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Kvant Math Problem 701

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem Lyuda, Marina, and Natasha drew an acute triangle $LMN$. Then Lyuda constructed her own triangle in which the lengths of two sides are $|LM|$ and $|LN|$, and the angle between them is $60^\circ$ greater than the angle $L$ of triangle $LMN$. Similarly, Marina constructed her own triangle with sides of lengths $|ML|$ and $|MN|$, the angle between...

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Kvant Math Problem 700

Verified: no Verdicts: SKIP + SKIP Solve time: 7m34s Source on kvant.digital Problem Can the set of all terminating decimal fractions be partitioned into (a) two, (b) three classes so that no two numbers whose difference is $10^m$ (for any integers $m=0$, $\pm1$, $\pm2$, $\ldots$) belong to the same class? A. Leiderman Exploration Consider the set of all terminating decimal fractions. Each such number can be expressed as $\frac{a}{10^k}$ for...

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Kvant Math Problem 698

Verified: no Verdicts: SKIP + SKIP Solve time: 7m57s Source on kvant.digital Problem On the sides $a$, $b$, $c$, $d$ of a cyclic quadrilateral, rectangles of dimensions $a\times c$, $b\times d$, $c\times a$, $d\times b$ are constructed outward. Prove that the centers of these rectangles are the vertices of a parallelogram, a rectangle. O. Penkin Exploration Let the cyclic quadrilateral be $ABCD$, with side lengths $$AB=a,\quad BC=b,\quad CD=c,\quad DA=d.$$ On...

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Kvant Math Problem 696

Verified: no Verdicts: SKIP + SKIP Solve time: 2m58s Source on kvant.digital Problem Is it possible to fill a $10 \times 10$-cell table with 100 distinct natural numbers so that for any $k \times k$-cell square ($2 \le k \le 10$) the sums, the products $k$ of the numbers on its diagonals are equal? A. Balinsky Exploration Consider the problem for small $k \times k$ squares. For a $2 \times...

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Kvant Math Problem 695

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem Is it possible to color all the cells of some rectangular table black and white so that the numbers of white and black cells are equal, while in each row and in each column more than $\dfrac34$ of the cells are of the same color? S. V. Konyagin All-Union School Olympiad, Grade 10, 1981 Exploration Let the...

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Kvant Math Problem 693

Verified: no Verdicts: SKIP + SKIP Solve time: 4m54s Source on kvant.digital Problem In a certain village, there are 1000 inhabitants. Each day, every inhabitant shares the news they learned yesterday with all their acquaintances. It is known that any piece of news eventually becomes known to all the villagers. Prove that it is possible to select 90 inhabitants such that if a piece of news is simultaneously told to...

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Kvant Math Problem 692

Verified: no Verdicts: SKIP + SKIP Solve time: 3m13s Source on kvant.digital Problem Points $C_1$, $A_1$, $B_1$ are taken on the sides $AB$, $BC$, $CA$, respectively, of triangle $ABC$ so that $$|AC_1|:|C_1B|=|BA_1|:|A_1C|=|CB_1|:|B_1A|=1:3.$$. Prove that the perimeter $P$ of triangle $ABC$ and the perimeter $P_1$ of triangle $A_1B_1C_1$ satisfy the inequalities $P_1\lt\dfrac34P$; $P_1\gt\dfrac12P$. V. Turchaninov All-Union Mathematical Olympiad for School Students (1981, Grade 8) Exploration Consider triangle $ABC$ with arbitrary side...

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Kvant Math Problem 690

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem Inside a convex polygon with area $S_1$ and perimeter $P_1$, there is a convex polygon with area $S_2$ and perimeter $P_2$. Prove the inequality $$2 \frac{S_1}{P_1} \gt \frac{S_2}{P_2}.$$. Formulate and prove an analogous statement for convex polyhedra. A. V. Kelarev Exploration For a convex polygon, the quantity $\dfrac{2S}{P}$ has a geometric meaning. If every side is shifted...

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Kvant Math Problem 688

Verified: no Verdicts: SKIP + SKIP Solve time: 3m16s Source on kvant.digital Problem Given natural numbers $a_1$, $a_2$, $\ldots$, $a_n$ such that $a_k \le k$ $(k=1,2, \ldots,n)$ and the sum $a_1+a_2+\ldots+a_n$ is even. Prove that one of the expressions $$a_1\pm a_2\pm a_3\pm \ldots \pm a_n$$ is equal to zero. S. Nenashev Exploration Let $$S=a_1+a_2+\cdots+a_n.$$ Since $a_k\le k$, we have $$S\le 1+2+\cdots+n=\frac{n(n+1)}2.$$ The condition that $S$ is even suggests rewriting the...

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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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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...

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Kvant Mathematics

Kvant Mathematics Solutions to mathematics problems from Kvant magazine (533 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...

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Kvant Math Problem 701

Kvant Math Problem 700

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Kvant Math Problem 696

Kvant Math Problem 695

Kvant Math Problem 693

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Kvant Math Problem 688

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