist N6 Category: Number Theory Problem For a positive integer n, let dpnq be the number of positive divisors of n, and let ϕpnq be the number of positive integers not exceeding n which are coprime to n. Does there exist a constant C such that ϕpdpnqq dpϕpnqq ď C for all n ě 1? (Cyprus)

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IMO 2006 Shortlist G8

IMO 2006 Shortlist G8 Category: Geometry Problem Let ABCD be a convex quadrilateral. A circle passing through the points A and D and a circle passing through the points B and C are externally tangent at a point P inside the quadrilateral. Suppose that ∠PAB + ∠PDC ≤ 90◦ and ∠PBA + ∠PCD ≤ 90◦ . Prove that AB + CD ≥ BC + AD. (Poland)

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IMO 2015 Shortlist N3

IMO 2015 Shortlist N3 Category: Number Theory Problem Let m and n be positive integers such that m ą n. Define xk “ pm kq{pn kq for k “ 1,2,...,n 1. Prove that if all the numbers x1,x2,...,xn 1 are integers, then x1x2 ¨¨¨xn`1 ´1 is divisible by an odd prime. (Austria)

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IMO 2016 Shortlist A7

IMO 2016 Shortlist A7 Category: Algebra Problem Denote by R the set of all real numbers. Find all functions f : R → R such that f(0) 6= 0 and f(x + y)2 = 2f(x)f(y) + max{f(x2 ) + f(y2 ),f(x2 y2 )} for all real numbers x and y.

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IMO 2017 Shortlist N7

IMO 2017 Shortlist N7 Category: Number Theory Problem Say that an ordered pair px,yq of integers is an irredu ible latti e point if x and y are relatively prime. For any nite set S of irredu ible latti e points, show that there is a homogenous polynomial in two variables, fpx,yq, with integer oe ients, of degree at least 1, su h that fpx,yq “ 1 for ea h...

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IMO 2016 Shortlist G7

IMO 2016 Shortlist G7 Category: Geometry Problem Let I be the incentre of a non-equilateral triangle ABC, IA be the A-excentre, I0 A be the reflection of IA in BC, and lA be the reflection of line AI0 A in AI. Define points IB,I0 B and line lB analogously. Let P be the intersection point of lA and lB. (a) Prove that P lies on line OI where O is...

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IMO 2019 Shortlist C5

IMO 2019 Shortlist C5 Category: Combinatorics Problem On a certain social network, there are 2019 users, some pairs of which are friends, where friendship is a symmetric relation. Initially, there are 1010 people with 1009 friends each and 1009 people with 1010 friends each. However, the friendships are rather unstable, so events of the following kind may happen repeatedly, one at a time: Let A, B, and C be people...

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IMO 2006 Shortlist N7

IMO 2006 Shortlist N7 Category: Number Theory Problem Prove that, for every positive integer n, there exists an integer m such that 2m m is divisible by n. (Estonia)

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IMO 2021 Shortlist N3

IMO 2021 Shortlist N3 Category: Number Theory Problem Find all positive integers n with the following property: the k positive divisors of n have a permutation pd1,d2,...,dkq such that for every i “1,2,...,k, the number d1 ¨¨¨ di is a perfect square.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A7

IMO 2007 Shortlist A7 Category: Algebra Problem Let n > 1 be an integer. In the space, consider the set S = (x,y,z) | x,y,z ∈ {0,1,...,n}, x + y + z > 0 . Find the smallest number of planes that jointly contain all (n+1)3 −1 points of S but none of them passes through the origin. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C7

IMO 2010 Shortlist C7 Category: Combinatorics Problem Let P1,...,Ps be arithmetic progressions of integers, the following conditions being satisfied: (i) each integer belongs to at least one of them; (ii) each progression contains a number which does not belong to other progressions. Denote by n the least common multiple of steps of these progressions; let n pα1 1 ...pαk k be its prime factorization. Prove that s ©1 k ¸...

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IMO 2015 Shortlist C5

IMO 2015 Shortlist C5 Category: Combinatorics Problem Consider an infinite sequence a1,a2,... of positive integers with ai ď 2015 for all i ě 1. Suppose that for any two distinct indices i and j we have i ai ‰ j aj. Prove that there exist two positive integers b and N such that ˇ ˇ ˇ ˇ ˇ n ÿ i“m`1 pai ´ bq ˇ ˇ ˇ ˇ ˇ ď...

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IMO 2021 Shortlist G5

IMO 2021 Shortlist G5 Category: Geometry Problem Let ABCD be a cyclic quadrilateral whose sides have pairwise different lengths. Let O be the circumcentre of ABCD. The internal anglebisectors of =ABC and =ADC meet AC at B1 and D1, respectively. Let OB be the centre of the circle which passes through B and is tangent to AC at D1. Similarly,let OD be the centre of the circle which passes through...

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IMO 2011 Shortlist G6

IMO 2011 Shortlist G6 Category: Geometry Problem Let ABC be a triangle with AB = AC, and let D be the midpoint of AC. The angle bisector of ∠BAC intersects the circle through D, B, and C in a point E inside the triangle ABC. The line BD intersects the circle through A, E, and B in two points B and F. The lines AF and BE meet at a...

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IMO 2007 Shortlist C5

IMO 2007 Shortlist C5 Category: Combinatorics Problem In the Cartesian coordinate plane define the strip Sn = {(x,y) | n ≤ x < n + 1} for every integer n. Assume that each strip Sn is colored either red or blue, and let a and b be two distinct positive integers. Prove that there exists a rectangle with side lengths a and b such that its vertices have the same...

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IMO 2012 Shortlist N1

IMO 2012 Shortlist N1 Category: Number Theory Problem Call admissible a set A of integers that has the following property: If x,y ∈ A (possibly x = y) then x2 kxy + y2 ∈ A for every integer k. Determine all pairs m,n of nonzero integers such that the only admissible set containing both m and n is the set of all integers. Solution. A pair of integers m,n fulfills...

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IMO 2019 Shortlist A3

IMO 2019 Shortlist A3 Category: Algebra Problem Let n ě 3 be a positive integer and let pa1,a2,...,anq be a strictly increasing sequence of n positive real numbers with sum equal to 2. Let X be a subset of t1,2,...,nu such that the value of ˇ ˇ ˇ ˇ ˇ 1 ´ ÿ iPX ai ˇ ˇ ˇ ˇ ˇ is minimised. Prove that there exists a strictly increasing sequence...

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IMO 2024 Shortlist C6

IMO 2024 Shortlist C6 Category: Combinatorics Problem Let n and T be positive integers. James has 4n marbles with weights 1, 2, ..., 4n. He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most T. Find, in terms...

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IMO 2019 Shortlist N8

IMO 2019 Shortlist N8 Category: Number Theory Problem Let a and b be two positive integers. Prove that the integer a2 ` R 4a2 b V is not a square. (Here rzs denotes the least integer greater than or equal to z.) (Russia) 10 Bath — UK, 11th–22nd July 2019

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IMO 2006 Shortlist C2

IMO 2006 Shortlist C2 Category: Combinatorics Problem A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles triangles with two odd sides. (Serbia)

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IMO 2019 Shortlist A5

IMO 2019 Shortlist A5 Category: Algebra Problem Let x1, x2, ..., xn be diﬀerent real numbers. Prove that ÿ 1ďiďn ź j‰i 1 ´ xixj xi ´ xj “ 0, if n is even; 1, if n is odd. (Kazakhstan) Shortlisted problems 5

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IMO 1997

IMO 1997 Official IMO 1997 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively black and white (as on a chessboard). For any pair of positive integers $m$ and $n$, consider a right-angled...

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IMO 2017 Shortlist A8

IMO 2017 Shortlist A8 Category: Algebra Problem Assume that a fun tion f : R Ñ R satises the following ondition: For every x,y P R su h that fpxq y ˘ fpyq x ˘ ą 0, we have fpxq y “ fpyq x. Prove that fpxq y ď fpyq x whenever x ą y. (Netherlands) 6 IMO 2017, Rio de Janeiro Combinatori s

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IMO 2016 Shortlist N5

IMO 2016 Shortlist N5 Category: Number Theory Problem Let a be a positive integer which is not a square number. Denote by A the set of all positive integers k such that k = x2 − a x2 − y2 (1) for some integers x and y with x > √ a. Denote by B the set of all positive integers k such that (1) is satisfied for some integers...

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IMO 2021 Shortlist A5

IMO 2021 Shortlist A5 Category: Algebra Problem Let n ě 2 be an integer, and let a1,a2,...,an be positive real numbers such that a1 a2 ¨¨¨ an “1. Prove that n ÿ k“1 ak 1 ´ak pa1 a2 ¨¨¨ ak´1q2 ă 1 .

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IMO 2024 Shortlist C8

IMO 2024 Shortlist C8 Category: Combinatorics Problem Let n be a positive integer. Given an nˆn board, the unit cell in the top left corner is initially coloured black, and the other cells are coloured white. We then apply a series of colouring operations to the board. In each operation, we choose a 2 ˆ 2 square with exactly one cell coloured black and we colour the remaining three cells...

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IMO 2010

IMO 2010 Official IMO 2010 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds $$ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor $$ where $\left\lfloor a\right\rfloor$ is greatest integer not greater than $a.$ Author: Pierre Bornsztein, France Problem 2 Given a triangle...

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IMO 2006 Shortlist N5

IMO 2006 Shortlist N5 Category: Number Theory Problem Find all integer solutions of the equation x7 − 1 x − 1 = y5 − 1. (Russia)

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IMO 2008 Shortlist G1

IMO 2008 Shortlist G1 Category: Geometry Problem In an acute-angled triangle ABC, point H is the orthocentre and A0, B0, C0 are the midpoints of the sides BC, CA, AB, respectively. Consider three circles passing through H: ωa around A0, ωb around B0 and ωc around C0. The circle ωa intersects the line BC at A1 and A2; ωb intersects CA at B1 and B2; ωc intersects AB at C1...

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IMO 2010 Shortlist N4

IMO 2010 Shortlist N4 Category: Number Theory Problem Let a,b be integers, and let Ppxqax3 bx. For any positive integer n we say that the pair pa,bq is n-good if n PpmqPpkq implies n m k for all integers m,k. We say that pa,bq is very good if pa,bq is n-good for infinitely many positive integers n. (a) Find a pair pa,bq which is 51-good, but...

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IMO 2006 Shortlist A1

IMO 2006 Shortlist A1 Category: Algebra Problem A sequence of real numbers a0,a1,a2,... is defined by the formula ai+1 = baic · haii for i ≥ 0; here a0 is an arbitrary real number, baic denotes the greatest integer not exceeding ai, and haii = ai − baic. Prove that ai = ai+2 for i sufficiently large. (Estonia)

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IMO 2008 Shortlist A4

IMO 2008 Shortlist A4 Category: Algebra Problem For an integer m, denote by t(m) the unique number in {1,2,3} such that m+ t(m) is a multiple of 3. A function f : Z → Z satisfies f(−1) = 0, f(0) = 1, f(1) = −1 and f(2n m) = f(2n − t(m)) − f(m) for all integers m,n ≥ 0 with 2n m. Prove that f(3p) ≥ 0 holds for...

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IMO 2009 Shortlist C7

IMO 2009 Shortlist C7 Category: Combinatorics Problem Variant 1. A grasshopper jumps along the real axis. He starts at point 0 and makes 2009 jumps to the right with lengths 1,2,...,2009 in an arbitrary order. Let M be a set of 2008 positive integers less than 1005 · 2009. Prove that the grasshopper can arrange his jumps in such a way that he never lands on a point from M....

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IMO 2007 Shortlist G7

IMO 2007 Shortlist G7 Category: Geometry Problem Given an acute triangle ABC with angles α, β and γ at vertices A, B and C, respectively, such that β > γ. Point I is the incenter, and R is the circumradius. Point D is the foot of the altitude from vertex A. Point K lies on line AD such that AK = 2R, and D separates A and K. Finally, lines...

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IMO 1983

IMO 1983 Official IMO 1983 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$. Problem 2 Let $A$ be one of the two distinct points of...

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IMO 2023 Shortlist C6

IMO 2023 Shortlist C6 Category: Combinatorics Problem Let N be a positive integer, and consider an N ˆ N grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the...

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IMO 2017 Shortlist C7

IMO 2017 Shortlist C7 Category: Combinatorics Problem For any nite sets X and Y of positive integers, denote by fXpkq the kth smallest positive integer not in X, and let X ˚ Y “ X Y tfXpyq: y P Y u. Let A be a set of a ą 0 positive integers, and let B be a set of b ą 0 positive integers. Prove that if A ˚ B...

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IMO 2015

IMO 2015 Official IMO 2015 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 We say that a finite set $\mathcal{S}$ in the plane is balanced if, for any two different points $A$, $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free...

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IMO 2008 Shortlist C2

IMO 2008 Shortlist C2 Category: Combinatorics Problem For every positive integer n determine the number of permutations (a1,a2,...,an) of the set {1,2,...,n} with the following property: 2(a1 + a2 + ··· + ak) is divisible by k for k = 1,2,...,n.

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IMO 2016 Shortlist G6

IMO 2016 Shortlist G6 Category: Geometry Problem Let ABCD be a convex quadrilateral with ∠ABC = ∠ADC < 90◦ . The internal angle bisectors of ∠ABC and ∠ADC meet AC at E and F respectively, and meet each other at point P. Let M be the midpoint of AC and let ω be the circumcircle of triangle BPD. Segments BM and DM intersect ω again at X and Y respectively....

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IMO 2011 Shortlist N8

IMO 2011 Shortlist N8 Category: Number Theory Problem Let k be a positive integer and set n = 2k Prove that n is a prime number if and only if the following holds: there is a permutation a1,...,an−1 of the numbers 1,2,...,n − 1 and a sequence of integers g1,g2,...,gn−1 such that n divides gai i −ai+1 for every i ∈ {1,2,...,n−1}, where we set an = a1.

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IMO 2011 Shortlist G8

IMO 2011 Shortlist G8 Category: Geometry Problem Let ABC be an acute triangle with circumcircle ω. Let t be a tangent line to ω. Let ta, tb, and tc be the lines obtained by reflecting t in the lines BC, CA, and AB, respectively. Show that the circumcircle of the triangle determined by the lines ta, tb, and tc is tangent to the circle ω.

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IMO 1987

IMO 1987 Official IMO 1987 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $p_n (k)$ be the number of permutations of the set ${ 1, \ldots , n } , ; n \ge 1$, which have exactly $k$ fixed points. Prove that $\sum_{k=0}^{n} k \cdot p_n (k) = n!$. (Remark:...

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IMO 2019 Shortlist N4

IMO 2019 Shortlist N4 Category: Number Theory Problem Let Zą0 be the set of positive integers. A positive integer constant C is given. Find all functions f : Zą0 Ñ Zą0 such that, for all positive integers a and b satisfying a b ą C, a fpbq | a2 ` bfpaq. (Croatia)

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IMO 2016 Shortlist G8

IMO 2016 Shortlist G8 Category: Geometry Problem Let A1,B1 and C1 be points on sides BC,CA and AB of an acute triangle ABC respectively, such that AA1,BB1 and CC1 are the internal angle bisectors of triangle ABC. Let I be the incentre of triangle ABC, and H be the orthocentre of triangle A1B1C1. Show that AH + BH + CH ⩾ AI + BI + CI. Shortlisted problems 9

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IMO 2017 Shortlist A4

IMO 2017 Shortlist A4 Category: Algebra Problem A sequen e of real numbers a1,a2,... satises the relation an “ ´ max i j“n pai ajq for all n ą 2017. Prove that this sequen e is bounded, i.e., there is a onstant M su h that |an| ď M for all positive integers n. (Russia) Shortlisted problems 5

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IMO 1978

IMO 1978 Official IMO 1978 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $m$ and $n$ be positive integers such that $1 \le m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$...

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IMO 2023 Shortlist N3

IMO 2023 Shortlist N3 Category: Number Theory Problem For positive integers n and k ě 2 define Ekpnq as the greatest exponent r such that kr divides n!. Prove that there are infinitely many n such that E10pnq ą E9pnq and infinitely many m such that E10pmq ă E9pmq. (Brazil)

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IMO 2024 Shortlist A5

IMO 2024 Shortlist A5 Category: Algebra Problem Find all periodic sequences a1, a2, ... of real numbers such that the following conditions hold for all n ě 1: an 2 a2 n “ an a2 n 1 and |an`1 ´ an| ď 1. (Kosovo)

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IMO 2023 Shortlist A7

IMO 2023 Shortlist A7 Category: Algebra Problem Let N be a positive integer. Prove that there exist three permutations a1,a2,...,aN; b1,b2,...,bN; and c1,c2,...,cN of 1,2,...,N such that ? ak a bk ? ck ´ 2 ? N ă 2023 for every k “ 1,2,...,N. (China) Shortlisted problems 5

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IMO 2017 Shortlist C5

IMO 2017 Shortlist C5 Category: Combinatorics Problem A hunter and an invisible rabbit play a game in the Eu lidean plane. The hunter's starting point H0 oin ides with the rabbit's starting point R0. In the nth round of the game (n ě 1), the following happens. (1) First the invisible rabbit moves se retly and unobserved from its urrent point Rn´1 to some new point Rn with Rn´1Rn “...

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IMO 2015 Shortlist G7

IMO 2015 Shortlist G7 Category: Geometry Problem Let ABCD be a convex quadrilateral, and let P, Q, R, and S be points on the sides AB, BC, CD, and DA, respectively. Let the line segments PR and QS meet at O. Suppose that each of the quadrilaterals APOS, BQOP, CROQ, and DSOR has an incircle. Prove that the lines AC, PQ, and RS are either concurrent or parallel to each...

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IMO 2010 Shortlist C4

IMO 2010 Shortlist C4 Category: Combinatorics Problem Six stacks S1,...,S6 of coins are standing in a row. In the beginning every stack contains a single coin. There are two types of allowed moves: Move 1: If stack Sk with 1 ¨k ¨5 contains at least one coin, you may remove one coin from Sk and add two coins to Sk 1. Move 2: If stack Sk with 1 ¨ k...

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IMO 2011 Shortlist A5

IMO 2011 Shortlist A5 Category: Algebra Problem Prove that for every positive integer n, the set {2,3,4,...,3n + 1} can be partitioned into n triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.

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IMO 2009 Shortlist N3

IMO 2009 Shortlist N3 Category: Number Theory Problem Let f be a non-constant function from the set of positive integers into the set of positive integers, such that a − b divides f(a) − f(b) for all distinct positive integers a,b. Prove that there exist infinitely many primes p such that p divides f(c) for some positive integer c.

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IMO 2018 Shortlist G2

IMO 2018 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with AB “ AC, and let M be the midpoint of BC. Let P be a point su h that PB ă PC and PA is parallel to BC. Let X and Y be points on the lines PB and PC, respe tively, so that B lies on the segment PX, C lies on the segment PY ,...

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IMO 1959

IMO 1959 Official IMO 1959 problems · 6/6 solved, 4 verified. # Status Time 1 ✓ verified 8m04s 2 solved 15m36s 3 ✓ verified 12m31s 4 ✓ verified 8m05s 5 ✓ verified 18m27s 6 solved 6m48s Problem 1 ✓ verified · 8m04s · Solution → Prove that $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$. Problem 2 solved · 15m36s · Solution → For what real values of $x$ is...

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IMO 2017 Shortlist G5

IMO 2017 Shortlist G5 Category: Geometry Problem Let ABCC1B1A1 be a onvex hexagon su h that AB “ BC, and suppose that the line segments AA1, BB1, and CC1 have the same perpendi ular bise tor. Let the diagonals AC1 and A1C meet at D, and denote by ω the ir le ABC. Let ω interse t the ir le A1BC1 again at E ‰ B. Prove that the lines...

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IMO 2020

IMO 2020 Official IMO 2020 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $$ \angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP :...

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IMO 2019 Shortlist G1

IMO 2019 Shortlist G1 Category: Geometry Problem Let ABC be a triangle. Circle Γ passes through A, meets segments AB and AC again at points D and E respectively, and intersects segment BC at F and G such that F lies between B and G. The tangent to circle BDF at F and the tangent to circle CEG at G meet at point T. Suppose that points A and T...

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IMO 2015 Shortlist G4

IMO 2015 Shortlist G4 Category: Geometry Problem Let ABC be an acute triangle, and let M be the midpoint of AC. A circle ω passing through B and M meets the sides AB and BC again at P and Q, respectively. Let T be the point such that the quadrilateral BPTQ is a parallelogram. Suppose that T lies on the circumcircle of the triangle ABC. Determine all possible values of...

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IMO 2015 Shortlist A3

IMO 2015 Shortlist A3 Category: Algebra Problem Let n be a fixed positive integer. Find the maximum possible value of ÿ 1ďrăsď2n ps ´ r ´ nqxrxs , where ´1 ď xi ď 1 for all i “ 1,2,...,2n. (Austria)

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IMO 2013 Shortlist A1

IMO 2013 Shortlist A1 Category: Algebra Problem Let n be a positive integer and let a1,...,an´1 be arbitrary real numbers. Define the sequences u0,...,un and v0,...,vn inductively by u0 “ u1 “ v0 “ v1 “ 1, and uk 1 “ uk akuk´1, vk 1 “ vk an´kvk´1 for k “ 1,...,n ´ 1. Prove that un “ vn. (France)

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IMO 2018 Shortlist G1

IMO 2018 Shortlist G1 Category: Geometry Problem Let ABC be an a ute-angled triangle with ir um ir le Γ. Let D and E be points on the segments AB and AC, respe tively, su h that AD “ AE. The perpendi ular bise tors of the segments BD and CE interse t the small ar s Ŋ AB and Ŋ AC at points F and G respe tively. Prove...

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IMO 2024 Shortlist N5

IMO 2024 Shortlist N5 Category: Number Theory Problem Let S be a finite nonempty set of prime numbers. Let 1 “ b1 ă b2 ă ¨¨¨ be the sequence of all positive integers whose prime divisors all belong to S. Prove that, for all but finitely many positive integers n, there exist positive integers a1, a2, ..., an such that a1 b1 a2 b2 ¨¨¨ an bn “ R b1...

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IMO 2011 Shortlist A7

IMO 2011 Shortlist A7 Category: Algebra Problem Let a, b, and c be positive real numbers satisfying min(a+b,b+c,c+a) > √ 2 and a2 +b2 +c2 = 3. Prove that a (b + c − a)2 + b (c + a − b)2 + c (a + b − c)2 ≥ (abc)2 .

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IMO 2021 Shortlist A3

IMO 2021 Shortlist A3 Category: Algebra Problem Given a positive integer n, find the smallest value of Ya1 ] Ya2 ] ¨¨¨` Yan n ] over all permutations pa1,a2,...,anq of p1,2,...,nq.

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IMO 2008 Shortlist G6

IMO 2008 Shortlist G6 Category: Geometry Problem There is given a convex quadrilateral ABCD. Prove that there exists a point P inside the quadrilateral such that ∠PAB + ∠PDC = ∠PBC + ∠PAD = ∠PCD + ∠PBA = ∠PDA + ∠PCB = 90◦ (1) if and only if the diagonals AC and BD are perpendicular.

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IMO 2017 Shortlist A3

IMO 2017 Shortlist A3 Category: Algebra Problem Let S be a nite set, and let A be the set of all fun tions from S to S. Let f be an element of A, and let T “ fpSq be the image of S under f. Suppose that f ˝ g ˝ f ‰ g ˝ f ˝ g for every g in A with g ‰ f. Show that...

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IMO 2020 Shortlist C2

IMO 2020 Shortlist C2 Category: Combinatorics Problem In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals Q1, ..., Q24 whose corners are vertices of the 100-gon, so that • the quadrilaterals Q1, ..., Q24 are pairwise disjoint, and • every quadrilateral Qi has three corners of one color and one corner of the other color....

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IMO 2010 Shortlist N1

IMO 2010 Shortlist N1 Category: Number Theory Problem Find the least positive integer n for which there exists a set ts1,s2,...,snu consisting of n distinct positive integers such that 1 1 s1 1 1 s2 ... 1 1 sn 51 .

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IMO 2006 Shortlist G2

IMO 2006 Shortlist G2 Category: Geometry Problem Let ABCD be a trapezoid with parallel sides AB > CD. Points K and L lie on the line segments AB and CD, respectively, so that AK/KB = DL/LC. Suppose that there are points P and Q on the line segment KL satisfying ∠APB = ∠BCD and ∠CQD = ∠ABC. Prove that the points P, Q, B and C are concyclic. (Ukraine)

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IMO 2018 Shortlist N6

IMO 2018 Shortlist N6 Category: Number Theory Problem Let f : t1,2,3,...u Ñ t2,3,...u be a fun tion su h that fpm nq | fpmq fpnq for all pairs m,n of positive integers. Prove that there exists a positive integer c ą 1 whi h divides all values of f. (Mexi o)

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IMO 2011 Shortlist N2

IMO 2011 Shortlist N2 Category: Number Theory Problem Consider a polynomial P(x) = (x + d1)(x + d2) · ... · (x + d9), where d1, d2,..., d9 are nine distinct integers. Prove that there exists an integer N such that for all integers x ≥ N the number P(x) is divisible by a prime number greater than 20.

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IMO 2016 Shortlist G2

IMO 2016 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with circumcircle Γ and incentre I. Let M be the midpoint of side BC. Denote by D the foot of perpendicular from I to side BC. The line through I per- pendicular to AI meets sides AB and AC at F and E respectively. Suppose the circumcircle of triangle AEF intersects Γ at a point X other than...

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IMO 2010 Shortlist G2

IMO 2010 Shortlist G2 Category: Geometry Problem Point P lies inside triangle ABC. Lines AP, BP, CP meet the circumcircle of ABC again at points K, L, M, respectively. The tangent to the circumcircle at C meets line AB at S. Prove that SC SP if and only if MK ML. (Poland)

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IMO 2014 Shortlist G5

IMO 2014 Shortlist G5 Category: Geometry Problem Let ABCD be a convex quadrilateral with =B “ =D “ 90˝ . Point H is the foot of the perpendicular from A to BD. The points S and T are chosen on the sides AB and AD, respectively, in such a way that H lies inside triangle SCT and =SHC ´ =BSC “ 90˝ , =THC ´ =DTC “ 90˝ . Prove...

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IMO 1965

IMO 1965 Official IMO 1965 problems · 1/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 solved 14m52s 6 — — Problem 1 Determine all values $x$ in the interval $0\leq x\leq 2\pi$ which satisfy the inequality $$ 2\cos x \leq \left| \sqrt{1+\sin 2x} - \sqrt{1-\sin 2x } \right| \leq \sqrt{2}. $$ Problem 2 Consider the system of equations $$ a_{11}x_1...

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IMO 2023 Shortlist N1

IMO 2023 Shortlist N1 Category: Number Theory Problem Determine all positive, composite integers n that satisfy the following property: if the positive divisors of n are 1 “ d1 ă d2 ă ¨¨¨ ă dk “ n, then di divides di 1 di`2 for every 1 ď i ď k ´ 2. (Colombia)

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IMO 2014 Shortlist G3

IMO 2014 Shortlist G3 Category: Geometry Problem Let Ω and O be the circumcircle and the circumcentre of an acute-angled triangle ABC with AB ą BC. The angle bisector of =ABC intersects Ω at M ‰ B. Let Γ be the circle with diameter BM. The angle bisectors of =AOB and =BOC intersect Γ at points P and Q, respectively. The point R is chosen on the line PQ so...

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IMO 2014 Shortlist N6

IMO 2014 Shortlist N6 Category: Number Theory Problem Let a1 ă a2 ă ¨¨¨ ă an be pairwise coprime positive integers with a1 being prime and a1 ě n ` 2. On the segment I “ r0,a1a2 ¨¨¨ans of the real line, mark all integers that are divisible by at least one of the numbers a1,...,an. These points split I into a number of smaller segments. Prove that the sum...

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IMO 2010 Shortlist G5

IMO 2010 Shortlist G5 Category: Geometry Problem Let ABCDE be a convex pentagon such that BC k AE, AB BC AE, and =ABC =CDE. Let M be the midpoint of CE, and let O be the circumcenter of triangle BCD. Given that =DMO 90, prove that 2=BDA =CDE. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A2

IMO 2007 Shortlist A2 Category: Algebra Problem Consider those functions f : N → N which satisfy the condition f(m + n) ≥ f(m) + f f(n) − 1 (1) for all m,n ∈ N. Find all possible values of f(2007). (N denotes the set of all positive integers.) (Bulgaria)

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IMO 2024 Shortlist A8

IMO 2024 Shortlist A8 Category: Algebra Problem Let p ‰ q be coprime positive integers. Determine all infinite sequences a1, a2, ... of positive integers such that the following conditions hold for all n ě 1: maxpan,an 1,...,an pq ´ minpan,an 1,...,an pq “ p and maxpan,an 1,...,an qq ´ minpan,an 1,...,an qq “ q. (Japan) Shortlisted problems 5

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IMO 2013 Shortlist G6

IMO 2013 Shortlist G6 Category: Geometry Problem Let the excircle of the triangle ABC lying opposite to A touch its side BC at the point A1. Define the points B1 and C1 analogously. Suppose that the circumcentre of the triangle A1B1C1 lies on the circumcircle of the triangle ABC. Prove that the triangle ABC is right-angled. (Russia) Shortlisted problems 7

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IMO 2024 Shortlist C1

IMO 2024 Shortlist C1 Category: Combinatorics Problem Let n be a positive integer. A class of n students run n races, in each of which they are ranked with no draws. A student is eligible for a rating pa,bq for positive integers a and b if they come in the top b places in at least a of the races. Their final score is the maximum possible value of a...

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IMO 1970

IMO 1970 Official IMO 1970 problems · 1/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 solved 19m45s Problem 1 Let $M$ be a point on the side $AB$ of $\triangle ABC$. Let $r_1, r_2$, and $r$ be the inscribed circles of triangles $AMC, BMC$, and $ABC$. Let $q_1, q_2$, and $q$ be the radii of the exscribed...

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IMO 2015 Shortlist A1

IMO 2015 Shortlist A1 Category: Algebra Problem Suppose that a sequence a1,a2,... of positive real numbers satisfies ak 1 ě kak a2 k pk ´ 1q for every positive integer k. Prove that a1 a2 ¨¨¨ ` an ě n for every n ě 2. (Serbia)

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IMO 2014 Shortlist G4

IMO 2014 Shortlist G4 Category: Geometry Problem Consider a fixed circle Γ with three fixed points A, B, and C on it. Also, let us fix a real number λ P p0,1q. For a variable point P R tA,B,Cu on Γ, let M be the point on the segment CP such that CM “ λ ¨ CP. Let Q be the second point of intersection of the circumcircles of the...

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IMO 2022 Shortlist A6

IMO 2022 Shortlist A6 Category: Algebra Problem Let R be the set of real numbers. We denote by F the set of all functions f : R Ñ R such that fpx fpyqq “ fpxq fpyq for every x,y P R. Find all rational numbers q such that for every function f P F, there exists some z P R satisfying fpzq “ qz. (Indonesia)

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IMO 2014 Shortlist N3

IMO 2014 Shortlist N3 Category: Number Theory Problem A coin is called a Cape Town coin if its value is 1{n for some positive integer n. Given a collection of Cape Town coins of total value at most 99 ` 1 , prove that it is possible to split this collection into at most 100 groups each of total value at most 1. (Luxembourg)

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IMO 1996

IMO 1996 Official IMO 1996 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one...

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IMO 2013 Shortlist C6

IMO 2013 Shortlist C6 Category: Combinatorics Problem In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible number of flights required to go from one of them to the other. It is known that for any city there are...

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IMO 2023 Shortlist A4

IMO 2023 Shortlist A4 Category: Algebra Problem Let Rą0 be the set of positive real numbers. Determine all functions f : Rą0 Ñ Rą0 such that x fpxq fpyq ˘ ě fpfpxqq y ˘ fpyq for every x,y P Rą0. (Belgium)

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IMO 2014 Shortlist G2

IMO 2014 Shortlist G2 Category: Geometry Problem Let ABC be a triangle. The points K, L, and M lie on the segments BC, CA, and AB, respectively, such that the lines AK, BL, and CM intersect in a common point. Prove that it is possible to choose two of the triangles ALM, BMK, and CKL whose inradii sum up to at least the inradius of the triangle ABC. (Estonia)

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IMO 2012 Shortlist N3

IMO 2012 Shortlist N3 Category: Number Theory Problem Determine all integers m ≥ 2 such that every n with m ≤ n ≤ m divides the binomial coefficient n m−2n . Solution. The integers in question are all prime numbers. First we check that all primes satisfy the condition, and even a stronger one. Namely, if p is a prime then every n with 1 ≤ n ≤ p...

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IMO 1991

IMO 1991 Official IMO 1991 problems · 4/6 solved. # Status Time 1 solved 18m14s 2 — — 3 solved 31m33s 4 solved 31m43s 5 solved 11m04s 6 — — Problem 1 solved · 18m14s · Solution → Given a triangle $,ABC,,$ let $,I,$ be the center of its inscribed circle. The internal bisectors of the angles $,A,B,C,$ meet the opposite sides in $,A^{\prime },B^{\prime },C^{\prime },$ respectively. Prove that...

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IMO 2017 Shortlist C1

IMO 2017 Shortlist C1 Category: Combinatorics Problem A re tangle R with odd integer side lengths is divided into small re tangles with integer side lengths. Prove that there is at least one among the small re tangles whose distan es from the four sides of R are either all odd or all even. (Singapore)

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IMO 2003

IMO 2003 Official IMO 2003 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 $S$ is the set ${1, 2, 3, \dots ,1000000}$. Show that for any subset $A$ of $S$ with $101$ elements we can find $100$ distinct elements $x_i$ of $S$, such that the sets ${a + x_i \mid a...

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IMO 2010 Shortlist N3

IMO 2010 Shortlist N3 Category: Number Theory Problem Find the smallest number n such that there exist polynomials f1, f2, ..., fn with rational coefficients satisfying x2 7 f1pxq2 f2pxq2 fnpxq2 . (Poland)

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IMO 2015 Shortlist A2

IMO 2015 Shortlist A2 Category: Algebra Problem Determine all functions f : Z Ñ Z with the property that f x ´ fpyq ˘ “ f fpxq ˘ ´ fpyq ´ 1 holds for all x,y P Z. (Croatia)

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IMO 2011 Shortlist G7

IMO 2011 Shortlist G7 Category: Geometry Problem Let ABCDEF be a convex hexagon all of whose sides are tangent to a circle ω with center O. Suppose that the circumcircle of triangle ACE is concentric with ω. Let J be the foot of the perpendicular from B to CD. Suppose that the perpendicular from B to DF intersects the line EO at a point K. Let L be the foot...

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IMO 2013

IMO 2013 Official IMO 2013 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1,m_2,...,m_k$ (not necessarily different) such that $1+\frac{2^k-1}{n}=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\cdots\left(1+\frac{1}{m_k}\right).$ Problem 2 A configuration of $4027$ points in the plane is called Colombian if it...

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IMO 2014 Shortlist G7

IMO 2014 Shortlist G7 Category: Geometry Problem Let ABC be a triangle with circumcircle Ω and incentre I. Let the line passing through I and perpendicular to CI intersect the segment BC and the arc BC (not containing A) of Ω at points U and V , respectively. Let the line passing through U and parallel to AI intersect AV at X, and let the line passing through V and...

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IMO 2010 Shortlist A1

IMO 2010 Shortlist A1 Category: Algebra Problem Determine all functions f : R ÑR such that the equality fprxsyqfpxqrfpyqs. (1) holds for all x,y PR. Here, by rxs we denote the greatest integer not exceeding x. (France)

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IMO 2010 Shortlist N5

IMO 2010 Shortlist N5 Category: Number Theory Problem Let N be the set of all positive integers. Find all functions f : N Ñ N such that the number fpmq n m fpnq is a square for all m,n PN. (U.S.A.)

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IMO 2023 Shortlist G2

IMO 2023 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with AC ą BC. Let ω be the circumcircle of triangle ABC and let r be the radius of ω. Point P lies on segment AC such that BC “ CP and point S is the foot of the perpendicular from P to line AB. Let ray BP intersect ω again at D and let Q lie on...

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IMO 2006 Shortlist C1

IMO 2006 Shortlist C1 Category: Combinatorics Problem We have n ≥ 2 lamps L1,...,Ln in a row, each of them being either on or off . Every second we simultaneously modify the state of each lamp as follows: — if the lamp Li and its neighbours (only one neighbour for i = 1 or i = n, two neighbours for other i) are in the same state, then Li is...

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IMO 2018 Shortlist A3

IMO 2018 Shortlist A3 Category: Algebra Problem Given any set S of positive integers, show that at least one of the following two assertions holds: (1) There exist distin t nite subsets F and G of S su h that ř xPF 1{x “ ř xPG 1{x; (2) There exists a positive rational number r ă 1 su h that ř xPF 1{x ‰ r for all nite subsets F...

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IMO 2021 Shortlist

IMO 2021 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

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IMO 2009 Shortlist G3

IMO 2009 Shortlist G3 Category: Geometry Problem Let ABC be a triangle. The incircle of ABC touches the sides AB and AC at the points Z and Y , respectively. Let G be the point where the lines BY and CZ meet, and let R and S be points such that the two quadrilaterals BCY R and BCSZ are parallelograms. Prove that GR = GS.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N1

IMO 2009 Shortlist N1 Category: Number Theory Problem A social club has n members. They have the membership numbers 1,2,...,n, respectively. From time to time members send presents to other members, including items they have already received as presents from other members. In order to avoid the embarrassing situation that a member might receive a present that he or she has sent to other members, the club adds the following...

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IMO 2014 Shortlist C2

IMO 2014 Shortlist C2 Category: Combinatorics Problem We have 2m sheets of paper, with the number 1 written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are a and b, then we erase these numbers and write the number a ` b on both sheets. Prove that after m2m´1 steps, the sum of the...

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IMO 2024 Shortlist N1

IMO 2024 Shortlist N1 Category: Number Theory Problem Find all positive integers n with the following property: for all positive divisors d of n, we have that d 1 | n or d 1 is prime. (Ghana)

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IMO

International Mathematical Olympiad Problems from the International Mathematical Olympiad , the world's premier mathematics competition for high-school students, held annually since 1959. This index covers all 65 competitions (74 problems solved, 35 verified). Year Solved Verified 2024 6/6 1/6 2023 0/6 0/0 2022 0/6 0/0 2021 0/6 0/0 2020 0/6 0/0 2019 0/6 0/0 2018 0/6 0/0 2017 0/6 0/0 2016 0/6 0/0 2015 0/6 0/0 2014 0/6 0/0 2013...

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IMO 2015 Shortlist N4

IMO 2015 Shortlist N4 Category: Number Theory Problem Suppose that a0,a1,... and b0,b1,... are two sequences of positive integers satisfying a0,b0 ě 2 and an 1 “ gcdpan,bnq 1, bn 1 “ lcmpan,bnq ´ 1 for all n ě 0. Prove that the sequence (an) is eventually periodic; in other words, there exist integers N ě 0 and t ą 0 such that an t “ an for all n...

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IMO 2010 Shortlist N6

IMO 2010 Shortlist N6 Category: Number Theory Problem The rows and columns of a 2n 2n table are numbered from 0 to 2n 1. The cells of the table have been colored with the following property being satisfied: for each 0 ¨i,j ¨2n 1, the jth cell in the ith row and the pi jqth cell in the jth row have the same color. (The indices of the cells in...

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IMO 2021 Shortlist G4

IMO 2021 Shortlist G4 Category: Geometry Problem Let ABCD be a quadrilateral inscribed in a circle Ω. Let the tangent to Ω at D intersect the rays BA and BC at points E and F, respectively. A point T is chosen inside the triangle ABC so that TE k CD and TF k AD. Let K ‰ D be a point on the segment DF such that TD “TK. Prove...

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IMO 2021 Shortlist A4

IMO 2021 Shortlist A4 Category: Algebra Problem Show that for all real numbers x1,...,xn the following inequality holds: n ÿ i“1 n ÿ j“1 b |xi ´xj|ď n ÿ i“1 n ÿ j“1 b |xi `xj|.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N3

IMO 2008 Shortlist N3 Category: Number Theory Problem Let a0,a1,a2,... be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, gcd(ai,ai+1) > ai−1. Prove that an ≥ 2n for all n ≥ 0.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G7

IMO 2008 Shortlist G7 Category: Geometry Problem Let ABCD be a convex quadrilateral with AB 6= BC. Denote by ω1 and ω2 the incircles of triangles ABC and ADC. Suppose that there exists a circle ω inscribed in angle ABC, tangent to the extensions of line segments AD and CD. Prove that the common external tangents of ω1 and ω2 intersect on ω.

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IMO 2013 Shortlist C2

IMO 2013 Shortlist C2 Category: Combinatorics Problem In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw k lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value...

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IMO 2023 Shortlist C2

IMO 2023 Shortlist C2 Category: Combinatorics Problem Determine the maximal length L of a sequence a1,...,aL of positive integers satisfying both the following properties: • every term in the sequence is less than or equal to 22023 , and • there does not exist a consecutive subsequence ai,ai 1,...,aj (where 1 ď i ď j ď L) with a choice of signs si,si 1,...,sj P t1,´1u for which siai si...

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IMO 2013 Shortlist N5

IMO 2013 Shortlist N5 Category: Number Theory Problem Fix an integer k ě 2. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer n ě k gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number m just written on the blackboard and replaces it by some number m1 with k ď...

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IMO 2021 Shortlist N6

IMO 2021 Shortlist N6 Category: Number Theory Problem Determine all integers n ě 2 with the following property: every n pairwise distinct integers whose sum is not divisible by n can be arranged in some order a1, a2, ..., an so that n divides 1 ¨a1 2 ¨a2 ¨¨¨`n ¨an.

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IMO 2013 Shortlist C8

IMO 2013 Shortlist C8 Category: Combinatorics Problem Players A and B play a paintful game on the real line. Player A has a pot of paint with four units of black ink. A quantity p of this ink suffices to blacken a (closed) real interval of length p. In every round, player A picks some positive integer m and provides 1{2m units of ink from the pot. Player B then...

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IMO 2015 Shortlist N5

IMO 2015 Shortlist N5 Category: Number Theory Problem Determine all triples pa,b,cq of positive integers for which ab ´ c, bc ´ a, and ca ´ b are powers of 2. Explanation: A power of 2 is an integer of the form 2n , where n denotes some nonnegative integer. (Serbia)

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IMO 2011 Shortlist N4

IMO 2011 Shortlist N4 Category: Number Theory Problem For each positive integer k, let t(k) be the largest odd divisor of k. Determine all positive integers a for which there exists a positive integer n such that all the differences t(n + a) − t(n), t(n + a + 1) − t(n + 1), ..., t(n + 2a − 1) − t(n + a − 1) are divisible by 4.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C4

IMO 2008 Shortlist C4 Category: Combinatorics Problem Let n and k be fixed positive integers of the same parity, k ≥ n. We are given 2n lamps numbered 1 through 2n; each of them can be on or off. At the beginning all lamps are off. We consider sequences of k steps. At each step one of the lamps is switched (from off to on or from on to off)....

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IMO 2012 Shortlist C7

IMO 2012 Shortlist C7 Category: Combinatorics Problem There are given 2500 points on a circle labeled 1,2,...,2500 in some order. Prove that one can choose 100 pairwise disjoint chords joining some of these points so that the 100 sums of the pairs of numbers at the endpoints of the chosen chords are equal. Solution. The proof is based on the following general fact. Lemma. In a graph G each vertex...

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IMO 2014 Shortlist A4

IMO 2014 Shortlist A4 Category: Algebra Problem Determine all functions f : Z Ñ Z satisfying f fpmq n ˘ fpmq “ fpnq fp3mq ` 2014 for all integers m and n. (Netherlands) Shortlisted problems 5

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IMO 1993

IMO 1993 Official IMO 1993 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $f(x)=x^n+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients. Problem 2 Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB=\angle...

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IMO 2016 Shortlist A5

IMO 2016 Shortlist A5 Category: Algebra Problem (a) Prove that for every positive integer n, there exists a fraction a b where a and b are integers satisfying 0 < b ⩽ √ n + 1 and √ n ⩽ a b ⩽ √ n + 1. (b) Prove that there are infinitely many positive integers n such that there is no fraction a b where a and b are...

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IMO 2017 Shortlist A2

IMO 2017 Shortlist A2 Category: Algebra Problem Let q be a real number. Gugu has a napkin with ten distin t real numbers written on it, and he writes the following three lines of real numbers on the bla kboard: • In the rst line, Gugu writes down every number of the form a´b, where a and b are two (not ne essarily distin t) numbers on his napkin. •...

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IMO 2024 Shortlist G6

IMO 2024 Shortlist G6 Category: Geometry Problem Let ABC be an acute triangle with AB ă AC, and let Γ be the circumcircle of ABC. Points X and Y lie on Γ so that XY and BC intersect on the external angle bisector of =BAC. Suppose that the tangents to Γ at X and Y intersect at a point T on the same side of BC as A, and that...

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IMO 2009 Shortlist G6

IMO 2009 Shortlist G6 Category: Geometry Problem Let the sides AD and BC of the quadrilateral ABCD (such that AB is not parallel to CD) intersect at point P. Points O1 and O2 are the circumcenters and points H1 and H2 are the orthocenters of triangles ABP and DCP, respectively. Denote the midpoints of segments O1H1 and O2H2 by E1 and E2, respectively. Prove that the perpendicular from E1 on...

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IMO 2004

IMO 2004 Official IMO 2004 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the...

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IMO 2006 Shortlist G10

IMO 2006 Shortlist G10 Category: Geometry Problem To each side a of a convex polygon we assign the maximum area of a triangle contained in the polygon and having a as one of its sides. Show that the sum of the areas assigned to all sides of the polygon is not less than twice the area of the polygon. (Serbia)

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IMO 2015 Shortlist N2

IMO 2015 Shortlist N2 Category: Number Theory Problem Let a and b be positive integers such that a!b! is a multiple of a! b!. Prove that 3a ě 2b 2. (United Kingdom)

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IMO 2015 Shortlist C4

IMO 2015 Shortlist C4 Category: Combinatorics Problem Let n be a positive integer. Two players A and B play a game in which they take turns choosing positive integers k ď n. The rules of the game are: piq A player cannot choose a number that has been chosen by either player on any previous turn. piiq A player cannot choose a number consecutive to any of those the player...

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IMO 2016 Shortlist C5

IMO 2016 Shortlist C5 Category: Combinatorics Problem Let n ⩾ 3 be a positive integer. Find the maximum number of diagonals of a regular n-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other. 6 IMO 2016 Hong Kong

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IMO 2016 Shortlist A3

IMO 2016 Shortlist A3 Category: Algebra Problem Find all integers n ⩾ 3 with the following property: for all real numbers a1,a2,...,an and b1,b2,...,bn satisfying |ak| + |bk| = 1 for 1 ⩽ k ⩽ n, there exist x1,x2,...,xn, each of which is either −1 or 1, such that n X k=1 xkak + n X k=1 xkbk ⩽ 1.

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IMO 2012 Shortlist C1

IMO 2012 Shortlist C1 Category: Combinatorics Problem Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers x and y such that x > y and x is to the left of y, and replaces the pair (x,y) by either (y + 1,x) or (x − 1,x). Prove that she can perform only finitely many such iterations. Solution 1. Note first that the allowed operation does...

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IMO 2019 Shortlist C3

IMO 2019 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. Harry has n coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are k coins showing heads and k ą 0, then he ﬂips the kth coin over; otherwise he stops the process. (For example, the process starting with THT would be THT Ñ HHT Ñ HTT...

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IMO 2011 Shortlist A2

IMO 2011 Shortlist A2 Category: Algebra Problem Determine all sequences (x1,x2,...,x2011) of positive integers such that for every positive inte- ger n there is an integer a with xn 1 + 2xn 2 + ··· + 2011xn 2011 = an+1

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IMO 1984

IMO 1984 Official IMO 1984 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$. Problem 2 Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$. Problem...

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IMO 2023 Shortlist G8

IMO 2023 Shortlist G8 Category: Geometry Problem Let ABC be an equilateral triangle. Points A1, B1, C1 lie inside triangle ABC such that triangle A1B1C1 is scalene, BA1 “ A1C, CB1 “ B1A, AC1 “ C1B and =BA1C =CB1A =AC1B “ 480˝ . Lines BC1 and CB1 intersect at A2; lines CA1 and AC1 intersect at B2; and lines AB1 and BA1 intersect at C2. Prove that the circumcircles of...

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IMO 2012 Shortlist A1

IMO 2012 Shortlist A1 Category: Algebra Problem Find all the functions f : Z → Z such that f(a)2 f(b)2 f(c)2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a) for all integers a, b, c satisfying a + b + c = 0. Solution. The substitution a = b = c = 0 gives 3f(0)2 = 6f(0)2 , hence f(0) = 0. (1) The substitution b = −a and c = 0...

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IMO 2016 Shortlist C7

IMO 2016 Shortlist C7 Category: Combinatorics Problem Let n ⩾ 2 be an integer. In the plane, there are n segments given in such a way that any two segments have an intersection point in the interior, and no three segments intersect at a single point. Jeff places a snail at one of the endpoints of each of the segments and claps his hands n−1 times. Each time when he...

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IMO 2007 Shortlist G4

IMO 2007 Shortlist G4 Category: Geometry Problem Consider five points A, B, C, D, E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let be a line passing through A, and let intersect segment DC and line BC at points F and G, respectively. Suppose that EF = EG = EC. Prove that ` is the bisector of angle DAB. (Luxembourg)

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IMO 2015 Shortlist G6

IMO 2015 Shortlist G6 Category: Geometry Problem Let ABC be an acute triangle with AB ą AC, and let Γ be its circumcircle. Let H, M, and F be the orthocenter of the triangle, the midpoint of BC, and the foot of the altitude from A, respectively. Let Q and K be the two points on Γ that satisfy =AQH “ 900 and =QKH “ 900 . Prove that the...

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IMO 2024 Shortlist G1

IMO 2024 Shortlist G1 Category: Geometry Problem Let ABCD be a cyclic quadrilateral such that AC ă BD ă AD and =DBA ă 90˝ . Point E lies on the line through D parallel to AB such that E and C lie on opposite sides of line AD, and AC “ DE. Point F lies on the line through A parallel to CD such that F and C lie on...

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IMO 2016 Shortlist

IMO 2016 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

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IMO 2023 Shortlist N2

IMO 2023 Shortlist N2 Category: Number Theory Problem Determine all pairs pa,pq of positive integers with p prime such that pa `a4 is a perfect square. (Bangladesh)

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IMO 2015 Shortlist C1

IMO 2015 Shortlist C1 Category: Combinatorics Problem In Lineland there are n ě 1 towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the 2n bulldozers are distinct. Every time when a right and a left...

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IMO 2019 Shortlist N6

IMO 2019 Shortlist N6 Category: Number Theory Problem Let H “ X i ? : i P Zą0 ( “ t1,2,4,5,7,...u, and let n be a positive integer. Prove that there exists a constant C such that, if A Ă t1,2,...,nu satisﬁes |A| ě C ? n, then there exist a,b P A such that a´b P H. (Here Zą0 is the set of positive integers, and tzu denotes the...

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IMO 2018 Shortlist C2

IMO 2018 Shortlist C2 Category: Combinatorics Problem Queenie and Horst play a game on a 20ˆ20 hessboard. In the beginning the board is empty. In every turn, Horst pla es a bla k knight on an empty square in su h a way that his new knight does not atta k any previous knights. Then Queenie pla es a white queen on an empty square. The game gets nished when...

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IMO 2022 Shortlist G8

IMO 2022 Shortlist G8 Category: Geometry Problem Let AA1 BCC1 B1 be a convex cyclic hexagon such that AC is tangent to the incircle of the triangle A1 B1 C1 , and A1 C1 is tangent to the incircle of the triangle ABC. Let the lines AB and A1 B1 meet at X and let the lines BC and B1 C1 meet at Y . Prove that if XBY B1...

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IMO 2024 Shortlist C7

IMO 2024 Shortlist C7 Category: Combinatorics Problem Let N be a positive integer and let a1, a2, ... be an infinite sequence of positive integers. Suppose that, for each n ą N, an is equal to the number of times an´1 appears in the list a1, a2, ..., an´1. Prove that at least one of the sequences a1, a3, a5, ... and a2, a4, a6, ... is eventually periodic. (Australia)

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IMO 2012 Shortlist G5

IMO 2012 Shortlist G5 Category: Geometry Problem Let ABC be a triangle with ∠BCA = 90◦ , and let C0 be the foot of the altitude from C. Choose a point X in the interior of the segment CC0, and let K,L be the points on the segments AX,BX for which BK = BC and AL = AC respectively. Denote by M the intersection of AL and BK. Show that...

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IMO 2009 Shortlist C1

IMO 2009 Shortlist C1 Category: Combinatorics Problem Consider 2009 cards, each having one gold side and one black side, lying in parallel on a long table. Initially all cards show their gold sides. Two players, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of 50 consecutive cards, the leftmost of which is showing gold, and turning...

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IMO 2007 Shortlist N4

IMO 2007 Shortlist N4 Category: Number Theory Problem For every integer k ≥ 2, prove that 23k divides the number 2k+1 2k − 2k 2k−1 (1) but 23k+1 does not. (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G4

IMO 2011 Shortlist G4 Category: Geometry Problem Let ABC be an acute triangle with circumcircle Ω. Let B0 be the midpoint of AC and let C0 be the midpoint of AB. Let D be the foot of the altitude from A, and let G be the centroid of the triangle ABC. Let ω be a circle through B0 and C0 that is tangent to the circle Ω at a point...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A1

IMO 2009 Shortlist A1 Category: Algebra Problem Find the largest possible integer k, such that the following statement is true: Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain b1 ≤ b2 ≤ ... ≤ b2009 the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G5

IMO 2016 Shortlist G5 Category: Geometry Problem Let D be the foot of perpendicular from A to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle ABC. A circle ω with centre S passes through A and D, and it intersects sides AB and AC at X and Y respectively. Let P be the foot of altitude from A to BC, and...

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IMO 2023 Shortlist G5

IMO 2023 Shortlist G5 Category: Geometry Problem Let ABC be an acute-angled triangle with circumcircle ω and circumcentre O. Points D ‰ B and E ‰ C lie on ω such that BD K AC and CE K AB. Let CO meet AB at X, and BO meet AC at Y . Prove that the circumcircles of triangles BXD and CY E have an intersection on line AO. (Malaysia)

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IMO 2023 Shortlist A2

IMO 2023 Shortlist A2 Category: Algebra Problem Let R be the set of real numbers. Let f : R Ñ R be a function such that fpx ` yqfpx ´ yq ě fpxq2 ´ fpyq2 for every x,y P R. Assume that the inequality is strict for some x0,y0 P R. Prove that fpxq ě 0 for every x P R or fpxq ď 0 for every x P R....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N1

IMO 2011 Shortlist N1 Category: Number Theory Problem For any integer d > 0, let f(d) be the smallest positive integer that has exactly d positive divisors (so for example we have f(1) = 1, f(5) = 16, and f(6) = 12). Prove that for every integer k ≥ 0 the number f(2k ) divides f(2k+1 ).

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IMO 2012 Shortlist A6

IMO 2012 Shortlist A6 Category: Algebra Problem Let f : N → N be a function, and let fm be f applied m times. Suppose that for every n ∈ N there exists a k ∈ N such that f2k (n) = n + k, and let kn be the smallest such k. Prove that the sequence k1,k2,... is unbounded. Solution. We restrict attention to the set S = {1,f(1),f2...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N1

IMO 2018 Shortlist N1 Category: Number Theory Problem Determine all pairs pn,kq of distin t positive integers su h that there exists a positive integer s for whi h the numbers of divisors of sn and of sk are equal. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist G4

IMO 2020 Shortlist G4 Category: Geometry Problem In the plane, there are n ě 6 pairwise disjoint disks D1,D2,...,Dn with radii R1 ě R2 ě ... ě Rn. For every i “ 1,2,...,n, a point Pi is chosen in disk Di. Let O be an arbitrary point in the plane. Prove that OP1 OP2 ... OPn ě R6 R7 ... Rn. (A disk is assumed to contain its boundary.) (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist C8

IMO 2009 Shortlist C8 Category: Combinatorics Problem For any integer n ≥ 2, we compute the integer h(n) by applying the following procedure to its decimal representation. Let r be the rightmost digit of n. (1) If r = 0, then the decimal representation of h(n) results from the decimal representation of n by removing this rightmost digit 0. (2) If 1 ≤ r ≤ 9 we split the decimal...

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IMO 1995

IMO 1995 Official IMO 1995 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the...

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IMO 2017 Shortlist N6

IMO 2017 Shortlist N6 Category: Number Theory Problem Find the smallest positive integer n, or show that no su h n exists, with the following property: there are innitely many distin t n-tuples of positive rational numbers pa1,a2,...,anq su h that both a1 a2 ¨¨¨ an and a1 a2 ¨¨¨ an are integers. (Singapore)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist C1

IMO 2019 Shortlist C1 Category: Combinatorics Problem The inﬁnite sequence a0, a1, a2, ... of (not necessarily diﬀerent) integers has the following properties: 0 ď ai ď i for all integers i ě 0, and ˆ k a0 ˙ ˆ k a1 ˙ ¨¨¨ ` ˆ k ak ˙ “ 2k for all integers k ě 0. Prove that all integers N ě 0 occur in the sequence (that is,...

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IMO 2012 Shortlist C4

IMO 2012 Shortlist C4 Category: Combinatorics Problem Players A and B play a game with N ≥ 2012 coins and 2012 boxes arranged around a circle. Initially A distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order B,A,B,A,... by the following rules: • On every move of his B passes 1 coin...

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IMO 2019 Shortlist C8

IMO 2019 Shortlist C8 Category: Combinatorics Problem Alice has a map of Wonderland, a country consisting of n ě 2 towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has oﬀered to help her....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist G2

IMO 2012 Shortlist G2 Category: Geometry Problem Let ABCD be a cyclic quadrilateral whose diagonals AC and BD meet at E. The extensions of the sides AD and BC beyond A and B meet at F. Let G be the point such that ECGD is a parallelogram, and let H be the image of E under reflection in AD. Prove that D, H, F, G are concyclic. Solution. We show...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A1

IMO 2011 Shortlist A1 Category: Algebra Problem For any set A = {a1,a2,a3,a4} of four distinct positive integers with sum sA = a1 +a2 +a3 +a4, let pA denote the number of pairs (i,j) with 1 ≤ i < j ≤ 4 for which ai +aj divides sA. Among all sets of four distinct positive integers, determine those sets A for which pA is maximal.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N2

IMO 2014 Shortlist N2 Category: Number Theory Problem Determine all pairs px,yq of positive integers such that a 7x2 ´ 13xy 7y2 “ |x ´ y| 1. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist G5

IMO 2018 Shortlist G5 Category: Geometry Problem Let ABC be a triangle with ir um ir le ω and in entre I. A line ℓ interse ts the lines AI, BI, and CI at points D, E, and F, respe tively, distin t from the points A, B, C, and I. The perpendi ular bise tors x, y, and z of the segments AD, BE, and CF, respe tively determine...

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IMO 2023 Shortlist G4

IMO 2023 Shortlist G4 Category: Geometry Problem Let ABC be an acute-angled triangle with AB ă AC. Denote its circumcircle by Ω and denote the midpoint of arc CAB by S. Let the perpendicular from A to BC meet BS and Ω at D and E ‰ A respectively. Let the line through D parallel to BC meet line BE at L and denote the circumcircle of triangle BDL by...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist G6

IMO 2014 Shortlist G6 Category: Geometry Problem Let ABC be a fixed acute-angled triangle. Consider some points E and F lying on the sides AC and AB, respectively, and let M be the midpoint of EF. Let the perpendicular bisector of EF intersect the line BC at K, and let the perpendicular bisector of MK intersect the lines AC and AB at S and T, respectively. We call the pair...

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IMO 2021 Shortlist G3

IMO 2021 Shortlist G3 Category: Geometry Problem Version 1. Let n be a fixed positive integer, and let S be the set of points px,yq on the Cartesian plane such that both coordinates x and y are nonnegative integers smaller than 2n (thus |S| “ 4n2 ). Assume that F is a set consisting of n2 quadrilaterals such that all their vertices lie in S, and each point in S...

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IMO 2019 Shortlist N1

IMO 2019 Shortlist N1 Category: Number Theory Problem Find all pairs pm,nq of positive integers satisfying the equation p2n ´ 1qp2n ´ 2qp2n ´ 4q¨¨¨p2n ´ 2n´1 q “ m! (El Salvador)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist G7

IMO 2012 Shortlist G7 Category: Geometry Problem Let ABCD be a convex quadrilateral with non-parallel sides BC and AD. Assume that there is a point E on the side BC such that the quadrilaterals ABED and AECD are circumscribed. Prove that there is a point F on the side AD such that the quadrilaterals ABCF and BCDF are circumscribed if and only if AB is parallel to CD. Solution. Let...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist C6

IMO 2012 Shortlist C6 Category: Combinatorics Problem Let k and n be fixed positive integers. In the liar’s guessing game, Amy chooses integers x and N with 1 ≤ x ≤ N. She tells Ben what N is, but not what x is. Ben may then repeatedly ask Amy whether x ∈ S for arbitrary sets S of integers. Amy will always answer with yes or no, but she might...

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IMO 2001

IMO 2001 Official IMO 2001 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Consider an acute triangle $\triangle ABC$. Let $P$ be the foot of the altitude of triangle $\triangle ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $\triangle ABC$. Assume that $\angle C \geq...

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IMO 2017 Shortlist G8

IMO 2017 Shortlist G8 Category: Geometry Problem There are 2017 mutually external ir les drawn on a bla kboard, su h that no two are tangent and no three share a ommon tangent. A tangent segment is a line segment that is a ommon tangent to two ir les, starting at one tangent point and ending at the other one. Lu iano is drawing tangent segments on the bla kboard,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N6

IMO 2009 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Show that if there exists a sequence a0,a1,... of integers satisfying the condition an = an−1 + nk n for all n ≥ 1, then k − 2 is divisible by 3.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G1

IMO 2016 Shortlist G1 Category: Geometry Problem In a convex pentagon ABCDE, let F be a point on AC such that ∠FBC = 90◦ . Suppose triangles ABF, ACD and ADE are similar isosceles triangles with ∠FAB = ∠FBA = ∠DAC = ∠DCA = ∠EAD = ∠EDA. Let M be the midpoint of CF. Point X is chosen such that AMXE is a parallelogram. Show that BD,EM and FX are...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C2

IMO 2010 Shortlist C2 Category: Combinatorics Problem On some planet, there are 2N countries (N © 4). Each country has a flag N units wide and one unit high composed of N fields of size 11, each field being either yellow or blue. No two countries have the same flag. We say that a set of N flags is diverse if these flags can be arranged into an N N...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist C3

IMO 2017 Shortlist C3 Category: Combinatorics Problem Sir Alex plays the following game on a row of 9 ells. Initially, all ells are empty. In ea h move, Sir Alex is allowed to perform exa tly one of the following two operations: (1) Choose any number of the form 2j , where j is a non-negative integer, and put it into an empty ell. (2) Choose two (not ne essarily...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist A2

IMO 2019 Shortlist A2 Category: Algebra Problem Let u1, u2, ..., u2019 be real numbers satisfying u1 u2 ¨¨¨ u2019 “ 0 and u2 1 u2 2 ¨¨¨ u2 2019 “ 1. Let a “ minpu1,u2,...,u2019q and b “ maxpu1,u2,...,u2019q. Prove that ab ď ´ . (Germany)

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IMO 1979

IMO 1979 Official IMO 1979 problems · 5/6 solved, 1 verified. # Status Time 1 ✓ verified 10m50s 2 solved 43m03s 3 solved 14m28s 4 solved 16m50s 5 solved 4m30s 6 — — Problem 1 ✓ verified · 10m50s · Solution → If $p$ and $q$ are natural numbers so that$$ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, $$prove that $p$ is divisible by $1979$. Problem 2 solved · 43m03s · Solution → We...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C3

IMO 2010 Shortlist C3 Category: Combinatorics Problem 2500 chess kings have to be placed on a 100 100 chessboard so that (i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); (ii) each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or sym- metry are supposed to be...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N5

IMO 2009 Shortlist N5 Category: Number Theory Problem Let P(x) be a non-constant polynomial with integer coefficients. Prove that there is no function T from the set of integers into the set of integers such that the number of integers x with Tn (x) = x is equal to P(n) for every n ≥ 1, where Tn denotes the n-fold application of T.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N3

IMO 2007 Shortlist N3 Category: Number Theory Problem Let X be a set of 10000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset Y of X such that a − b + c − d + e is not divisible by 47 for any a,b,c,d,e ∈ Y . (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N3

IMO 2018 Shortlist N3 Category: Number Theory Problem Dene the sequen e a0,a1,a2,... by an “ 2n ` 2tn{2u . Prove that there are innitely many terms of the sequen e whi h an be expressed as a sum of (two or more) distin t terms of the sequen e, as well as innitely many of those whi h annot be expressed in su h a way. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A5

IMO 2018 Shortlist A5 Category: Algebra Problem Determine all fun tions f : p0,8q Ñ R satisfying ˆ x x ˙ fpyq “ fpxyq f ´y x ¯ for all x,y ą 0. (South Korea)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G7

IMO 2006 Shortlist G7 Category: Geometry Problem In a triangle ABC, let Ma,Mb,Mc be respectively the midpoints of the sides BC, CA, AB and Ta,Tb,Tc be the midpoints of the arcs BC, CA, AB of the circumcircle of ABC, not containing the opposite vertices. For i ∈ {a,b,c}, let ωi be the circle with MiTi as diameter. Let pi be the common external tangent to ωj, ωk ({i,j,k} = {a,b,c})...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A6

IMO 2009 Shortlist A6 Category: Algebra Problem Suppose that s1,s2,s3,... is a strictly increasing sequence of positive integers such that the subsequences ss1, ss2, ss3, ... and ss1+1, ss2+1, ss3+1, ... are both arithmetic progressions. Prove that s1,s2,s3,... is itself an arithmetic progression.

Practice › Mathematics › IMO ›

IMO 2024

IMO 2024 Official IMO 2024 problems · 6/6 solved, 1 verified. # Status Time 1 ✓ verified 5m00s 2 solved 33m38s 3 solved 8m59s 4 solved 1h06m 5 solved 11m42s 6 solved 17m40s Problem 1 ✓ verified · 5m00s · Solution → Determine all real numbers $\alpha$ such that, for every positive integer $n$, the integer $$ \lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor $$...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist G1

IMO 2021 Shortlist G1 Category: Geometry Problem Let ABCD be a parallelogram such that AC “ BC. A point P is chosen on the extension of the segment AB beyond B. The circumcircle of the triangle ACD meets the segment PD again at Q, and the circumcircle of the triangle APQ meets the segment PC again at R. Prove that the lines CD, AQ, and BR are concurrent.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist C3

IMO 2015 Shortlist C3 Category: Combinatorics Problem For a finite set A of positive integers, we call a partition of A into two disjoint nonempty subsets A1 and A2 good if the least common multiple of the elements in A1 is equal to the greatest common divisor of the elements in A2. Determine the minimum value of n such that there exists a set of n positive integers with exactly...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist G1

IMO 2023 Shortlist G1 Category: Geometry Problem Let ABCDE be a convex pentagon such that =ABC “ =AED “ 90˝ . Suppose that the midpoint of CD is the circumcentre of triangle ABE. Let O be the circumcentre of triangle ACD. Prove that line AO passes through the midpoint of segment BE. (Slovakia)

Practice › Mathematics › IMO ›

IMO 2008

IMO 2008 Official IMO 2008 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects line $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N6

IMO 2012 Shortlist N6 Category: Number Theory Problem Let x and y be positive integers. If x2n − 1 is divisible by 2n y + 1 for every positive integer n, prove that x = 1. Solution. First we prove the following fact: For every positive integer y there exist infinitely many primes p ≡ 3 (mod 4) such that p divides some number of the form 2n y +...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist C2

IMO 2022 Shortlist C2 Category: Combinatorics Problem The Bank of Oslo issues coins made out of two types of metal: aluminium (denoted A) and copper (denoted C). Morgane has n aluminium coins, and n copper coins, and arranges her 2n coins in a row in some arbitrary initial order. Given a fixed positive integer k ď 2n, she repeatedly performs the following operation: identify the largest subsequence containing the k-th...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G5

IMO 2009 Shortlist G5 Category: Geometry Problem Let P be a polygon that is convex and symmetric to some point O. Prove that for some parallelogram R satisfying P ⊂ R we have |R| |P| ≤ √ where |R| and |P| denote the area of the sets R and P, respectively.

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IMO 2019 Shortlist A6

IMO 2019 Shortlist A6 Category: Algebra Problem A polynomial Ppx,y,zq in three variables with real coeﬃcients satisﬁes the identities Ppx,y,zq “ Ppx,y,xy ´ zq “ Ppx,zx ´ y,zq “ Ppyz ´ x,y,zq. Prove that there exists a polynomial Fptq in one variable such that Ppx,y,zq “ Fpx2 y2 z2 ´ xyzq. (Russia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A5

IMO 2008 Shortlist A5 Category: Algebra Problem Let a,b,c,d be positive real numbers such that abcd = 1 and a + b + c + d > a b + b c + c d + d a . Prove that a + b + c + d < b a + c b + d c + a d .

Practice › Mathematics › IMO ›

IMO 1968

IMO 1968 Official IMO 1968 problems · 1/6 solved, 1 verified. # Status Time 1 — — 2 — — 3 — — 4 ✓ verified 11m45s 5 — — 6 — — Problem 1 Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another. Problem 2 Find all natural numbers $x$ such that...

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IMO 2014 Shortlist A6

IMO 2014 Shortlist A6 Category: Algebra Problem Find all functions f : Z Ñ Z such that n2 ` 4fpnq “ fpfpnqq2 for all n P Z. (United Kingdom) 6 IMO 2014 South Africa

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist A4

IMO 2016 Shortlist A4 Category: Algebra Problem Denote by R+ the set of all positive real numbers. Find all functions f : R+ → R+ such that xf(x2 )f(f(y)) + f(yf(x)) = f(xy) Ä f(f(x2 )) + f(f(y2 )) ä for all positive real numbers x and y.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N1

IMO 2007 Shortlist N1 Category: Number Theory Problem Find all pairs (k,n) of positive integers for which 7k − 3n divides k4 n2 . (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N2

IMO 2020 Shortlist N2 Category: Number Theory Problem For each prime p, there is a kingdom of p-Landia consisting of p islands numbered 1, 2, ..., p. Two distinct islands numbered n and m are connected by a bridge if and only if p divides pn2 ´ m 1qpm2 ´ n 1q. The bridges may pass over each other, but cannot cross. Prove that for infinitely many p there are...

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IMO 2021 Shortlist A6

IMO 2021 Shortlist A6 Category: Algebra Problem Let A be a finite set of (not necessarily positive) integers, and let m ě2 be an integer. Assume that there exist non-empty subsets B1,B2,B3,...,Bm of A whose elements add up to the sums m1 ,m2 ,m3 ,...,mm , respectively. Prove that A contains at least m{2 elements.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N6

IMO 2016 Shortlist N6 Category: Number Theory Problem Denote by N the set of all positive integers. Find all functions f : N → N such that for all positive integers m and n, the integer f(m) + f(n) − mn is nonzero and divides mf(m) + nf(n). 10 IMO 2016 Hong Kong

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N6

IMO 2013 Shortlist N6 Category: Number Theory Problem Determine all functions f : Q ÝÑ Z satisfying f ˆ fpxq a b ˙ “ f ´x a b ¯ for all x P Q, a P Z, and b P Zą0. (Here, Zą0 denotes the set of positive integers.) (Israel)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N8

IMO 2017 Shortlist N8 Category: Number Theory Problem Let p be an odd prime number and Zą0 be the set of positive integers. Suppose that a fun tion f : Zą0 ˆ Zą0 Ñ t0,1u satises the following properties: • fp1,1q “ 0; • fpa,bq fpb,aq “ 1 for any pair of relatively prime positive integers pa,bq not both equal to 1; • fpa b,bq “ fpa,bq for any pair...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C1

IMO 2007 Shortlist C1 Category: Combinatorics Problem Let n > 1 be an integer. Find all sequences a1,a2,...,an2+n satisfying the following conditions: (a) ai ∈ {0,1} for all 1 ≤ i ≤ n2 n; (b) ai+1 + ai+2 + ... + ai+n < ai+n+1 + ai+n+2 + ... + ai+2n for all 0 ≤ i ≤ n2 − n. (Serbia)

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IMO 2023 Shortlist

IMO 2023 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number...

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IMO 2019 Shortlist N7

IMO 2019 Shortlist N7 Category: Number Theory Problem Prove that there is a constant c ą 0 and inﬁnitely many positive integers n with the following property: there are inﬁnitely many positive integers that cannot be expressed as the sum of fewer than cnlogpnq pairwise coprime nth powers. (Canada)

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IMO 2014 Shortlist C6

IMO 2014 Shortlist C6 Category: Combinatorics Problem We are given an infinite deck of cards, each with a real number on it. For every real number x, there is exactly one card in the deck that has x written on it. Now two players draw disjoint sets A and B of 100 cards each from this deck. We would like to define a rule that declares one of them a...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A3

IMO 2011 Shortlist A3 Category: Algebra Problem Determine all pairs (f,g) of functions from the set of real numbers to itself that satisfy g(f(x + y)) = f(x) + (2x + y)g(y) for all real numbers x and y.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist A7

IMO 2019 Shortlist A7 Category: Algebra Problem Let Z be the set of integers. We consider functions f : Z Ñ Z satisfying f fpx yq y ˘ “ f fpxq ` y ˘ for all integers x and y. For such a function, we say that an integer v is f-rare if the set Xv “ tx P Z: fpxq “ vu is ﬁnite and nonempty. (a) Prove that...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G8

IMO 2009 Shortlist G8 Category: Geometry Problem Let ABCD be a circumscribed quadrilateral. Let g be a line through A which meets the segment BC in M and the line CD in N. Denote by I1, I2, and I3 the incenters of 4ABM, 4MNC, and 4NDA, respectively. Show that the orthocenter of 4I1I2I3 lies on g.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist A7

IMO 2022 Shortlist A7 Category: Algebra Problem For a positive integer n we denote by spnq the sum of the digits of n. Let Ppxq “ xn an´1xn´1 ¨¨¨ a1x a0 be a polynomial, where n ě 2 and ai is a positive integer for all 0 ď i ď n ´ 1. Could it be the case that, for all positive integers k, spkq and spPpkqq have the same...

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IMO 2018 Shortlist C4

IMO 2018 Shortlist C4 Category: Combinatorics Problem An anti-Pas al pyramid is a nite set of numbers, pla ed in a triangle-shaped array so that the rst row of the array ontains one number, the se ond row ontains two numbers, the third row ontains three numbers and so on; and, ex ept for the numbers in the bottom row, ea h number equals the absolute value of the dieren...

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Mathematics

Mathematics Worked solutions to several mathematical problem collections. Each collection has its own index with the full list of problems. The International Mathematical Olympiad sets the hardest problems given to high-school students anywhere, six a year since 1959. Around the contest sit its two feeder collections: the IMO Shortlist , the pool from which each year's jury picks the final six, and the IMO Longlist , the wider set of...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N7

IMO 2007 Shortlist N7 Category: Number Theory Problem For a prime p and a positive integer n, denote by νp(n) the exponent of p in the prime factorization of n!. Given a positive integer d and a finite set {p1,...,pk} of primes. Show that there are infinitely many positive integers n such that d νpi (n) for all 1 ≤ i ≤ k. (India)

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IMO 1976

IMO 1976 Official IMO 1976 problems · 2/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 solved 45m37s 6 solved 44m41s Problem 1 In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have....

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IMO 2022 Shortlist

IMO 2022 Shortlist 33 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics C9 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number...

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IMO 2021 Shortlist N2

IMO 2021 Shortlist N2 Category: Number Theory Problem Let n ě100 be an integer. The numbers n,n 1,...,2n are written on n 1 cards, one number per card. The cards are shuffled and divided into two piles. Prove that one of the piles contains two cards such that the sum of their numbers is a perfect square.

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IMO 2022 Shortlist A5

IMO 2022 Shortlist A5 Category: Algebra Problem Find all positive integers n ě 2 for which there exist n real numbers a1 ă ¨¨¨ ă an and a real number r ą 0 such that the 1 npn´1q differences aj ´ai for 1 ď i ă j ď n are equal, in some order, to the numbers r1 ,r2 ,...,r npn´1q . (Czech Republic)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G1

IMO 2009 Shortlist G1 Category: Geometry Problem Let ABC be a triangle with AB = AC. The angle bisectors of A and B meet the sides BC and AC in D and E, respectively. Let K be the incenter of triangle ADC. Suppose that ∠BEK = 45◦ . Find all possible values of ∠BAC.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A2

IMO 2008 Shortlist A2 Category: Algebra Problem (a) Prove the inequality x2 (x − 1)2 + y2 (y − 1)2 + z2 (z − 1)2 ≥ 1 for real numbers x,y,z 6= 1 satisfying the condition xyz = 1. (b) Show that there are infinitely many triples of rational numbers x, y, z for which this inequality turns into equality.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N8

IMO 2016 Shortlist N8 Category: Number Theory Problem Find all polynomials P(x) of odd degree d and with integer coefficients satisfying the following property: for each positive integer n, there exist n positive integers x1,x2,...,xn such that 1 < P(xi) P(xj) < 2 and P(xi) P(xj) is the d-th power of a rational number for every pair of indices i and j with 1 ⩽ i,j ⩽ n. Shortlisted problems...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist A2

IMO 2014 Shortlist A2 Category: Algebra Problem Define the function f : p0,1q Ñ p0,1q by fpxq “ x ` 1 if x ă 1 , x2 if x ě 1 . Let a and b be two real numbers such that 0 ă a ă b ă 1. We define the sequences an and bn by a0 “ a, b0 “ b, and an “ fpan´1q, bn “ fpbn´1q...

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IMO 2017 Shortlist G4

IMO 2017 Shortlist G4 Category: Geometry Problem In triangle ABC, let ω be the ex ir le opposite A. Let D, E, and F be the points where ω is tangent to lines BC, CA, and AB, respe tively. The ir le AEF interse ts line BC at P and Q. Let M be the midpoint of AD. Prove that the ir le MPQ is tangent to ω. (Denmark)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist C6

IMO 2017 Shortlist C6 Category: Combinatorics Problem Let n ą 1 be an integer. An n ˆ n ˆ n ube is omposed of n3 unit ubes. Ea h unit ube is painted with one olor. For ea h nˆnˆ1 box onsisting of n2 unit ubes (of any of the three possible orientations), we onsider the set of the olors present in that box (ea h olor is listed only...

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IMO 2017 Shortlist

IMO 2017 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

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IMO 2023 Shortlist N7

IMO 2023 Shortlist N7 Category: Number Theory Problem Let a,b,c,d be positive integers satisfying ab a b cd c d “ pa bqpc dq a b c d . Determine all possible values of a b c ` d. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist C4

IMO 2019 Shortlist C4 Category: Combinatorics Problem On a ﬂat plane in Camelot, King Arthur builds a labyrinth L consisting of n walls, each of which is an inﬁnite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners:...

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IMO 2022 Shortlist A4

IMO 2022 Shortlist A4 Category: Algebra Problem Let n ě 3 be an integer, and let x1,x2,...,xn be real numbers in the interval r0,1s. Let s “ x1 x2 ...`xn, and assume that s ě 3. Prove that there exist integers i and j with 1 ď i ă j ď n such that 2j´i xixj ą 2s´3 . (Trinidad and Tobago)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist G2

IMO 2013 Shortlist G2 Category: Geometry Problem Let ω be the circumcircle of a triangle ABC. Denote by M and N the midpoints of the sides AB and AC, respectively, and denote by T the midpoint of the arc BC of ω not containing A. The circumcircles of the triangles AMT and ANT intersect the perpendicular bisectors of AC and AB at points X and Y , respectively; assume that...

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IMO 2022 Shortlist N3

IMO 2022 Shortlist N3 Category: Number Theory Problem Let a ą 1 be a positive integer, and let d ą 1 be a positive integer coprime to a. Let x1 “ 1 and, for k ě 1, define xk`1 “ xk ` d if a doesn’t divide xk, xk{a if a divides xk. Find the greatest positive integer n for which there exists an index k such that xk is...

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IMO 2008 Shortlist

IMO 2008 Shortlist 26 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory

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Kvant Mathematics

Kvant Mathematics Solutions to mathematics problems from Kvant magazine (524 solved, 0 verified). # Status Time 1 solved 3m46s 2 solved 2m19s 3 solved 4m10s 4 solved 3m05s 5 solved 3m02s 6 solved 3m07s 7 solved 5m50s 8 solved 3m33s 9 solved 3m50s 10 solved 4m14s 11 solved 4m23s 13 solved 1m21s 14 solved 3m28s 15 solved 2m51s 16 solved 1m14s 17 solved 1m43s 18 solved 1m36s 19 solved 1m20s...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C7

IMO 2007 Shortlist C7 Category: Combinatorics Problem Let α < 3 − √ be a positive real number. Prove that there exist positive integers n and p > α · 2n for which one can select 2p pairwise distinct subsets S1, ..., Sp, T1, ..., Tp of the set {1,2,...,n} such that Si ∩ Tj 6= ∅ for all 1 ≤ i,j ≤ p. (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C3

IMO 2007 Shortlist C3 Category: Combinatorics Problem Find all positive integers n, for which the numbers in the set S = {1,2,...,n} can be colored red and blue, with the following condition being satisfied: the set S × S × S contains exactly 2007 ordered triples (x,y,z) such that (i) x,y,z are of the same color and (ii) x+y +z is divisible by n. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist C1

IMO 2022 Shortlist C1 Category: Combinatorics Problem A ˘1-sequence is a sequence of 2022 numbers a1,...,a2022, each equal to either 1 or ´1. Determine the largest C so that, for any ˘1-sequence, there exists an integer k and indices 1 ď t1 ă ... ă tk ď 2022 so that ti 1 ´ ti ď 2 for all i, and k ÿ i“1 ati ě C. (Czech Republic)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist G6

IMO 2021 Shortlist G6 Category: Geometry Problem Determine all integers n ě 3 satisfying the following property: every convex n-gon whose sides all have length 1 contains an equilateral triangle of side length 1. (Every polygon is assumed to contain its boundary.) Shortlisted problems 9

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist G1

IMO 2015 Shortlist G1 Category: Geometry Problem Let ABC be an acute triangle with orthocenter H. Let G be the point such that the quadrilateral ABGH is a parallelogram. Let I be the point on the line GH such that AC bisects HI. Suppose that the line AC intersects the circumcircle of the triangle GCI at C and J. Prove that IJ “ AH. (Australia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist G8

IMO 2020 Shortlist G8 Category: Geometry Problem Let Γ and I be the circumcircle and the incenter of an acute-angled triangle ABC. Two circles ωB and ωC passing through B and C, respectively, are tangent at I. Let ωB meet the shorter arc AB of Γ and segment AB again at P and M, respectively. Similarly, let ωC meet the shorter arc AC of Γ and segment AC again at...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A5

IMO 2009 Shortlist A5 Category: Algebra Problem Let f be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers x and y such that f (x − f(y)) > yf(x) + x.

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IMO 2022

IMO 2022 Official IMO 2022 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G8

IMO 2007 Shortlist G8 Category: Geometry Problem Point P lies on side AB of a convex quadrilateral ABCD. Let ω be the incircle of triangle CPD, and let I be its incenter. Suppose that ω is tangent to the incircles of triangles APD and BPC at points K and L, respectively. Let lines AC and BD meet at E, and let lines AK and BL meet at F. Prove that...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist C6

IMO 2018 Shortlist C6 Category: Combinatorics Problem Let a and b be distin t positive integers. The following innite pro ess takes pla e on an initially empty board. piq If there is at least a pair of equal numbers on the board, we hoose su h a pair and in rease one of its omponents by a and the other by b. piiq If no su h pair exists,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist C7

IMO 2014 Shortlist C7 Category: Combinatorics Problem Let M be a set of n ě 4 points in the plane, no three of which are collinear. Initially these points are connected with n segments so that each point in M is the endpoint of exactly two segments. Then, at each step, one may choose two segments AB and CD sharing a common interior point and replace them by the segments...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist A5

IMO 2020 Shortlist A5 Category: Algebra Problem A magician intends to perform the following trick. She announces a positive integer n, along with 2n real numbers x1 ă ... ă x2n, to the audience. A member of the audience then secretly chooses a polynomial Ppxq of degree n with real coefficients, computes the 2n values Ppx1q,...,Ppx2nq, and writes down these 2n values on the blackboard in non-decreasing order. After that...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist G1

IMO 2013 Shortlist G1 Category: Geometry Problem Let ABC be an acute-angled triangle with orthocenter H, and let W be a point on side BC. Denote by M and N the feet of the altitudes from B and C, respectively. Denote by ω1 the circumcircle of BWN, and let X be the point on ω1 which is diametrically opposite to W. Analogously, denote by ω2 the circumcircle of CWM, and...

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IMO 2021 Shortlist A8

IMO 2021 Shortlist A8 Category: Algebra Problem Determine all functions f : R ÑR that satisfy fpaq´fpbq ˘ fpbq´fpcq ˘ fpcq´fpaq ˘ “fpab2 bc2 ca2 q´fpa2 b b2 c `c2 aq for all real numbers a,b,c. 6 Saint-Petersburg — Russia, 16th–24th July 2021

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IMO 1988

IMO 1988 Official IMO 1988 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Consider 2 concentric circle radii $R$ and $r$ ($R > r$) with centre $O.$ Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large...

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IMO 1981

IMO 1981 Official IMO 1981 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 $\displaystyle P$ is a point inside a given triangle $\displaystyle ABC$. $\displaystyle D, E, F$ are the feet of the perpendiculars from $\displaystyle P$ to the lines $\displaystyle BC, CA, AB$, respectively. Find all $\displaystyle P$ for which...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C41

IMO 2010 Shortlist C41 Category: Combinatorics Problem . Same as Problem C4, but the constant 201020102010 is replaced by 20102010 . (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C5

IMO 2021 Shortlist C5 Category: Combinatorics Problem Let n and k be two integers with n ą k ě 1. There are 2n 1 students standing in a circle. Each student S has 2k neighbours— namely, the k students closest to S on the right, and the k students closest to S on the left. Suppose that n 1 of the students are girls, and the other n are boys....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist C4

IMO 2017 Shortlist C4 Category: Combinatorics Problem Let N ě 2 be an integer. NpN ` 1q so er players, no two of the same height, stand in a row in some order. Coa h Ralph wants to remove NpN ´ 1q people from this row so that in the remaining row of 2N players, no one stands between the two tallest ones, no one stands between the third and...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist G8

IMO 2024 Shortlist G8 Category: Geometry Problem Let ABC be a triangle with AB ă AC ă BC, and let D be a point in the interior of segment BC. Let E be a point on the circumcircle of triangle ABC such that A and E lie on opposite sides of line BC and =BAD “ =EAC. Let I, IB, IC, JB, and JC be the incentres of triangles ABC,...

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IMO 2006 Shortlist

IMO 2006 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry G9 Geometry G10 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N4

IMO 2021 Shortlist N4 Category: Number Theory Problem Alice is given a rational number r ą 1 and a line with two points B ‰ R, where point R contains a red bead and point B contains a blue bead. Alice plays a solitaire game by performinga sequence of moves. In every move, she chooses a (not necessarily positive) integer k, and a bead to move. If that bead is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist A2

IMO 2016 Shortlist A2 Category: Algebra Problem Find the smallest real constant C such that for any positive real numbers a1,a2,a3,a4 and a5 (not necessarily distinct), one can always choose distinct subscripts i,j,k and l such that ai aj − ak al ⩽ C.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A4

IMO 2009 Shortlist A4 Category: Algebra Problem Let a, b, c be positive real numbers such that ab + bc + ca ≤ 3abc. Prove that r a2 + b2 a + b + r b2 + c2 b + c + r c2 + a2 c + a 3 ≤ √ √ a + b + √ b + c + √ c + a .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist C3

IMO 2016 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer relatively prime to 6. We paint the vertices of a regular n-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist A6

IMO 2015 Shortlist A6 Category: Algebra Problem Let n be a fixed integer with n ě 2. We say that two polynomials P and Q with real coefficients are block-similar if for each i P t1,2,...,nu the sequences Pp2015iq,Pp2015i´ 1q,...,Pp2015i ´ 2014q and Qp2015iq,Qp2015i´ 1q,...,Qp2015i ´ 2014q are permutations of each other. paq Prove that there exist distinct block-similar polynomials of degree n ` 1. pbq Prove that there do...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A3

IMO 2009 Shortlist A3 Category: Algebra Problem Determine all functions f from the set of positive integers into the set of positive integers such that for all x and y there exists a non degenerated triangle with sides of lengths x, f(y) and f(y + f(x) − 1).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist G3

IMO 2018 Shortlist G3 Category: Geometry Problem A ir le ω of radius 1 is given. A olle tion T of triangles is alled good, if the following onditions hold: piq ea h triangle from T is ins ribed in ω; piiq no two triangles from T have a ommon interior point. Determine all positive real numbers t su h that, for ea h positive integer n, there exists a...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist A2

IMO 2020 Shortlist A2 Category: Algebra Problem Let A denote the set of all polynomials in three variables x,y,z with integer coefficients. Let B denote the subset of A formed by all polynomials which can be expressed as px y zqPpx,y,zq pxy yz zxqQpx,y,zq xyzRpx,y,zq with P,Q,R P A. Find the smallest non-negative integer n such that xi yj zk P B for all non- negative integers i,j,k satisfying i...

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IMO 2022 Shortlist C9

IMO 2022 Shortlist C9 Category: Combinatorics Problem Let Zě0 be the set of non-negative integers, and let f : Zě0 ˆ Zě0 Ñ Zě0 be a bijection such that whenever fpx1,y1q ą fpx2,y2q, we have fpx1 1,y1q ą fpx2 1,y2q and fpx1,y1 1q ą fpx2,y2 1q. Let N be the number of pairs of integers px,yq, with 0 ď x,y ă 100, such that fpx,yq is odd. Find the smallest...

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IMO 2018 Shortlist

IMO 2018 Shortlist 28 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory N7...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist A4

IMO 2010 Shortlist A4 Category: Algebra Problem A sequence x1,x2,... is defined by x1 1 and x2k xk, x2k1 p1qk 1 xk for all k ©1. Prove that x1 x2 xn ©0 for all n ©1. (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N1

IMO 2008 Shortlist N1 Category: Number Theory Problem Let n be a positive integer and let p be a prime number. Prove that if a, b, c are integers (not necessarily positive) satisfying the equations an pb = bn pc = cn pa, then a = b = c.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist A4

IMO 2024 Shortlist A4 Category: Algebra Problem Let Zą0 be the set of all positive integers. Determine all subsets S of t20 ,21 ,22 ,...u for which there exists a function f : Zą0 Ñ Zą0 such that S “ tfpa ` bq ´ fpaq ´ fpbq | a,b P Zą0u. (Thailand)

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IMO 2019 Shortlist

IMO 2019 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics C9 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C7

IMO 2006 Shortlist C7 Category: Combinatorics Problem Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let A be the number of antipodal pairs of vertices, and...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist C8

IMO 2016 Shortlist C8 Category: Combinatorics Problem Let n be a positive integer. Determine the smallest positive integer k with the following property: it is possible to mark k cells on a 2n × 2n board so that there exists a unique partition of the board into 1 × 2 and 2 × 1 dominoes, none of which contains two marked cells. Shortlisted problems 7

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IMO 2024 Shortlist N6

IMO 2024 Shortlist N6 Category: Number Theory Problem Let n be a positive integer. We say that a polynomial P with integer coeffi- cients is n-good if there exists a polynomial Q of degree 2 with integer coefficients such that QpkqpPpkq ` Qpkqq is never divisible by n for any integer k. Determine all integers n such that every polynomial with integer coefficients is an n-good polynomial. (France)

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IMO 2021 Shortlist C7

IMO 2021 Shortlist C7 Category: Combinatorics Problem Consider a checkered 3m ˆ3m square, where m is an integer greater than 1. A frog sits on the lower left corner cell S and wants to get to the upper right corner cell F. The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be sticky, and the frog...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G7

IMO 2010 Shortlist G7 Category: Geometry Problem Three circular arcs γ1, γ2, and γ3 connect the points A and C. These arcs lie in the same half-plane defined by line AC in such a way that arc γ2 lies between the arcs γ1 and γ3. Point B lies on the segment AC. Let h1, h2, and h3 be three rays starting at B, lying in the same half-plane, h2 being...

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IMO 2006

IMO 2006 Official IMO 2006 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $ABC$ be a triangle with incentre $I.$ A point $P$ in the interior of the triangle satisfies $\angle PBA + \angle PCA = \angle PBC + \angle PCB$. Show that $AP \ge AI,$ and that equality holds...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G3

IMO 2016 Shortlist G3 Category: Geometry Problem Let B = (−1,0) and C = (1,0) be fixed points on the coordinate plane. A nonempty, bounded subset S of the plane is said to be nice if (i) there is a point T in S such that for every point Q in S, the segment TQ lies entirely in S; and (ii) for any triangle P1P2P3, there exists a unique point...

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IMO 2022 Shortlist N2

IMO 2022 Shortlist N2 Category: Number Theory Problem Find all positive integers n ą 2 such that n! ˇ ˇ ˇ ˇ ź păqďn, p,q primes pp ` qq. (Nigeria)

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IMO 2022 Shortlist G4

IMO 2022 Shortlist G4 Category: Geometry Problem Let ABC be an acute-angled triangle with AC ą AB, let O be its circumcentre, and let D be a point on the segment BC. The line through D perpendicular to BC intersects the lines AO, AC and AB at W, X and Y , respectively. The circumcircles of triangles AXY and ABC intersect again at Z ‰ A. Prove that if OW...

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IMO 2023 Shortlist G3

IMO 2023 Shortlist G3 Category: Geometry Problem Let ABCD be a cyclic quadrilateral with =BAD ă =ADC. Let M be the midpoint of the arc CD not containing A. Suppose there is a point P inside ABCD such that =ADB “ =CPD and =ADP “ =PCB. Prove that lines AD,PM,BC are concurrent. (Slovakia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N4

IMO 2016 Shortlist N4 Category: Number Theory Problem Let n,m,k and l be positive integers with n 6= 1 such that nk +mnl +1 divides nk+l −1. Prove that • m = 1 and l = 2k; or • l|k and m = nk−l−1 nl−1 .

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IMO 2019 Shortlist N5

IMO 2019 Shortlist N5 Category: Number Theory Problem Let a be a positive integer. We say that a positive integer b is a-good if an b ˘ ´ 1 is divisible by an 1 for all positive integers n with an ě b. Suppose b is a positive integer such that b is a-good, but b 2 is not a-good. Prove that b 1 is prime. (Netherlands)

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IMO 2024 Shortlist A7

IMO 2024 Shortlist A7 Category: Algebra Problem Let Q be the set of rational numbers. Let f : Q Ñ Q be a function such that the following property holds: for all x, y P Q, fpx fpyqq “ fpxq y or fpfpxq yq “ x fpyq. Determine the maximum possible number of elements of tfpxq ` fp´xq | x P Qu. (Japan) 4 Bath, United Kingdom, 10th –22nd July...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist C1

IMO 2016 Shortlist C1 Category: Combinatorics Problem The leader of an IMO team chooses positive integers n and k with n > k, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an n-digit binary string, and the deputy leader writes down all n-digit binary strings which differ from the leader’s in exactly k positions. (For example, if n = 3...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist A2

IMO 2006 Shortlist A2 Category: Algebra Problem The sequence of real numbers a0,a1,a2,... is defined recursively by a0 = −1, n X k=0 an−k k + 1 = 0 for n ≥ 1. Show that an > 0 for n ≥ 1. (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist A2

IMO 2024 Shortlist A2 Category: Algebra Problem Let n be a positive integer. Find the minimum possible value of S “ 20 x2 0 21 x2 1 ¨¨¨ 2n x2 n, where x0, x1, ..., xn are nonnegative integers such that x0 x1 ¨¨¨ xn “ n. (China)

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IMO 1989

IMO 1989 Official IMO 1989 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Prove that in the set ${1,2, \ldots, 1989}$ can be expressed as the disjoint union of subsets $A_i, {i = 1,2, \ldots, 117}$ such that i.) each $A_i$ contains 17 elements ii.) the sum of all the elements...

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IMO 2010 Shortlist

IMO 2010 Shortlist 31 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C41 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G61 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist A2

IMO 2012 Shortlist A2 Category: Algebra Problem Let Z and Q be the sets of integers and rationals respectively. a) Does there exist a partition of Z into three non-empty subsets A,B,C such that the sets A + B, B + C, C + A are disjoint? b) Does there exist a partition of Q into three non-empty subsets A,B,C such that the sets A + B, B + C,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist G1

IMO 2017 Shortlist G1 Category: Geometry Problem Let ABCDE be a onvex pentagon su h that AB “ BC “ CD, =EAB “ =BCD, and =EDC “ =CBA. Prove that the perpendi ular line from E to BC and the line seg- ments AC and BD are on urrent. (Italy)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist C7

IMO 2020 Shortlist C7 Category: Combinatorics Problem Consider any rectangular table having finitely many rows and columns, with a real number apr,cq in the cell in row r and column c. A pair pR,Cq, where R is a set of rows and C a set of columns, is called a saddle pair if the following two conditions are satisfied: piq For each row r1 , there is r P R...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G3

IMO 2008 Shortlist G3 Category: Geometry Problem Let ABCD be a convex quadrilateral and let P and Q be points in ABCD such that PQDA and QPBC are cyclic quadrilaterals. Suppose that there exists a point E on the line segment PQ such that ∠PAE = ∠QDE and ∠PBE = ∠QCE. Show that the quadrilateral ABCD is cyclic.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N3

IMO 2016 Shortlist N3 Category: Number Theory Problem Define P(n) = n2 n + 1. For any positive integers a and b, the set {P(a),P(a + 1),P(a + 2),...,P(a + b)} is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G1

IMO 2007 Shortlist G1 Category: Geometry Problem In triangle ABC, the angle bisector at vertex C intersects the circumcircle and the per- pendicular bisectors of sides BC and CA at points R, P, and Q, respectively. The midpoints of BC and CA are S and T, respectively. Prove that triangles RQT and RPS have the same area. (Czech Republic)

Practice › Mathematics › IMO ›

IMO 2019

IMO 2019 Official IMO 2019 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$, $$ f(2a) + 2f(b) = f(f(a + b)). $$ Problem 2 In triangle $ABC$, point...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N1

IMO 2017 Shortlist N1 Category: Number Theory Problem The sequen e a0,a1,a2,... of positive integers satises an 1 “ #? an, if ? an is an integer an 3, otherwise for every n ě 0. Determine all values of a0 ą 1 for whi h there is at least one number a su h that an “ a for innitely many values of n. (South Afri a)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G3

IMO 2010 Shortlist G3 Category: Geometry Problem Let A1A2 ...An be a convex polygon. Point P inside this polygon is chosen so that its projections P1, ..., Pn onto lines A1A2, ..., AnA1 respectively lie on the sides of the polygon. Prove that for arbitrary points X1, ..., Xn on sides A1A2, ..., AnA1 respectively, max " X1X2 P1P2 ,..., XnX1 PnP1 * ©1. (Armenia)

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IMO 1972

IMO 1972 Official IMO 1972 problems · 6/6 solved, 2 verified. # Status Time 1 ✓ verified 20m29s 2 solved 21m31s 3 solved 7m11s 4 solved 6m59s 5 ✓ verified 8m24s 6 solved 7m52s Problem 1 ✓ verified · 20m29s · Solution → Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same...

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IMO 2020 Shortlist N1

IMO 2020 Shortlist N1 Category: Number Theory Problem Given a positive integer k, show that there exists a prime p such that one can choose distinct integers a1,a2,...,ak 3 P t1,2,...,p ´ 1u such that p divides aiai 1ai 2ai 3 ´ i for all i “ 1,2,...,k. (South Africa)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N5

IMO 2022 Shortlist N5 Category: Number Theory Problem For each 1 ď i ď 9 and T P N, define dipTq to be the total number of times the digit i appears when all the multiples of 1829 between 1 and T inclusive are written out in base 10. Show that there are infinitely many T P N such that there are precisely two distinct values among d1pTq,d2pTq,...,d9pTq. (United Kingdom)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N2

IMO 2018 Shortlist N2 Category: Number Theory Problem Let n ą 1 be a positive integer. Ea h ell of an n ˆ n table ontains an integer. Suppose that the following onditions are satised: piq Ea h number in the table is ongruent to 1 modulo n; piiq The sum of numbers in any row, as well as the sum of numbers in any olumn, is ongruent to n...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A4

IMO 2018 Shortlist A4 Category: Algebra Problem Let a0,a1,a2,... be a sequen e of real numbers su h that a0 “ 0, a1 “ 1, and for every n ě 2 there exists 1 ď k ď n satisfying an “ an´1 ¨¨¨ an´k k . Find the maximal possible value of a2018 ´ a2017. (Belgium)

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IMO 2021 Shortlist C2

IMO 2021 Shortlist C2 Category: Combinatorics Problem Let n ě3 be an integer. An integer m ěn 1 is called n-colourful if, given infinitely many marbles in each of n colours C1,C2,...,Cn, it is possible to place m of them around a circle so that in any group of n 1 consecutive marbles there is at least one marble of colour Ci for each i “1,...,n. Prove that there are...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N7

IMO 2013 Shortlist N7 Category: Number Theory Problem Let ν be an irrational positive number, and let m be a positive integer. A pair pa,bq of positive integers is called good if arbνs ´ btaνu “ m. A good pair pa,bq is called excellent if neither of the pairs pa´b,bq and pa,b´aq is good. (As usual, by txu and rxs we denote the integer numbers such that x ´ 1...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist C2

IMO 2019 Shortlist C2 Category: Combinatorics Problem You are given a set of n blocks, each weighing at least 1; their total weight is 2n. Prove that for every real number r with 0 ď r ď 2n ´ 2 you can choose a subset of the blocks whose total weight is at least r but at most r ` 2. (Thailand)

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IMO 2021 Shortlist C6

IMO 2021 Shortlist C6 Category: Combinatorics Problem A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes acolouringofthecellswithfinitelymanycolours. The rabbitthen secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cellthat ithas not visited before (two cellsare adjacent if they share a side). The...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N5

IMO 2018 Shortlist N5 Category: Number Theory Problem Four positive integers x, y, z, and t satisfy the relations xy ´ zt “ x y “ z t. Is it possible that both xy and zt are perfe t squares? (Russia)

Practice › Mathematics › IMO ›

IMO 1975

IMO 1975 Official IMO 1975 problems · 2/6 solved. # Status Time 1 solved 15m43s 2 solved 35m15s 3 — — 4 — — 5 — — 6 — — Problem 1 solved · 15m43s · Solution → Let $x_i, y_i~(i = 1, 2, \ldots, n)$ be real numbers such that $$ x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n. $$...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N7

IMO 2009 Shortlist N7 Category: Number Theory Problem Let a and b be distinct integers greater than 1. Prove that there exists a positive integer n such that (an − 1)(bn − 1) is not a perfect square.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist C2

IMO 2009 Shortlist C2 Category: Combinatorics Problem For any integer n ≥ 2, let N(n) be the maximal number of triples (ai,bi,ci), i = 1,...,N(n), consisting of nonnegative integers ai,bi and ci such that the following two conditions are satis- fied: (1) ai + bi + ci = n for all i = 1,...,N(n), (2) If i 6= j, then ai 6= aj,bi 6= bj and ci 6= cj. Determine...

Practice › Mathematics › IMO ›

IMO 2014

IMO 2014 Official IMO 2014 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Let $a_0<a_1<a_2<\cdots \quad$ be an infinite sequence of positive integers, Prove that there exists a unique integer $n\ge1$ such that $$ a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}. $$ Problem 2 Let $n\ge2$ be an integer. Consider an $n\times n$ chessboard...

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IMO 2019 Shortlist G5

IMO 2019 Shortlist G5 Category: Geometry Problem Let ABCDE be a convex pentagon with CD “ DE and =EDC ‰ 2 ¨ =ADB. Suppose that a point P is located in the interior of the pentagon such that AP “ AE and BP “ BC. Prove that P lies on the diagonal CE if and only if areapBCDq areapADEq “ areapABDq areapABPq. (Hungary)

Practice › Mathematics › IMO ›

IMO 2017

IMO 2017 Official IMO 2017 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$ a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \ a_n + 3 & \text{otherwise.} \end{cases}...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist G2

IMO 2020 Shortlist G2 Category: Geometry Problem Let ABCD be a convex quadrilateral. Suppose that P is a point in the interior of ABCD such that =PAD : =PBA : =DPA “ 1 : 2 : 3 “ =CBP : =BAP : =BPC. The internal bisectors of angles ADP and PCB meet at a point Q inside the triangle ABP. Prove that AQ “ BQ. (Poland)

Practice › Mathematics › IMO ›

IMO 2012

IMO 2012 Official IMO 2012 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Given $\triangle ABC$ the point $J$ is the centre of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C7

IMO 2011 Shortlist C7 Category: Combinatorics Problem On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number k of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G2

IMO 2011 Shortlist G2 Category: Geometry Problem Let A1A2A3A4 be a non-cyclic quadrilateral. Let O1 and r1 be the circumcenter and the circumradius of the triangle A2A3A4. Define O2, O3, O4 and r2, r3, r4 in a similar way. Prove that O1A2 1 − r2 + O2A2 2 − r2 + O3A2 3 − r2 + O4A2 4 − r2 = 0.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G4

IMO 2010 Shortlist G4 Category: Geometry Problem Let I be the incenter of a triangle ABC and Γ be its circumcircle. Let the line AI intersect Γ at a point D A. Let F and E be points on side BC and arc BDC respectively such that =BAF =CAE 1 =BAC. Finally, let G be the midpoint of the segment IF. Prove that the lines DG and EI...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist C6

IMO 2019 Shortlist C6 Category: Combinatorics Problem Let n ą 1 be an integer. Suppose we are given 2n points in a plane such that no three of them are collinear. The points are to be labelled A1, A2, ..., A2n in some order. We then consider the 2n angles =A1A2A3, =A2A3A4, ..., =A2n´2A2n´1A2n, =A2n´1A2nA1, =A2nA1A2. We measure each angle in the way that gives the smallest positive value (i.e....

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IMO 2013 Shortlist N3

IMO 2013 Shortlist N3 Category: Number Theory Problem Prove that there exist infinitely many positive integers n such that the largest prime divisor of n4 n2 1 is equal to the largest prime divisor of pn 1q4 pn 1q2 1. (Belgium)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N7

IMO 2021 Shortlist N7 Category: Number Theory Problem Let a1,a2,a3,... be an infinite sequence of positive integers such that an 2m divides an an m for allpositive integers n and m. Prove that this sequence is eventually periodic, i.e. there exist positive integers N and d such that an “an d for all n ąN.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C8

IMO 2021 Shortlist C8 Category: Combinatorics Problem Determine the largest N for which there exists a table T of integers with N rows and 100 columns that has the following properties: (i) Every row contains the numbers 1, 2, ..., 100 in some order. (ii) For any two distinct rows r and s, there is a column c such that |Tpr,cq´Tps,cq|ě2. Here Tpr,cq means the number at the intersection of...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist A3

IMO 2012 Shortlist A3 Category: Algebra Problem Let a2,...,an be n − 1 positive real numbers, where n ≥ 3, such that a2a3 ···an = 1. Prove that (1 + a2)2 (1 + a3)3 ···(1 + an)n nn . Solution. The substitution a2 = x2 x1 , a3 = x3 x2 , ..., an = x1 xn−1 transforms the original problem into the inequality (x1 + x2)2 (x2 + x3)3...

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IMO 2024 Shortlist G5

IMO 2024 Shortlist G5 Category: Geometry Problem Let ABC be a triangle with incentre I, and let Ω be the circumcircle of triangle BIC. Let K be a point in the interior of segment BC such that =BAK ă =KAC. The angle bisector of =BKA intersects Ω at points W and X such that A and W lie on the same side of BC, and the angle bisector of =CKA...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N3

IMO 2011 Shortlist N3 Category: Number Theory Problem Let n ≥ 1 be an odd integer. Determine all functions f from the set of integers to itself such that for all integers x and y the difference f(x) − f(y) divides xn − yn .

Practice › Mathematics › IMO ›

IMO 1985

IMO 1985 Official IMO 1985 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$. Problem 2 Let $n$ and $k$ be given relatively...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N5

IMO 2007 Shortlist N5 Category: Number Theory Problem Find all surjective functions f : N → N such that for every m,n ∈ N and every prime p, the number f(m + n) is divisible by p if and only if f(m) + f(n) is divisible by p. (N is the set of all positive integers.) (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A3

IMO 2008 Shortlist A3 Category: Algebra Problem Let S ⊆ R be a set of real numbers. We say that a pair (f,g) of functions from S into S is a Spanish Couple on S, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. f(x) < f(y) and g(x) < g(y) for all x,y ∈ S with x < y; (ii) The inequality f(g(g(x))) < g(f(x))...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist C5

IMO 2009 Shortlist C5 Category: Combinatorics Problem Five identical empty buckets of 2-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighboring buckets, empties them into the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist A4

IMO 2013 Shortlist A4 Category: Algebra Problem Let n be a positive integer, and consider a sequence a1,a2,...,an of positive integers. Extend it periodically to an infinite sequence a1,a2,... by defining an i “ ai for all i ě 1. If a1 ď a2 ď ¨¨¨ ď an ď a1 n and aai ď n i ´ 1 for i “ 1,2,...,n, prove that a1 ¨¨¨ ` an ď n2...

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IMO 1990

IMO 1990 Official IMO 1990 problems · 1/6 solved. # Status Time 1 solved 1m49s 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 solved · 1m49s · Solution → Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the...

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IMO 2024 Shortlist N4

IMO 2024 Shortlist N4 Category: Number Theory Problem Determine all positive integers a and b such that there exists a positive integer g such that gcdpan b,bn aq “ g for all sufficiently large n. (Indonesia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N7

IMO 2011 Shortlist N7 Category: Number Theory Problem Let p be an odd prime number. For every integer a, define the number Sa = a + a2 ··· + ap−1 p − 1 . Let m and n be integers such that S3 + S4 − 3S2 = m n . Prove that p divides m.

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IMO 2022 Shortlist N6

IMO 2022 Shortlist N6 Category: Number Theory Problem Let Q be a set of prime numbers, not necessarily finite. For a positive integer n consider its prime factorisation; define ppnq to be the sum of all the exponents and qpnq to be the sum of the exponents corresponding only to primes in Q. A positive integer n is called special if ppnq ppn 1q and qpnq qpn 1q are both...

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IMO 2013 Shortlist A6

IMO 2013 Shortlist A6 Category: Algebra Problem Let m ‰ 0 be an integer. Find all polynomials Ppxq with real coefficients such that px3 ´ mx2 1qPpx 1q px3 mx2 1qPpx ´ 1q “ 2px3 ´ mx 1qPpxq for all real numbers x. (Serbia) 4 IMO 2013 Colombia

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IMO 2021 Shortlist A7

IMO 2021 Shortlist A7 Category: Algebra Problem Let n ě 1 be an integer, and let x0,x1,...,xn 1 be n 2 non-negative real numbers that satisfy xixi 1 ´x2 i´1 ě1 for all i “1,2,...,n. Show that x0 x1 ¨¨¨ xn xn 1 ą ˆ 2n ˙3{2 .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C3

IMO 2008 Shortlist C3 Category: Combinatorics Problem In the coordinate plane consider the set S of all points with integer coordinates. For a positive integer k, two distinct points A,B ∈ S will be called k-friends if there is a point C ∈ S such that the area of the triangle ABC is equal to k. A set T ⊂ S will be called a k-clique if every two points...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A7

IMO 2018 Shortlist A7 Category: Algebra Problem Find the maximal value of S “ 3 c a b 7 c b c 7 3 c c d 7 c d a 7 , where a,b,c,d are nonnegative real numbers whi h satisfy a b c d “ 100. (Taiwan) 4 Cluj-Napo a Romania, 314 July 2018 Combinatori s

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IMO 2020 Shortlist N3

IMO 2020 Shortlist N3 Category: Number Theory Problem Let n be an integer with n ě 2. Does there exist a sequence pa1,...,anq of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G4

IMO 2009 Shortlist G4 Category: Geometry Problem Given a cyclic quadrilateral ABCD, let the diagonals AC and BD meet at E and the lines AD and BC meet at F. The midpoints of AB and CD are G and H, respectively. Show that EF is tangent at E to the circle through the points E, G, and H.

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IMO 2024 Shortlist

IMO 2024 Shortlist 31 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

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IMO 2021 Shortlist N5

IMO 2021 Shortlist N5 Category: Number Theory Problem Prove that there are only finitely many quadruples pa,b,c,nq of positive integers such that n! “an´1 bn´1 cn´1 .

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IMO 2015 Shortlist C2

IMO 2015 Shortlist C2 Category: Combinatorics Problem Let V be a finite set of points in the plane. We say that V is balanced if for any two distinct points A,B P V, there exists a point C P V such that AC “ BC. We say that V is center-free if for any distinct points A,B,C P V, there does not exist a point P P V such that...

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IMO 2022 Shortlist G3

IMO 2022 Shortlist G3 Category: Geometry Problem Let ABCD be a cyclic quadrilateral. Assume that the points Q, A, B, P are collinear in this order, in such a way that the line AC is tangent to the circle ADQ, and the line BD is tangent to the circle BCP. Let M and N be the midpoints of BC and AD, respectively. Prove that the following three lines are concurrent:...

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IMO 2015 Shortlist

IMO 2015 Shortlist 29 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory N7...

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IMO 2014 Shortlist C5

IMO 2014 Shortlist C5 Category: Combinatorics Problem Consider n ě 3 lines in the plane such that no two lines are parallel and no three have a common point. These lines divide the plane into polygonal regions; let F be the set of regions having finite area. Prove that it is possible to colour Pa n{2 T of the lines blue in such a way that no region in F...

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IMO 2013 Shortlist A5

IMO 2013 Shortlist A5 Category: Algebra Problem Let Zě0 be the set of all nonnegative integers. Find all the functions f : Zě0 Ñ Zě0 satisfying the relation fpfpfpnqqq “ fpn 1q 1 for all n P Zě0. (Serbia)

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IMO 2023 Shortlist N4

IMO 2023 Shortlist N4 Category: Number Theory Problem Let a1,a2,...,an,b1,b2,...,bn be 2n positive integers such that the n ` 1 products a1a2a3 ¨¨¨an, b1a2a3 ¨¨¨an, b1b2a3 ¨¨¨an, . . . b1b2b3 ¨¨¨bn form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression. (Canada)

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IMO 2015 Shortlist A4

IMO 2015 Shortlist A4 Category: Algebra Problem Find all functions f : R Ñ R satisfying the equation f x fpx yq ˘ fpxyq “ x fpx yq ` yfpxq for all real numbers x and y. (Albania)

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IMO 2013 Shortlist A2

IMO 2013 Shortlist A2 Category: Algebra Problem Prove that in any set of 2000 distinct real numbers there exist two pairs a ą b and c ą d with a ‰ c or b ‰ d, such that ˇ ˇ ˇ ˇ a ´ b c ´ d ´ 1 ˇ ˇ ˇ ˇ ă . (Lithuania)

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IMO 2008 Shortlist G4

IMO 2008 Shortlist G4 Category: Geometry Problem In an acute triangle ABC segments BE and CF are altitudes. Two circles passing through the points A and F are tangent to the line BC at the points P and Q so that B lies between C and Q. Prove that the lines PE and QF intersect on the circumcircle of triangle AEF.

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IMO 2017 Shortlist N2

IMO 2017 Shortlist N2 Category: Number Theory Problem Let p ě 2 be a prime number. Eduardo and Fernando play the following game making moves alternately: in ea h move, the urrent player hooses an index i in the set t0,1,...,p´1u that was not hosen before by either of the two players and then hooses an element ai of the set t0,1,2,3,4,5,6,7,8,9u. Eduardo has the rst move. The game ends...

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IMO 2019 Shortlist G8

IMO 2019 Shortlist G8 Category: Geometry Problem Let L be the set of all lines in the plane and let f be a function that assigns to each line ℓ P L a point fpℓq on ℓ. Suppose that for any point X, and for any three lines ℓ1, ℓ2, ℓ3 passing through X, the points fpℓ1q, fpℓ2q, fpℓ3q and X lie on a circle. Prove that there is a...

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IMO 2007 Shortlist C4

IMO 2007 Shortlist C4 Category: Combinatorics Problem Let A0 = (a1,...,an) be a finite sequence of real numbers. For each k ≥ 0, from the sequence Ak = (x1,...,xn) we construct a new sequence Ak+1 in the following way. We choose a partition {1,...,n} = I ∪ J, where I and J are two disjoint sets, such that the expression X i∈I xi − X j∈J xj attains the smallest...

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IMO 2020 Shortlist A1

IMO 2020 Shortlist A1 Category: Algebra Problem Version 1. Let n be a positive integer, and set N “ 2n . Determine the smallest real number an such that, for all real x, N c x2N 1 ď anpx ´ 1q2 x. Version 2. For every positive integer N, determine the smallest real number bN such that, for all real x, N c x2N 1 ď bNpx ´ 1q2 x....

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IMO 2014 Shortlist C3

IMO 2014 Shortlist C3 Category: Combinatorics Problem Let n ě 2 be an integer. Consider an n ˆ n chessboard divided into n2 unit squares. We call a configuration of n rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer k such that for every happy configuration of rooks, we can find a k ˆ k square without a...

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IMO 2024 Shortlist G2

IMO 2024 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with AB ă AC ă BC, incentre I and incircle ω. Let X be the point in the interior of side BC such that the line through X parallel to AC is tangent to ω. Similarly, let Y be the point in the interior of side BC such that the line through Y parallel to AB is tangent...

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IMO 2012 Shortlist C5

IMO 2012 Shortlist C5 Category: Combinatorics Problem The columns and the rows of a 3n × 3n square board are numbered 1,2,...,3n. Every square (x,y) with 1 ≤ x,y ≤ 3n is colored asparagus, byzantium or citrine according as the modulo 3 remainder of x+y is 0, 1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are 3n2 tokens of...

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IMO 2012 Shortlist G6

IMO 2012 Shortlist G6 Category: Geometry Problem Let ABC be a triangle with circumcenter O and incenter I. The points D, E and F on the sides BC, CA and AB respectively are such that BD + BF = CA and CD + CE = AB. The circumcircles of the triangles BFD and CDE intersect at P 6= D. Prove that OP = OI. Solution. By Miquel’s theorem the circles...

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IMO 2023 Shortlist A3

IMO 2023 Shortlist A3 Category: Algebra Problem Let x1,x2,...,x2023 be distinct real positive numbers such that an “ d px1 x2 ¨¨¨ xnq ˆ x1 x2 ¨¨¨ xn ˙ is an integer for every n “ 1,2,...,2023. Prove that a2023 ě 3034. (Netherlands)

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IMO 2014 Shortlist N4

IMO 2014 Shortlist N4 Category: Number Theory Problem Let n ą 1 be a given integer. Prove that infinitely many terms of the sequence pakqkě1, defined by ak “ Z nk k ^ , are odd. (For a real number x, txu denotes the largest integer not exceeding x.) (Hong Kong)

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IMO 2020 Shortlist A6

IMO 2020 Shortlist A6 Category: Algebra Problem Determine all functions f : Z Ñ Z such that fa2 b2 pa bq “ afpaq bfpbq for every a,b P Z. Here, fn denotes the nth iteration of f, i.e., f0 pxq “ x and fn 1 pxq “ fpfn pxqq for all n ě 0. (Slovakia) Shortlisted problems 5

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N4

IMO 2008 Shortlist N4 Category: Number Theory Problem Let n be a positive integer. Show that the numbers 2n − 1 , 2n − 1 , 2n − 1 , ... , 2n − 1 2n−1 − 1 are congruent modulo 2n to 1,3,5,...,2n −1 in some order.

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IMO 2017 Shortlist A5

IMO 2017 Shortlist A5 Category: Algebra Problem An integer n ě 3 is given. We all an n-tuple of real numbers px1,x2,...,xnq Shiny if for ea h permutation y1,y2,...,yn of these numbers we have n´1 ÿ i“1 yiyi 1 “ y1y2 y2y3 y3y4 ¨¨¨ ` yn´1yn ě ´1. Find the largest onstant K “ Kpnq su h that ÿ 1ďiăjďn xixj ě K holds for every Shiny n-tuple px1,x2,...,xnq. (Serbia)

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IMO 2011 Shortlist N6

IMO 2011 Shortlist N6 Category: Number Theory Problem Let P(x) and Q(x) be two polynomials with integer coefficients such that no nonconstant polynomial with rational coefficients divides both P(x) and Q(x). Suppose that for every positive integer n the integers P(n) and Q(n) are positive, and 2Q(n) − 1 divides 3P(n) − 1. Prove that Q(x) is a constant polynomial.

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IMO 2020 Shortlist C6

IMO 2020 Shortlist C6 Category: Combinatorics Problem 4n coins of weights 1,2,3,...,4n are given. Each coin is colored in one of n colors and there are four coins of each color. Show that all these coins can be partitioned into two sets with the same total weight, such that each set contains two coins of each color. (Hungary) Shortlisted problems 7

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IMO 2016 Shortlist C4

IMO 2016 Shortlist C4 Category: Combinatorics Problem Find all positive integers n for which we can fill in the entries of an n × n table with the following properties: • each entry can be one of I, M and O; • in each row and each column, the letters I, M and O occur the same number of times; and • in any diagonal whose number of entries is...

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IMO 2014 Shortlist C9

IMO 2014 Shortlist C9 Category: Combinatorics Problem There are n circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he...

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IMO 2011 Shortlist C3

IMO 2011 Shortlist C3 Category: Combinatorics Problem Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line ℓ going through a point P ∈ S. Rotate ℓ clockwise around the pivot P until the line contains another point Q of S. The point Q...

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IMO 2017 Shortlist N3

IMO 2017 Shortlist N3 Category: Number Theory Problem Determine all integers n ě 2 with the following property: for any integers a1,a2,...,an whose sum is not divisible by n, there exists an index 1 ď i ď n su h that none of the numbers ai,ai ai 1,...,ai ai 1 ¨¨¨ ai`n´1 is divisible by n. (We let ai “ ai´n when i ą n.) (Thailand)

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IMO 2024 Shortlist A6

IMO 2024 Shortlist A6 Category: Algebra Problem Let a0, a1, a2, ... be an infinite strictly increasing sequence of positive integers such that for each n ě 1 we have an P !an´1 an 1 , ? an´1 ¨ an`1 ) . Let b1, b2, ... be an infinite sequence of letters defined as bn “ A, if an “ 1 pan´1 an 1q; G, otherwise. Prove that there exist...

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IMO 2022 Shortlist N7

IMO 2022 Shortlist N7 Category: Number Theory Problem Let k be a positive integer and let S be a finite set of odd prime numbers. Prove that there is at most one way (modulo rotation and reflection) to place the elements of S around a circle such that the product of any two neighbors is of the form x2 x k for some positive integer x. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A1

IMO 2008 Shortlist A1 Category: Algebra Problem Find all functions f : (0,∞) → (0,∞) such that f(p)2 f(q)2 f(r2) + f(s2) = p2 q2 r2 + s2 for all p,q,r,s > 0 with pq = rs.

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IMO 2006 Shortlist A5

IMO 2006 Shortlist A5 Category: Algebra Problem Let a,b,c be the sides of a triangle. Prove that √ b + c − a √ b + √ c − √ a + √ c + a − b √ c + √ a − √ b + √ a + b − c √ a + √ b − √ c ≤ 3. (Korea)

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IMO 2012 Shortlist G3

IMO 2012 Shortlist G3 Category: Geometry Problem In an acute triangle ABC the points D, E and F are the feet of the altitudes through A, B and C respectively. The incenters of the triangles AEF and BDF are I1 and I2 respectively; the circumcenters of the triangles ACI1 and BCI2 are O1 and O2 respectively. Prove that I1I2 and O1O2 are parallel. Solution. Let ∠CAB = α, ∠ABC =...

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IMO 2015 Shortlist N6

IMO 2015 Shortlist N6 Category: Number Theory Problem Let Zą0 denote the set of positive integers. Consider a function f : Zą0 Ñ Zą0. For any m,n P Zą0 we write fn pmq “ fpfp...f looomooon n pmq...qq. Suppose that f has the following two properties: piq If m,n P Zą0, then fn pmq ´ m n P Zą0; piiq The set Zą0 z tfpnq|n P Zą0u is finite. Prove...

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IMO 2008 Shortlist C6

IMO 2008 Shortlist C6 Category: Combinatorics Problem For n ≥ 2, let S1,S2,...,S2n be 2n subsets of A = {1,2,3,...,2n+1 } that satisfy the following property: There do not exist indices a and b with a < b and elements x,y,z ∈ A with x < y < z such that y,z ∈ Sa and x,z ∈ Sb. Prove that at least one of the sets S1,S2,...,S2n contains no more...

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IMO 2011 Shortlist N5

IMO 2011 Shortlist N5 Category: Number Theory Problem Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n, the difference f(m) − f(n) is divisible by f(m − n). Prove that for all integers m, n with f(m) ≤ f(n) the number f(n) is divisible by f(m).

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IMO 2022 Shortlist C6

IMO 2022 Shortlist C6 Category: Combinatorics Problem Let n be a positive integer. We start with n piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. For each positive integer n, find the smallest number of non-empty piles that one can...

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IMO 2024 Shortlist C2

IMO 2024 Shortlist C2 Category: Combinatorics Problem Let n be a positive integer. The integers 1, 2, 3, ..., n2 are to be written in the cells of an n ˆ n board such that each integer is written in exactly one cell and each cell contains exactly one integer. For every integer d with d | n, the d-division of the board is the division of the board into...

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IMO 2022 Shortlist C5

IMO 2022 Shortlist C5 Category: Combinatorics Problem Let m,n ě 2 be integers, let X be a set with n elements, and let X1,X2,...,Xm be pairwise distinct non-empty, not necessary disjoint subsets of X. A function f : X Ñ t1,2,...,n ` 1u is called nice if there exists an index k such that ÿ xPXk fpxq ą ÿ xPXi fpxq for all i ‰ k. Prove that the number...

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IMO 2019 Shortlist G2

IMO 2019 Shortlist G2 Category: Geometry Problem Let ABC be an acute-angled triangle and let D, E, and F be the feet of altitudes from A, B, and C to sides BC, CA, and AB, respectively. Denote by ωB and ωC the incircles of triangles BDF and CDE, and let these circles be tangent to segments DF and DE at M and N, respectively. Let line MN meet circles ωB...

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IMO 2010 Shortlist A3

IMO 2010 Shortlist A3 Category: Algebra Problem Let x1,...,x100 be nonnegative real numbers such that xi xi 1 xi 2 ¨ 1 for all i 1,...,100 (we put x101 x1, x102 x2). Find the maximal possible value of the sum S 100 ¸ i1 xixi 2. (Russia) Answer. .

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IMO 1971

IMO 1971 Official IMO 1971 problems · 6/6 solved, 2 verified. # Status Time 1 solved 43m36s 2 solved 9m05s 3 ✓ verified 3m43s 4 solved 12m37s 5 solved 7m23s 6 ✓ verified 24m01s Problem 1 solved · 43m36s · Solution → Prove that the following assertion is true for $n = 3$ and $n = 5$, and that it is false for every other natural number $n > 2$:...

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IMO 2008 Shortlist N5

IMO 2008 Shortlist N5 Category: Number Theory Problem For every n ∈ N let d(n) denote the number of (positive) divisors of n. Find all func- tions f : N → N with the following properties: (i) d(f(x)) = x for all x ∈ N; (ii) f(xy) divides (x − 1)yxy−1 f(x) for all x,y ∈ N.

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IMO 1999

IMO 1999 Official IMO 1999 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition: For any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an...

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IMO 2012 Shortlist G8

IMO 2012 Shortlist G8 Category: Geometry Problem Let ABC be a triangle with circumcircle ω and ℓ a line without common points with ω. Denote by P the foot of the perpendicular from the center of ω to ℓ. The side-lines BC,CA,AB intersect ℓ at the points X,Y,Z different from P. Prove that the circumcircles of the triangles AXP,BY P and CZP have a common point different from P or...

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IMO 2008 Shortlist N2

IMO 2008 Shortlist N2 Category: Number Theory Problem Let a1,a2,...,an be distinct positive integers, n ≥ 3. Prove that there exist distinct indices i and j such that ai + aj does not divide any of the numbers 3a1,3a2,...,3an.

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IMO 2024 Shortlist N3

IMO 2024 Shortlist N3 Category: Number Theory Problem Determine all sequences a1, a2, ... of positive integers such that, for any pair of positive integers m ď n, the arithmetic and geometric means am am 1 ¨¨¨ an n ´ m 1 and pamam 1 ¨¨¨anq n´m`1 are both integers. (Singapore)

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IMO 2011 Shortlist C1

IMO 2011 Shortlist C1 Category: Combinatorics Problem Let n > 0 be an integer. We are given a balance and n weights of weight 20 , 21 ,...,2n−1 . In a sequence of n moves we place all weights on the balance. In the first move we choose a weight and put it on the left pan. In each of the following moves we choose one of the remaining weights...

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IMO 2018 Shortlist G4

IMO 2018 Shortlist G4 Category: Geometry Problem A point T is hosen inside a triangle ABC. Let A1, B1, and C1 be the ree tions of T in BC, CA, and AB, respe tively. Let Ω be the ir um ir le of the triangle A1B1C1. The lines A1T, B1T, and C1T meet Ω again at A2, B2, and C2, respe tively. Prove that the lines AA2, BB2, and CC2...

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IMO 2010 Shortlist A2

IMO 2010 Shortlist A2 Category: Algebra Problem Let the real numbers a,b,c,d satisfy the relations a b c d 6 and a2 b2 c2 d2 12. Prove that 36 ¨4pa3 b3 c3 d3 qpa4 b4 c4 d4 q¨48. (Ukraine)

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IMO 2007

IMO 2007 Official IMO 2007 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Real numbers $a_1, a_2, \dots , a_n$ are given. For each $i$ ($1\le i\le n$) define $$ d_i=\max{a_j:1\le j\le i}-\min{a_j:i\le j\le n} $$ and let $$ d=\max{d_i:1\le i\le n}. $$ (a) Prove that, for any real numbers $x_1\le...

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IMO 2021 Shortlist C1

IMO 2021 Shortlist C1 Category: Combinatorics Problem Let S be an infiniteset of positive integers, such that there exist four pairwise distinct a,b,c,d PS with gcdpa,bq‰gcdpc,dq. Prove that there exist three pairwise distinct x,y,z PS such that gcdpx,yq“gcdpy,zq‰gcdpz,xq.

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IMO 2010 Shortlist A7

IMO 2010 Shortlist A7 Category: Algebra Problem Let a1,...,ar be positive real numbers. For n ¡r, we inductively define an max 1¨k¨n1 pak ankq. (1) Prove that there exist positive integers ℓ ¨r and N such that an anℓ aℓ for all n ©N. (Iran)

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IMO 2015 Shortlist G2

IMO 2015 Shortlist G2 Category: Geometry Problem Let ABC be a triangle inscribed into a circle Ω with center O. A circle Γ with center A meets the side BC at points D and E such that D lies between B and E. Moreover, let F and G be the common points of Γ and Ω. We assume that F lies on the arc AB of Ω not containing C,...

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IMO 2007 Shortlist A4

IMO 2007 Shortlist A4 Category: Algebra Problem Find all functions f : R+ → R+ such that f x + f(y) = f(x + y) + f(y) (1) for all x,y ∈ R+ . (Symbol R+ denotes the set of all positive real numbers.) (Thaliand)

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IMO 2020 Shortlist C5

IMO 2020 Shortlist C5 Category: Combinatorics Problem Let p be an odd prime, and put N “ 1 pp3 ´ pq ´ 1. The numbers 1,2,...,N are painted arbitrarily in two colors, red and blue. For any positive integer n ď N, denote by rpnq the fraction of integers in t1,2,...,nu that are red. Prove that there exists a positive integer a P t1,2,...,p ´ 1u such that rpnq ‰...

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IMO 2014 Shortlist G1

IMO 2014 Shortlist G1 Category: Geometry Problem The points P and Q are chosen on the side BC of an acute-angled triangle ABC so that =PAB “ =ACB and =QAC “ =CBA. The points M and N are taken on the rays AP and AQ, respectively, so that AP “ PM and AQ “ QN. Prove that the lines BM and CN intersect on the circumcircle of the triangle ABC....

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IMO 2024 Shortlist G7

IMO 2024 Shortlist G7 Category: Geometry Problem Let ABC be a triangle with incentre I such that AB ă AC ă BC. The second intersections of AI, BI, and CI with the circumcircle of triangle ABC are MA, MB, and MC, respectively. Lines AI and BC intersect at D and lines BMC and CMB intersect at X. Suppose the circumcircles of triangles XMBMC and XBC intersect again at S ‰...

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IMO 2022 Shortlist G2

IMO 2022 Shortlist G2 Category: Geometry Problem In the acute-angled triangle ABC, the point F is the foot of the altitude from A, and P is a point on the segment AF. The lines through P parallel to AC and AB meet BC at D and E, respectively. Points X ‰ A and Y ‰ A lie on the circles ABD and ACE, respectively, such that DA “ DX and...

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IMO 2021 Shortlist A1

IMO 2021 Shortlist A1 Category: Algebra Problem Let n be an integer, and let A be a subset of t0,1,2,3,...,5n u consisting of 4n 2 numbers. Prove that there exist a,b,c PA such that a ăb ăc and c 2a ą3b.

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IMO 2006 Shortlist G1

IMO 2006 Shortlist G1 Category: Geometry Problem Let ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies ∠PBA + ∠PCA = ∠PBC + ∠PCB. Show that AP ≥ AI and that equality holds if and only if P coincides with I. (Korea)

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IMO 2023 Shortlist N6

IMO 2023 Shortlist N6 Category: Number Theory Problem A sequence of integers a0,a1,a2,... is called kawaii, if a0 “ 0,a1 “ 1, and, for any positive integer n, we have pan 1 ´ 3an 2an´1qpan 1 ´ 4an 3an´1q “ 0. An integer is called kawaii if it belongs to a kawaii sequence. Suppose that two consecutive positive integers m and m`1 are both kawaii (not necessarily belonging to the...

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IMO 2011 Shortlist C6

IMO 2011 Shortlist C6 Category: Combinatorics Problem Let n be a positive integer and let W = ...x−1x0x1x2 ... be an infinite periodic word consisting of the letters a and b. Suppose that the minimal period N of W is greater than 2n . A finite nonempty word U is said to appear in W if there exist indices k ≤ ℓ such that U = xkxk+1 ...xℓ. A finite...

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IMO 2017 Shortlist C8

IMO 2017 Shortlist C8 Category: Combinatorics Problem Let n be a given positive integer. In the Cartesian plane, ea h latti e point with nonnegative oordinates initially ontains a buttery, and there are no other butter- ies. The neighborhood of a latti e point c onsists of all latti e points within the axis-aligned p2n 1qˆp2n 1q square entered at c, apart from c itself. We all a buttery lonely,...

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IMO 2021 Shortlist A2

IMO 2021 Shortlist A2 Category: Algebra Problem Forevery integern ě1 consider the nˆn tablewithentry Z ij n `1 ^ atthe intersection of row i and column j, for every i “1,...,n and j “1,...,n. Determine all integers n ě1 for which the sum of the n2 entries in the table is equal to 1 n2 pn ´1q.

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IMO 2013 Shortlist N4

IMO 2013 Shortlist N4 Category: Number Theory Problem Determine whether there exists an infinite sequence of nonzero digits a1,a2,a3,... and a positive integer N such that for every integer k ą N, the number akak´1 ...a1 is a perfect square. (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A2

IMO 2009 Shortlist A2 Category: Algebra Problem Let a, b, c be positive real numbers such that a + b + c = a + b + c. Prove that (2a + b + c)2 + (2b + c + a)2 + (2c + a + b)2 ≤ .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist A5

IMO 2012 Shortlist A5 Category: Algebra Problem Find all functions f : R → R that satisfy the conditions f(1 + xy) − f(x + y) = f(x)f(y) for all x,y ∈ R and f(−1) 6= 0. Solution. The only solution is the function f(x) = x − 1, x ∈ R. We set g(x) = f(x)+1 and show that g(x) = x for all real x. The conditions take...

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IMO 2017 Shortlist C2

IMO 2017 Shortlist C2 Category: Combinatorics Problem Let n be a positive integer. Dene a hameleon to be any sequen e of 3n letters, with exa tly n o urren es of ea h of the letters a, b, and c. Dene a swap to be the transposition of two adja ent letters in a hameleon. Prove that for any hameleon X, there exists a hameleon Y su h that...

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IMO 2020 Shortlist G9

IMO 2020 Shortlist G9 Category: Geometry Problem Prove that there exists a positive constant c such that the following statement is true: Assume that n is an integer with n ě 2, and let S be a set of n points in the plane such that the distance between any two distinct points in S is at least 1. Then there is a line ℓ separating S such that the...

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IMO 2023 Shortlist C1

IMO 2023 Shortlist C1 Category: Combinatorics Problem Let m and n be positive integers greater than 1. In each unit square of an mˆn grid lies a coin with its tail-side up. A move consists of the following steps: select a 2 ˆ 2 square in the grid; flip the coins in the top-left and bottom-right unit squares; flip the coin in either the top-right or bottom-left unit square. Determine...

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IMO 2019 Shortlist C9

IMO 2019 Shortlist C9 Category: Combinatorics Problem For any two diﬀerent real numbers x and y, we deﬁne Dpx,yq to be the unique integer d satisfying 2d ď |x ´ y| ă 2d`1 . Given a set of reals F, and an element x P F, we say that the scales of x in F are the values of Dpx,yq for y P F with x ‰ y. Let k...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G4

IMO 2006 Shortlist G4 Category: Geometry Problem A point D is chosen on the side AC of a triangle ABC with ∠C < ∠A < 90◦ in such a way that BD = BA. The incircle of ABC is tangent to AB and AC at points K and L, respectively. Let J be the incentre of triangle BCD. Prove that the line KL intersects the line segment AJ at its...

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IMO 2022 Shortlist G7

IMO 2022 Shortlist G7 Category: Geometry Problem Let ABC and A1 B1 C1 be two triangles having the same circumcircle ω, and the same orthocentre H. Let Ω be the circumcircle of the triangle determined by the lines AA1 , BB1 and CC1 . Prove that H, the centre of ω, and the centre of Ω are collinear. (Denmark)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist G7

IMO 2021 Shortlist G7 Category: Geometry Problem A point D is chosen inside an acute-angled triangle ABC with AB ą AC so that =BAD “ =DAC. A point E is constructed on the segment AC so that =ADE “ =DCB. Similarly, a point F is constructed on the segment AB so that =ADF “ =DBC. A point X is chosen on the line AC so that CX “ BX. Let...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist C2

IMO 2012 Shortlist C2 Category: Combinatorics Problem Let n ≥ 1 be an integer. What is the maximum number of disjoint pairs of elements of the set {1,2,...,n} such that the sums of the different pairs are different integers not exceeding n? Solution. Consider x such pairs in {1,2,...,n}. The sum S of the 2x numbers in them is at least 1+2+···+2x since the pairs are disjoint. On the other...

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IMO 2020 Shortlist G3

IMO 2020 Shortlist G3 Category: Geometry Problem Let ABCD be a convex quadrilateral with =ABC ą 90˝ , =CDA ą 90˝ , and =DAB “ =BCD. Denote by E and F the reflections of A in lines BC and CD, respectively. Suppose that the segments AE and AF meet the line BD at K and L, respectively. Prove that the circumcircles of triangles BEK and DFL are tangent to each...

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IMO 2024 Shortlist N7

IMO 2024 Shortlist N7 Category: Number Theory Problem Let Zą0 denote the set of positive integers. Let f : Zą0 Ñ Zą0 be a function satisfying the following property: for m, n P Zą0, the equation fpmnq2 “ fpm2 qfpfpnqqfpmfpnqq holds if and only if m and n are coprime. For each positive integer n, determine all the possible values of fpnq. (Japan) 10 Bath, United Kingdom, 10th –22nd July...

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IMO 1992

IMO 1992 Official IMO 1992 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Find all integers $a$, $b$, $c$ satisfying $1 < a < b < c$ such that $(a - 1)(b -1)(c - 1)$ is a divisor of $abc - 1$. Problem 2 Let $\mathbb{R}$ denote the set of all...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist A6

IMO 2006 Shortlist A6 Category: Algebra Problem Determine the smallest number M such that the inequality ab(a2 − b2 ) + bc(b2 − c2 ) + ca(c2 − a2 ) ≤ M a2 b2 c2 2 holds for all real numbers a, b, c. (Ireland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G9

IMO 2006 Shortlist G9 Category: Geometry Problem Points A1,B1,C1 are chosen on the sides BC, CA, AB of a triangle ABC, respectively. The circumcircles of triangles AB1C1, BC1A1, CA1B1 intersect the circumcircle of triangle ABC again at points A2,B2,C2, respectively (A2 6= A, B2 6= B, C2 6= C). Points A3,B3,C3 are symmetric to A1,B1,C1 with respect to the midpoints of the sides BC, CA, AB respectively. Prove that the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C6

IMO 2010 Shortlist C6 Category: Combinatorics Problem Given a positive integer k and other two integers b ¡ w ¡ 1. There are two strings of pearls, a string of b black pearls and a string of w white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step: (i) The strings are...

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IMO 2022 Shortlist A2

IMO 2022 Shortlist A2 Category: Algebra Problem Let k ě 2 be an integer. Find the smallest integer n ě k ` 1 with the property that there exists a set of n distinct real numbers such that each of its elements can be written as a sum of k other distinct elements of the set. (Slovakia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist G7

IMO 2018 Shortlist G7 Category: Geometry Problem Let O be the ir um entre, and Ω be the ir um ir le of an a ute-angled triangle ABC. Let P be an arbitrary point on Ω, distin t from A, B, C, and their antipodes in Ω. Denote the ir um entres of the triangles AOP, BOP, and COP by OA, OB, and OC, respe tively. The lines ℓA, ℓB,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A7

IMO 2008 Shortlist A7 Category: Algebra Problem Prove that for any four positive real numbers a,b,c,d the inequality (a − b)(a − c) a + b + c + (b − c)(b − d) b + c + d + (c − d)(c − a) c + d + a + (d − a)(d − b) d + a + b ≥ 0 holds. Determine all cases of equality.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist C7

IMO 2015 Shortlist C7 Category: Combinatorics Problem In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least 3, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most 2015 unsociable groups, prove that it is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist A5

IMO 2015 Shortlist A5 Category: Algebra Problem Let 2Z 1 denote the set of odd integers. Find all functions f : Z Ñ 2Z 1 satisfying f x fpxq y ˘ f x ´ fpxq ´ y ˘ “ fpx yq ` fpx ´ yq for every x,y P Z. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist A5

IMO 2010 Shortlist A5 Category: Algebra Problem Denote by Q the set of all positive rational numbers. Determine all functions f : Q ÑQ which satisfy the following equation for all x,y PQ : f fpxq2 y x3 fpxyq. (1) (Switzerland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist G7

IMO 2020 Shortlist G7 Category: Geometry Problem Let P be a point on the circumcircle of an acute-angled triangle ABC. Let D, E, and F be the reflections of P in the midlines of triangle ABC parallel to BC, CA, and AB, respectively. Denote by ωA, ωB, and ωC the circumcircles of triangles ADP, BEP, and CFP, respectively. Denote by ω the circumcircle of the triangle formed by the perpendicular...

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IMO 2022 Shortlist A8

IMO 2022 Shortlist A8 Category: Algebra Problem For a positive integer n, an n-sequence is a sequence pa0,...,anq of non-negative integers satisfying the following condition: if i and j are non-negative integers with i j ď n, then ai aj ď n and aai aj “ ai j. Let fpnq be the number of n-sequences. Prove that there exist positive real numbers c1, c2 and λ such that c1λn ă...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G5

IMO 2008 Shortlist G5 Category: Geometry Problem Let k and n be integers with 0 ≤ k ≤ n−2. Consider a set L of n lines in the plane such that no two of them are parallel and no three have a common point. Denote by I the set of intersection points of lines in L. Let O be a point in the plane not lying on any line of...

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IMO 2023 Shortlist C3

IMO 2023 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. We arrange 1 2 ¨¨¨ ` n circles in a triangle with n rows, such that the ith row contains exactly i circles. The following figure shows the case n “ 6. n “ 6 In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N8

IMO 2012 Shortlist N8 Category: Number Theory Problem Prove that for every prime p > 100 and every integer r there exist two integers a and b such that p divides a2 b5 − r. Solution 1. Throughout the solution, all congruence relations are meant modulo p. Fix p, and let P = {0,1,...,p−1} be the set of residue classes modulo p. For every r ∈ P, let Sr =...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N5

IMO 2017 Shortlist N5 Category: Number Theory Problem Find all pairs pp,qq of prime numbers with p ą q for whi h the number pp qqp q pp ´ qqp´q ´ 1 pp qqp´qpp ´ qqp q ´ 1 is an integer. (Japan) Shortlisted problems 11

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist C2

IMO 2016 Shortlist C2 Category: Combinatorics Problem Find all positive integers n for which all positive divisors of n can be put into the cells of a rectangular table under the following constraints: • each cell contains a distinct divisor; • the sums of all rows are equal; and • the sums of all columns are equal.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist A3

IMO 2014 Shortlist A3 Category: Algebra Problem For a sequence x1,x2,...,xn of real numbers, we define its price as max 1ďiďn |x1 ¨¨¨ xi|. Given n real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price D. Greedy George, on the other hand, chooses x1 such that |x1| is as small as...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist C7

IMO 2023 Shortlist C7 Category: Combinatorics Problem The Imomi archipelago consists of n ě 2 islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of k companies. It is known that if any one of the k companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist C8

IMO 2022 Shortlist C8 Category: Combinatorics Problem Alice fills the fields of an n ˆ n board with numbers from 1 to n2 , each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that: (i) The first field in the sequence is one that is only adjacent...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist A7

IMO 2017 Shortlist A7 Category: Algebra Problem Let a0,a1,a2,... be a sequen e of integers and b0,b1,b2,... be a sequen e of positive integers su h that a0 “ 0,a1 “ 1, and an`1 “ anbn ` an´1, if bn´1 “ 1 anbn ´ an´1, if bn´1 ą 1 for n “ 1,2,.... Prove that at least one of the two numbers a2017 and a2018 must be greater than or...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist A1

IMO 2022 Shortlist A1 Category: Algebra Problem Let panqně1 be a sequence of positive real numbers with the property that pan 1q2 anan 2 ď an an`2 for all positive integers n. Show that a2022 ď 1. (Nigeria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A6

IMO 2018 Shortlist A6 Category: Algebra Problem Let m,n ě 2 be integers. Let fpx1,...,xnq be a polynomial with real oe ients su h that fpx1,...,xnq “ Yx1 ... xn m ] for every x1,...,xn P 0,1,...,m ´ 1 ( . Prove that the total degree of f is at least n. (Brazil)

Practice › Mathematics › IMO ›

IMO 1966

IMO 1966 Official IMO 1966 problems · 1/6 solved. # Status Time 1 — — 2 — — 3 solved 11m33s 4 — — 5 — — 6 — — Problem 1 In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A6

IMO 2008 Shortlist A6 Category: Algebra Problem Let f : R → N be a function which satisfies f x + f(y) = f y + f(x) for all x,y ∈ R. (1) Prove that there is a positive integer which is not a value of f.

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IMO 2018 Shortlist N4

IMO 2018 Shortlist N4 Category: Number Theory Problem Let a1, a2, ..., an, ... be a sequen e of positive integers su h that a1 a2 a2 a3 ¨¨¨ an´1 an an a1 is an integer for all n ě k, where k is some positive integer. Prove that there exists a positive integer m su h that an “ an`1 for all n ě m. (Mongolia)

Practice › Mathematics › IMO ›

IMO 1982

IMO 1982 Official IMO 1982 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $$ f(m+n)-f(m)-f(n)=0 \text{ or } 1. $$ Determine $f(1982)$. Problem 2 A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$,...

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IMO 2024 Shortlist A3

IMO 2024 Shortlist A3 Category: Algebra Problem Decide whether for every sequence panq of positive real numbers, 3a1 3a2 ¨¨¨ 3an p2a1 2a2 ¨¨¨ 2anq2 ă is true for at least one positive integer n. (China)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist G6

IMO 2022 Shortlist G6 Category: Geometry Problem In an acute-angled triangle ABC, point H is the foot of the altitude from A. Let P be a moving point such that the bisectors k and of angles PBC and PCB, respectively, intersect each other on the line segment AH. Let k and AC meet at E, let and AB meet at F, and let EF and AH meet at Q. Prove...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist C1

IMO 2018 Shortlist C1 Category: Combinatorics Problem Let n ě 3 be an integer. Prove that there exists a set S of 2n positive integers satisfying the following property: For every m “ 2,3,...,n the set S an be partitioned into two subsets with equal sums of elements, with one of subsets of ardinality m. (I eland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N1

IMO 2015 Shortlist N1 Category: Number Theory Problem Determine all positive integers M for which the sequence a0,a1,a2,..., defined by a0 “ 2M 1 and ak 1 “ aktaku for k “ 0,1,2,..., contains at least one integer term. (Luxembourg)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist A1

IMO 2019 Shortlist A1 Category: Algebra Problem Let Z be the set of integers. Determine all functions f : Z Ñ Z such that, for all integers a and b, fp2aq 2fpbq “ fpfpa bqq. (South Africa)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N1

IMO 2022 Shortlist N1 Category: Number Theory Problem A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) (Cyprus)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N7

IMO 2012 Shortlist N7 Category: Number Theory Problem Find all n ∈ N for which there exist nonnegative integers a1,a2,...,an such that 2a1 + 2a2 ··· + 2an = 3a1 3a2 ··· + n 3an = 1. Solution. Such numbers a1,a2,...,an exist if and only if n ≡ 1 (mod 4) or n ≡ 2 (mod 4). Let Pn k=1 k 3ak = 1 with a1,a2,...,an nonnegative integers. Then 1·x1+2·x2+···+n·xn...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A6

IMO 2007 Shortlist A6 Category: Algebra Problem Let a1,a2,...,a100 be nonnegative real numbers such that a2 1 +a2 2 +...+a2 100 = 1. Prove that a2 1a2 + a2 2a3 + ... + a2 100a1 < . (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist G7

IMO 2019 Shortlist G7 Category: Geometry Problem The incircle ω of acute-angled scalene triangle ABC has centre I and meets sides BC, CA, and AB at D, E, and F, respectively. The line through D perpendicular to EF meets ω again at R. Line AR meets ω again at P. The circumcircles of triangles PCE and PBF meet again at Q ‰ P. Prove that lines DI and PQ meet...

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IMO 2019 Shortlist G3

IMO 2019 Shortlist G3 Category: Geometry Problem In triangle ABC, let A1 and B1 be two points on sides BC and AC, and let P and Q be two points on segments AA1 and BB1, respectively, so that line PQ is parallel to AB. On ray PB1, beyond B1, let P1 be a point so that =PP1C “ =BAC. Similarly, on ray QA1, beyond A1, let Q1 be a point...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G1

IMO 2010 Shortlist G1 Category: Geometry Problem Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N2

IMO 2010 Shortlist N2 Category: Number Theory Problem Find all pairs pm,nq of nonnegative integers for which m2 2 3n m 2n 1 1 . (1) (Australia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G3

IMO 2006 Shortlist G3 Category: Geometry Problem Let ABCDE be a convex pentagon such that ∠BAC = ∠CAD = ∠DAE and ∠ABC = ∠ACD = ∠ADE. The diagonals BD and CE meet at P. Prove that the line AP bisects the side CD. (USA)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C6

IMO 2007 Shortlist C6 Category: Combinatorics Problem In a mathematical competition some competitors are friends; friendship is always mutual. Call a group of competitors a clique if each two of them are friends. The number of members in a clique is called its size. It is known that the largest size of cliques is even. Prove that the competitors can be arranged in two rooms such that the largest size...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C6

IMO 2006 Shortlist C6 Category: Combinatorics Problem A holey triangle is an upward equilateral triangle of side length n with n upward unit triangular holes cut out. A diamond is a 60◦ −120◦ unit rhombus. Prove that a holey triangle T can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length k in T contains at most k holes, for...

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IMO 2024 Shortlist N2

IMO 2024 Shortlist N2 Category: Number Theory Problem Determine all finite, nonempty sets S of positive integers such that for every a, b P S there exists c P S with a | b ` 2c. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C4

IMO 2021 Shortlist C4 Category: Combinatorics Problem ThekingdomofAnisotropyconsistsofn cities. Foreverytwocitiesthereexistsexactly one direct one-way road between them. We say that a path from X to Y is a sequence of roads such that one can move from X to Y along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection. Let A...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G7

IMO 2009 Shortlist G7 Category: Geometry Problem Let ABC be a triangle with incenter I and let X, Y and Z be the incenters of the triangles BIC, CIA and AIB, respectively. Let the triangle XY Z be equilateral. Prove that ABC is equilateral too.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N8

IMO 2023 Shortlist N8 Category: Number Theory Problem Let Zą0 be the set of positive integers. Determine all functions f : Zą0 Ñ Zą0 such that fbfpaq pa 1q “ pa 1qfpbq holds for all a,b P Zą0, where fk pnq “ fpfp¨¨¨fpnq¨¨¨qq denotes the composition of f with itself k times. (Taiwan) 10 Chiba, Japan, 2nd–13th July 2023 Shortlisted problems – solutions 11

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist C7

IMO 2019 Shortlist C7 Category: Combinatorics Problem There are 60 empty boxes B1, ..., B60 in a row on a table and an unlimited supply of pebbles. Given a positive integer n, Alice and Bob play the following game. In the ﬁrst round, Alice takes n pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer k...

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IMO 2015 Shortlist C6

IMO 2015 Shortlist C6 Category: Combinatorics Problem Let S be a nonempty set of positive integers. We say that a positive integer n is clean if it has a unique representation as a sum of an odd number of distinct elements from S. Prove that there exist infinitely many positive integers that are not clean. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G2

IMO 2008 Shortlist G2 Category: Geometry Problem Given trapezoid ABCD with parallel sides AB and CD, assume that there exist points E on line BC outside segment BC, and F inside segment AD, such that ∠DAE = ∠CBF. Denote by I the point of intersection of CD and EF, and by J the point of intersection of AB and EF. Let K be the midpoint of segment EF; assume it...

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IMO 2009 Shortlist C4

IMO 2009 Shortlist C4 Category: Combinatorics Problem For an integer m ≥ 1, we consider partitions of a 2m ×2m chessboard into rectangles consisting of cells of the chessboard, in which each of the 2m cells along one diagonal forms a separate rectangle of side length 1. Determine the smallest possible sum of rectangle perimeters in such a partition.

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IMO 2022 Shortlist C3

IMO 2022 Shortlist C3 Category: Combinatorics Problem In each square of a garden shaped like a 2022 ˆ 2022 board, there is initially a tree of height 0. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: • The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at...

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IMO 2018 Shortlist N7

IMO 2018 Shortlist N7 Category: Number Theory Problem Let n ě 2018 be an integer, and let a1,a2,...,an,b1,b2,...,bn be pairwise distin t positive integers not ex eeding 5n. Suppose that the sequen e a1 b1 , a2 b2 , ..., an bn forms an arithmeti progression. Prove that the terms of the sequen e are equal. (Thailand) 8 Cluj-Napo a Romania, 314 July 2018 Shortlisted problems solutions 9

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IMO 2015 Shortlist G5

IMO 2015 Shortlist G5 Category: Geometry Problem Let ABC be a triangle with CA ‰ CB. Let D, F, and G be the midpoints of the sides AB, AC, and BC, respectively. A circle Γ passing through C and tangent to AB at D meets the segments AF and BG at H and I, respectively. The points H1 and I1 are symmetric to H and I about F and G,...

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IMO 2012 Shortlist A7

IMO 2012 Shortlist A7 Category: Algebra Problem We say that a function f : Rk → R is a metapolynomial if, for some positive integers m and n, it can be represented in the form f(x1,...,xk) = max i=1,...,m min j=1,...,n Pi,j(x1,...,xk) where Pi,j are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial. Solution. We use the notation f(x) = f(x1,...,xk) for x = (x1,...,xk)...

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IMO 1973

IMO 1973 Official IMO 1973 problems · 6/6 solved, 5 verified. # Status Time 1 ✓ verified 22m51s 2 ✓ verified 14m44s 3 ✓ verified 11m52s 4 solved 33m43s 5 ✓ verified 36m23s 6 ✓ verified 22m47s Problem 1 ✓ verified · 22m51s · Solution → Point $O$ lies on line $g$; $\overrightarrow{OP_1}, \overrightarrow{OP_2}, \ldots, \overrightarrow{OP_n}$ are unit vectors such that points $P_1, P_2, \ldots, P_n$ all lie in a...

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IMO 2023 Shortlist A5

IMO 2023 Shortlist A5 Category: Algebra Problem Let a1,a2,...,a2023 be positive integers such that • a1,a2,...,a2023 is a permutation of 1,2,...,2023, and • |a1 ´ a2|,|a2 ´ a3|,...,|a2022 ´ a2023| is a permutation of 1,2,...,2022. Prove that max ` a1,a2023 ˘ ě 507. (Australia) 4 Chiba, Japan, 2nd–13th July 2023

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IMO 1998

IMO 1998 Official IMO 1998 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside...

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IMO 2023

IMO 2023 Official IMO 2023 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Determine all composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\dots,d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\dots<d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1\le i \le k-2$. Problem 2 Let $ABC$ be an acute-angled...

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IMO 2006 Shortlist N3

IMO 2006 Shortlist N3 Category: Number Theory Problem The sequence f(1),f(2),f(3),... is defined by f(n) = n jn k + jn k ··· + jn n k , where bxc denotes the integer part of x. (a) Prove that f(n + 1) > f(n) infinitely often. (b) Prove that f(n + 1) < f(n) infinitely often. (South Africa)

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IMO 2016 Shortlist A1

IMO 2016 Shortlist A1 Category: Algebra Problem Let a,b and c be positive real numbers such that min{ab,bc,ca} ⩾ 1. Prove that » (a2 + 1)(b2 + 1)(c2 + 1) ⩽ Ç a + b + c å2

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IMO 2020 Shortlist

IMO 2020 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry G9 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number...

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IMO 2022 Shortlist N8

IMO 2022 Shortlist N8 Category: Number Theory Problem Prove that 5n ´ 3n is not divisible by 2n ` 65 for any positive integer n. (Belgium) 8 Oslo, Norway, 6th–16th July 2022 Shortlisted problems – solutions 9

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G4

IMO 2016 Shortlist G4 Category: Geometry Problem Let ABC be a triangle with AB = AC 6= BC and let I be its incentre. The line BI meets AC at D, and the line through D perpendicular to AC meets AI at E. Prove that the reflection of I in AC lies on the circumcircle of triangle BDE.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A1

IMO 2007 Shortlist A1 Category: Algebra Problem Given a sequence a1,a2,...,an of real numbers. For each i (1 ≤ i ≤ n) define di = max{aj : 1 ≤ j ≤ i} − min{aj : i ≤ j ≤ n} and let d = max{di : 1 ≤ i ≤ n}. (a) Prove that for arbitrary real numbers x1 ≤ x2 ≤ ... ≤ xn, max |xi − ai|...

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IMO 2014 Shortlist N5

IMO 2014 Shortlist N5 Category: Number Theory Problem Find all triples pp,x,yq consisting of a prime number p and two positive integers x and y such that xp´1 y and x yp´1 are both powers of p. (Belgium)

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IMO 1986

IMO 1986 Official IMO 1986 problems · 1/6 solved. # Status Time 1 solved 10m25s 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 solved · 10m25s · Solution → Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set ${2,5,13,d}$ such that $ab-1$ is not a perfect square....

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IMO 2011 Shortlist C4

IMO 2011 Shortlist C4 Category: Combinatorics Problem Determine the greatest positive integer k that satisfies the following property: The set of positive integers can be partitioned into k subsets A1,A2,...,Ak such that for all integers n ≥ 15 and all i ∈ {1,2,...,k} there exist two distinct elements of Ai whose sum is n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N2

IMO 2012 Shortlist N2 Category: Number Theory Problem Find all triples (x,y,z) of positive integers such that x ≤ y ≤ z and x3 (y3 z3 ) = 2012(xyz + 2). Solution. First note that x divides 2012·2 = 23 ·503. If 503 | x then the right-hand side of the equation is divisible by 5033 , and it follows that 5032 | xyz + 2. This is false as...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C5

IMO 2006 Shortlist C5 Category: Combinatorics Problem An (n,k)-tournament is a contest with n players held in k rounds such that: (i) Each player plays in each round, and every two players meet at most once. (ii) If player A meets player B in round i, player C meets player D in round i, and player A meets player C in round j, then player B meets player D in...

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IMO 2011

IMO 2011 Official IMO 2011 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 Given any set $A = {a_1, a_2, a_3, a_4}$ of four distinct positive integers, let $s(A)=a_1+a_2+a_3+a_4$ be the sum of the positive integers in $A$. Let $n(A)$ denote the number of pairs $(i,j)$ with $1 \le i <...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist C7

IMO 2013 Shortlist C7 Category: Combinatorics Problem Let n ě 2 be an integer. Consider all circular arrangements of the numbers 0,1,...,n; the n 1 rotations of an arrangement are considered to be equal. A circular arrangement is called beautiful if, for any four distinct numbers 0 ď a,b,c,d ď n with a c “ b d, the chord joining numbers a and c does not intersect the chord joining...

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IMO 2002

IMO 2002 Official IMO 2002 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 $S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h + k < n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h' \le h,k' \le...

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IMO 2007 Shortlist G5

IMO 2007 Shortlist G5 Category: Geometry Problem Let ABC be a fixed triangle, and let A1, B1, C1 be the midpoints of sides BC, CA, AB, respectively. Let P be a variable point on the circumcircle. Let lines PA1, PB1, PC1 meet the circumcircle again at A0 , B0 , C0 respectively. Assume that the points A, B, C, A0 , B0 , C0 are distinct, and lines AA0 ,...

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IMO 2013 Shortlist

IMO 2013 Shortlist 27 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory N7 Number Theory

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G6

IMO 2007 Shortlist G6 Category: Geometry Problem Determine the smallest positive real number k with the following property. Let ABCD be a convex quadrilateral, and let points A1, B1, C1 and D1 lie on sides AB, BC, CD and DA, respectively. Consider the areas of triangles AA1D1, BB1A1, CC1B1, and DD1C1; let S be the sum of the two smallest ones, and let S1 be the area of quadrilateral A1B1C1D1....

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IMO 2020 Shortlist N5

IMO 2020 Shortlist N5 Category: Number Theory Problem Determine all functions f defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: piq fpnq ‰ 0 for at least one n; piiq fpxyq “ fpxq ` fpyq for every positive integers x and y; piiiq there are infinitely many positive integers n such that fpkq “ fpn ´ kq for all k ă...

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IMO 2010 Shortlist N11

IMO 2010 Shortlist N11 Category: Number Theory Problem . Same as Problem N1, but the constant is replaced by . (Canada) Answer for Problem N1. n 39. Solution for Problem N1. Suppose that for some n there exist the desired numbers; we may assume that s1 s2 sn. Surely s1 ¡ 1 since otherwise 1 1 s1 0. So we have 2 ¨s1 ¨s2 1 ¨¨sn pn...

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IMO 1964

IMO 1964 Official IMO 1964 problems · 1/6 solved, 1 verified. # Status Time 1 — — 2 — — 3 ✓ verified 3m04s 4 — — 5 — — 6 — — Problem 1 (a) Find all positive integers $n$ for which $2^n-1$ is divisible by $7$. (b) Prove that there is no positive integer $n$ for which $2^n+1$ is divisible by $7$. Problem 2 Suppose $a, b, c$...

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IMO 2020 Shortlist A3

IMO 2020 Shortlist A3 Category: Algebra Problem Suppose that a, b, c, d are positive real numbers satisfying pa cqpb dq “ ac bd. Find the smallest possible value of a b b c c d d a . (Israel)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C5

IMO 2008 Shortlist C5 Category: Combinatorics Problem Let S = {x1,x2,...,xk+ } be a (k + )-element set of real numbers contained in the interval [0,1]; k and are positive integers. A k-element subset A ⊂ S is called nice if k X xi∈A xi − X xj∈S\A xj ≤ k + 2k . Prove that the number of nice subsets is at least k + k + k...

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IMO 2019 Shortlist G6

IMO 2019 Shortlist G6 Category: Geometry Problem Let I be the incentre of acute-angled triangle ABC. Let the incircle meet BC, CA, and AB at D, E, and F, respectively. Let line EF intersect the circumcircle of the triangle at P and Q, such that F lies between E and P. Prove that =DPA ` =AQD “ =QIP. (Slovakia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N6

IMO 2008 Shortlist N6 Category: Number Theory Problem Prove that there exist infinitely many positive integers n such that n2 1 has a prime divisor greater than 2n + √ 2n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist A4

IMO 2006 Shortlist A4 Category: Algebra Problem Prove the inequality X i<j aiaj ai + aj ≤ n 2(a1 + a2 + ··· + an) X i<j aiaj for positive real numbers a1,a2,...,an. (Serbia)

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IMO 1960

IMO 1960 Official IMO 1960 problems · 7/7 solved, 6 verified. # Status Time 1 solved 27m41s 2 ✓ verified 7m41s 3 ✓ verified 17m36s 4 ✓ verified 33m08s 5 ✓ verified 4m30s 6 ✓ verified 3m00s 7 ✓ verified 5m59s Problem 1 solved · 27m41s · Solution → Find all three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the...

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IMO 2017 Shortlist A1

IMO 2017 Shortlist A1 Category: Algebra Problem Let a1,a2,...,an,k, and M be positive integers su h that a1 a2 ¨¨¨ an “ k and a1a2 ...an “ M. If M ą 1, prove that the polynomial Ppxq “ Mpx 1qk ´ px a1qpx a2q¨¨¨px ` anq has no positive roots. (Trinidad and Tobago)

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IMO 2016 Shortlist A8

IMO 2016 Shortlist A8 Category: Algebra Problem Determine the largest real number a such that for all n ⩾ 1 and for all real numbers x0,x1,...,xn satisfying 0 = x0 < x1 < x2 < ··· < xn, we have x1 − x0 + x2 − x1 ··· + xn − xn−1 ⩾ a Ç x1 x2 ··· + n + 1 xn å . Shortlisted problems 5

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IMO 2020 Shortlist G6

IMO 2020 Shortlist G6 Category: Geometry Problem Let I and IA be the incenter and the A-excenter of an acute-angled triangle ABC with AB ă AC. Let the incircle meet BC at D. The line AD meets BIA and CIA at E and F, respectively. Prove that the circumcircles of triangles AID and IAEF are tangent to each other. (Slovakia) Shortlisted problems 9

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G61

IMO 2010 Shortlist G61 Category: Geometry Problem . The vertices X, Y , Z of an equilateral triangle XY Z lie respectively on the sides BC, CA, AB of a triangle ABC. Prove that if the incenter of triangle ABC lies outside triangle XY Z, then one of the angles of triangle ABC is greater than 120. (Bulgaria)

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IMO 1977

IMO 1977 Official IMO 1977 problems · 0/6 solved. # Status Time 1 — — 2 — — 3 — — 4 — — 5 — — 6 — — Problem 1 In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL,...

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IMO 2017 Shortlist G6

IMO 2017 Shortlist G6 Category: Geometry Problem Let n ě 3 be an integer. Two regular n-gons A and B are given in the plane. Prove that the verti es of A that lie inside B or on its boundary are onse utive. (That is, prove that there exists a line separating those verti es of A that lie inside B or on its boundary from the other verti es...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A2

IMO 2018 Shortlist A2 Category: Algebra Problem Find all positive integers n ě 3 for whi h there exist real numbers a1,a2,...,an, an 1 “ a1, an 2 “ a2 su h that aiai 1 1 “ ai`2 for all i “ 1,2,...,n. (Slovakia)

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IMO 2006 Shortlist A3

IMO 2006 Shortlist A3 Category: Algebra Problem The sequence c0,c1,...,cn,... is defined by c0 = 1, c1 = 0 and cn+2 = cn+1 + cn for n ≥ 0. Consider the set S of ordered pairs (x,y) for which there is a finite set J of positive integers such that x = P j∈J cj, y = P j∈J cj−1. Prove that there exist real numbers α,β and m,M with...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N2

IMO 2007 Shortlist N2 Category: Number Theory Problem Let b,n > 1 be integers. Suppose that for each k > 1 there exists an integer ak such that b − an k is divisible by k. Prove that b = An for some integer A. (Canada)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N2

IMO 2016 Shortlist N2 Category: Number Theory Problem Let τ(n) be the number of positive divisors of n. Let τ1(n) be the number of positive divisors of n which have remainders 1 when divided by 3. Find all possible integral values of the fraction τ(10n) τ1(10n) .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C1

IMO 2010 Shortlist C1 Category: Combinatorics Problem In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C4

IMO 2006 Shortlist C4 Category: Combinatorics Problem A cake has the form of an n × n square composed of n2 unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement A. Let B be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer...

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IMO 1974

IMO 1974 Official IMO 1974 problems · 5/6 solved, 4 verified. # Status Time 1 ✓ verified 10m39s 2 solved 32m46s 3 ✓ verified 5m13s 4 ✓ verified 10m53s 5 — — 6 ✓ verified 7m06s Problem 1 ✓ verified · 10m39s · Solution → Three players $A, B$ and $C$ play the following game: On each of three cards an integer is written. These three numbers $p, q, r$...

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IMO 2015 Shortlist G3

IMO 2015 Shortlist G3 Category: Geometry Problem Let ABC be a triangle with =C “ 900 , and let H be the foot of the altitude from C. A point D is chosen inside the triangle CBH so that CH bisects AD. Let P be the intersection point of the lines BD and CH. Let ω be the semicircle with diameter BD that meets the segment CB at an interior...

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IMO 2024 Shortlist G4

IMO 2024 Shortlist G4 Category: Geometry Problem Let ABCD be a quadrilateral with AB parallel to CD and AB ă CD. Lines AD and BC intersect at a point P. Point X ‰ C on the circumcircle of triangle ABC is such that PC “ PX. Point Y ‰ D on the circumcircle of triangle ABD is such that PD “ PY . Lines AX and BY intersect at Q....

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IMO 2016 Shortlist A6

IMO 2016 Shortlist A6 Category: Algebra Problem The equation (x − 1)(x − 2)···(x − 2016) = (x − 1)(x − 2)···(x − 2016) is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve...

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IMO 2022 Shortlist C4

IMO 2022 Shortlist C4 Category: Combinatorics Problem Let n ą 3 be a positive integer. Suppose that n children are arranged in a circle, and n coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbours on the right and left. Determine all initial distributions of coins from which...

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IMO 2023 Shortlist G7

IMO 2023 Shortlist G7 Category: Geometry Problem Let ABC be an acute, scalene triangle with orthocentre H. Let a be the line through the reflection of B with respect to CH and the reflection of C with respect to BH. Lines b and c are defined similarly. Suppose lines a, b, and c determine a triangle T . Prove that the orthocentre of T , the circumcentre of T and...

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IMO 2024 Shortlist C3

IMO 2024 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. There are 2n knights sitting at a round table. They consist of n pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of...

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IMO 2020 Shortlist G1

IMO 2020 Shortlist G1 Category: Geometry Problem Let ABC be an isosceles triangle with BC “ CA, and let D be a point inside side AB such that AD ă DB. Let P and Q be two points inside sides BC and CA, respectively, such that =DPB “ =DQA “ 90˝ . Let the perpendicular bisector of PQ meet line segment CQ at E, and let the circumcircles of triangles...

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IMO 1963

IMO 1963 Official IMO 1963 problems · 2/6 solved. # Status Time 1 solved 11m46s 2 — — 3 — — 4 — — 5 solved 5m45s 6 — — Problem 1 solved · 11m46s · Solution → Find all real roots of the equation $\sqrt{x^2-p}+2\sqrt{x^2-1}=x$, where $p$ is a real parameter. Problem 2 Point $A$ and segment $BC$ are given. Determine the locus of points in space which are...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist A6

IMO 2010 Shortlist A6 Category: Algebra Problem Suppose that f and g are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations fpgpnqq fpnq 1 and gpfpnqq gpnq 1 hold for all positive integers. Prove that fpnqgpnq for all positive integer n. (Germany)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G6

IMO 2010 Shortlist G6 Category: Geometry Problem The vertices X, Y , Z of an equilateral triangle XY Z lie respectively on the sides BC, CA, AB of an acute-angled triangle ABC. Prove that the incenter of triangle ABC lies inside triangle XY Z.

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IMO 2020 Shortlist N7

IMO 2020 Shortlist N7 Category: Number Theory Problem Let S be a set consisting of n ě 3 positive integers, none of which is a sum of two other distinct members of S. Prove that the elements of S may be ordered as a1,a2,...,an so that ai does not divide ai´1 ai 1 for all i “ 2,3,...,n ´ 1. (Ukraine) Shortlisted problems 11 12 Saint-Petersburg — Russia, 18th–28th September...

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IMO 2010 Shortlist A8

IMO 2010 Shortlist A8 Category: Algebra Problem Given six positive numbers a,b,c,d,e,f such that a b c d e f. Let a c e S and b d f T. Prove that 2ST ¡ b 3pS Tq Spbd bf dfq Tpac ae ceq . (1) (South Korea)

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IMO 2006 Shortlist N2

IMO 2006 Shortlist N2 Category: Number Theory Problem For x ∈ (0,1) let y ∈ (0,1) be the number whose nth digit after the decimal point is the (2n )th digit after the decimal point of x. Show that if x is rational then so is y. (Canada)

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IMO 2014 Shortlist N7

IMO 2014 Shortlist N7 Category: Number Theory Problem Let c ě 1 be an integer. Define a sequence of positive integers by a1 “ c and an 1 “ a3 n ´ 4c ¨ a2 n 5c2 ¨ an ` c for all n ě 1. Prove that for each integer n ě 2 there exists a prime number p dividing an but none of the numbers a1,...,an´1. (Austria)

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IMO 2011 Shortlist

IMO 2011 Shortlist 30 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number...

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IMO 2014 Shortlist A5

IMO 2014 Shortlist A5 Category: Algebra Problem Consider all polynomials Ppxq with real coefficients that have the following property: for any two real numbers x and y one has |y2 ´ Ppxq| ď 2|x| if and only if |x2 ´ Ppyq| ď 2|y|. Determine all possible values of Pp0q. (Belgium)

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IMO 1962

IMO 1962 Official IMO 1962 problems · 6/7 solved, 4 verified. # Status Time 1 ✓ verified 7m17s 2 ✓ verified 25m30s 3 ✓ verified 2m18s 4 ✓ verified 21m14s 5 solved 12m07s 6 — — 7 solved 15m54s Problem 1 ✓ verified · 7m17s · Solution → Find the smallest natural number $n$ with the following properties: (a) Its decimal representation ends in 6. (b) If the final digit...

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IMO 2007 Shortlist N6

IMO 2007 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Prove that the number (4k2 − 1)2 has a positive divisor of the form 8kn − 1 if and only if k is even. (United Kingdom)

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IMO 2018 Shortlist C3

IMO 2018 Shortlist C3 Category: Combinatorics Problem Let n be a given positive integer. Sisyphus performs a sequen e of turns on a board onsisting of n 1 squares in a row, numbered 0 to n from left to right. Initially, n stones are put into square 0, and the other squares are empty. At every turn, Sisyphus hooses any nonempty square, say with k stones, takes one of those...

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IMO 2024 Shortlist A1

IMO 2024 Shortlist A1 Category: Algebra Problem Determine all real numbers α such that the number tαu t2αu ¨¨¨ ` tnαu is a multiple of n for every positive integer n. (Here tzu denotes the greatest integer less than or equal to z.) (Colombia)

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IMO 2020 Shortlist G5

IMO 2020 Shortlist G5 Category: Geometry Problem Let ABCD be a cyclic quadrilateral with no two sides parallel. Let K, L, M, and N be points lying on sides AB, BC, CD, and DA, respectively, such that KLMN is a rhombus with KL k AC and LM k BD. Let ω1, ω2, ω3, and ω4 be the incircles of triangles ANK, BKL, CLM, and DMN, respectively. Prove that the internal...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist C7

IMO 2018 Shortlist C7 Category: Combinatorics Problem Consider 2018 pairwise rossing ir les no three of whi h are on urrent. These ir les subdivide the plane into regions bounded by ir ular edges that meet at verti es. Noti e that there are an even number of verti es on ea h ir le. Given the ir le, alternately olour the verti es on that ir le red and...

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IMO 2020 Shortlist N4

IMO 2020 Shortlist N4 Category: Number Theory Problem For any odd prime p and any integer n, let dppnq P t0,1,...,p ´ 1u denote the remainder when n is divided by p. We say that pa0,a1,a2,...q is a p-sequence, if a0 is a positive integer coprime to p, and an 1 “ an dppanq for n ě 0. (a) Do there exist infinitely many primes p for which there exist...

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