#longlist

IMO 1967 LL USS56

IMO 1967 LL USS56 Origin: USS Problem In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi. Solution We shall prove by induction on n the following statement: If in some group of...

IMO 1987 LL MOR44

IMO 1987 LL MOR44 Origin: MOR Problem Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot + sin \thetan = 0. Prove that | sin \theta1 + 2 sin \theta2 + \cdot \cdot \cdot + n sin \thetan| \leq n2 .

IMO 1987 LL BEL7

IMO 1987 LL BEL7 Origin: BEL Problem Let f : (0, +\infty) \toR be a function having the property that f(x) = f(1/x) for all x > 0. Prove that there exists a function u : [1, +\infty) \toR satisfying u  x+1/x = f(x) for all x > 0.

IMO 1979 LL FRA21

IMO 1979 LL FRA21 Origin: FRA Problem Let E be the set of all bijective mappings from R to R satisfying (\forallt \inR) f(t) + f −1(t) = 2t, where f −1 is the mapping inverse to f. Find all elements of E that are monotonic mappings.

IMO 1979 LL NET48

IMO 1979 LL NET48 Origin: NET Problem In the plane a circle C of unit radius is given. For any line l a number s(l) is defined in the following way: If l and C intersect in two points, s(l) is their distance; otherwise, s(l) = 0. Let P be a point at distance r from the center of C. One defines M(r) to be the maximum value of the...

IMO 1985 LL CZS20

IMO 1985 LL CZS20 Origin: CZS Problem Let T be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points (x, y, z) and (u, v, w) are called neighbors if |x −u| + |y −v| + |z −w| = 1. Show that there exists a subset S of T such that for each p \inT , there is exactly one point...

IMO 1972 LL BUL1

IMO 1972 LL BUL1 Origin: BUL Problem Find all integer solutions of the equation 1 + x + x2 + x3 + x4 = y4.

IMO 1985 LL FRA25

IMO 1985 LL FRA25 Origin: FRA Problem Find eight positive integers n1, n2, . . . , n8 with the follow- ing property: For every integer k, −1985 \leqk \leq1985, there are eight integers \alpha1, \alpha2, . . . , \alpha8, each belonging to the set {−1, 0, 1}, such that k = 8 i=1 \alphaini.

IMO 1978 LL FRA18

IMO 1978 LL FRA18 Origin: FRA Problem Given a natural number n, prove that the number M(n) of points with integer coordinates inside the circle (O(0, 0), \sqrtn) satisfies \pin −5\sqrtn + 1 < M(n) < \pin + 4\sqrtn + 1.

IMO 1986 LL FRA21

IMO 1986 LL FRA21 Origin: FRA Problem Let AB be a segment of unit length and let C, D be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set {A, B, C, D}.

IMO 1982 LL FIN23

IMO 1982 LL FIN23 Origin: FIN Problem Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.

IMO 1970 LL ROM48

IMO 1970 LL ROM48 Origin: ROM Problem Let a polynomial p(x) with integer coefficients take the value 5 for five different integer values of x. Prove that p(x) does not take the value 8 for any integer x.

IMO 1982 LL FRA28

IMO 1982 LL FRA28 Origin: FRA Problem Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn, define vk = k\sqrtu1u2 \cdot \cdot \cdot uk. Prove that n  k=1 vk \leqe \cdot n  k=1 uk. (e is the base of the natural logarithm).

IMO 1966 LL ROM17

IMO 1966 LL ROM17 Origin: ROM Problem Suppose ABCD and A′B′C′D′ are two parallelograms arbi- trarily arranged in space, and let points M, N, P, Q divide the segments AA′, BB′, CC′, DD′ respectively in equal ratios. (a) Show that MNPQ is a parallelogram; (b) Find the locus of MNPQ as M varies along the segment AA′.

IMO 1969 LL CZS14

IMO 1969 LL CZS14 Origin: CZS Problem Let a and b be two positive real numbers. If x is a real solution of the equation x2 + px + q = 0 with real coefficients p and q such that |p| \leqa, |q| \leqb, prove that |x| \leq1  a + a2 + 4b . (1) Conversely, if x satisfies (1), prove that there exist real numbers p and q...

IMO 1967 LL HUN22

IMO 1967 LL HUN22 Origin: HUN Problem The distance between the centers of the circles k1 and k2 with radii r is equal to r. Points A and B are on the circle k1, symmetric with respect to the line connecting the centers of the circles. Point P is an arbitrary point on k2. Prove that PA2 + PB2 \geq2r2. When does equality hold? Solution Let O1 and O2 be...

IMO 1983 LL LUX43

IMO 1983 LL LUX43 Origin: LUX Problem Given a square ABCD, let P, Q, R, and S be four variable points on the sides AB, BC, CD, and DA, respectively. Determine the positions of the points P, Q, R, and S for which the quadrilateral PQRS is a parallelogram, a rectangle, a square, or a trapezoid.

IMO 1983 LL LUX45

IMO 1983 LL LUX45 Origin: LUX Problem Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass 2, and then take the same spoon of the new mixture from glass 2 and pour it back into the first glass. What happens...

IMO 1988 LL GRE29

IMO 1988 LL GRE29 Origin: GRE Problem Find positive integers x1, x2, . . . , x29, at least one of which is greater than 1988, such that x2 1 + x2 2 + \cdot \cdot \cdot + x2 29 = 29x1x2 . . . x29.

IMO 1986 LL GBR31

IMO 1986 LL GBR31 Origin: GBR Problem Let P and Q be distinct points in the plane of a triangle ABC such that AP : AQ = BP : BQ = CP : CQ. Prove that the line PQ passes through the circumcenter of the triangle.

IMO 1976 LL USS45

IMO 1976 LL USS45 Origin: USS Problem We are given n (n \geq5) circles in a plane. Suppose that every three of them have a common point. Prove that all n circles have a common point.

IMO 1986 LL GRE39

IMO 1986 LL GRE39 Origin: GRE Problem Let S be a k-element set. (a) Find the number of mappings f : S \toS such that (i) f(x) ̸= x for x \inS, (ii) f(f(x)) = x for x \inS. (b) The same with the condition (i) left out.

IMO 1967 LL CZS12

IMO 1967 LL CZS12 Origin: CZS Problem Given a segment AB of the length 1, define the set M of points in the following way: it contains the two points A, B, and also all points obtained from A, B by iterating the following rule: (∗) for every pair of points X, Y in M, the set M also contains the point Z of the segment XY for which Y...

IMO 1979 LL BEL5

IMO 1979 LL BEL5 Origin: BEL Problem Describe which natural numbers do not belong to the set E = {[n + \sqrtn + 1/2] | n \inN}.

IMO 1966 LL BUL3

IMO 1966 LL BUL3 Origin: BUL Problem A regular triangular prism has height h and a base of side length a. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the...

IMO 1966 LL CZS11

IMO 1966 LL CZS11 Origin: CZS Problem Does there exist an integer z that can be written in two different ways as z = x! + y!, where x, y are natural numbers with x \leqy?

IMO 1988 LL INA43

IMO 1988 LL INA43 Origin: INA Problem (a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find the value of k. (b) If p, q, and r are distinct roots of x3 −x2 + x −2 = 0, find the value of p3 + q3 + r3. (c) If r is the remainder when each of the numbers 1059, 1417, and 2312 is...

IMO 1982 LL BEL6

IMO 1982 LL BEL6 Origin: BEL Problem On the three distinct lines a, b, and c three points A, B, and C are given, respectively. Construct three collinear points X, Y, Z on lines a, b, c, respectively, such that BY AX = 2 and CZ AX = 3.

IMO 1989 LL USA103

IMO 1989 LL USA103 Origin: USA Problem An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle...

IMO 1982 LL BEL5

IMO 1982 LL BEL5 Origin: BEL Problem Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.

IMO 1977 LL SWE41

IMO 1977 LL SWE41 Origin: SWE Problem A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers 1, 2, 3, . . ., N are marked, and on the ring N integers a1, a2, . . . , aN of sum 1 are marked (see the figure). The ring can be turned into N differ- ent positions in which the numbers on...

IMO 1992 LL ITA39

IMO 1992 LL ITA39 Origin: ITA Problem Let n \geq2 be an integer. Find the minimum k for which there exists a partition of {1, 2, . . ., k} into n subsets X1, X2, . . . , Xn such that the following condition holds: for any i, j, 1 \leqi < j \leqn, there exist x1 \inX1, x2 \inX2 such that |xi −xj| = 1.

IMO 1988 LL HKG31

IMO 1988 LL HKG31 Origin: HKG Problem The circle x2 + y2 = r2 meets the coordinate axes at A = (r, 0), B = (−r, 0), C = (0, r), and D = (0, −r). Let P = (u, v) and Q = (−u, v) be two points on the circumference of the circle. Let N be the point of intersection of PQ and the y-axis, and let M...

IMO 1983 LL LUX44

IMO 1983 LL LUX44 Origin: LUX Problem We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.

IMO 1976 LL BUL4

IMO 1976 LL BUL4 Origin: BUL Problem Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1 is the square of some integer.

IMO 1979 LL VIE76

IMO 1979 LL VIE76 Origin: VIE Problem Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter 6.25. Find the sides of the triangle.

IMO 1966 LL ROM39

IMO 1966 LL ROM39 Origin: ROM Problem In a plane, a circle with center O and radius R and two points A, B are given. (a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P, say P1, P2, and find the distance between them if OA = a,...

IMO 1966 LL BUL32

IMO 1966 LL BUL32 Origin: BUL Problem The sides a, b, c of a triangle ABC form an arithmetic progression; the sides of another triangle A1B1C1 also form an arithmetic progression. Suppose that \angleA = \angleA1. Prove that the triangles ABC and A1B1C1 are similar.

IMO 1969 LL SWE63

IMO 1969 LL SWE63 Origin: SWE Problem Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

IMO 1985 LL MOR56

IMO 1985 LL MOR56 Origin: MOR Problem Let ABCD be a rhombus with angle \angleA = 60◦. Let E be a point, different from D, on the line AD. The lines CE and AB intersect at F. The lines DF and BE intersect at M. Determine the angle ∡BMD as a function of the position of E on AD.

IMO 1976 LL USS44

IMO 1976 LL USS44 Origin: USS Problem A circle of radius 1 rolls around a circle of radius \sqrt

IMO 1971 LL SWE45

IMO 1971 LL SWE45 Origin: SWE Problem Let m and n denote integers greater than 1, and let \nu(n) be the number of primes less than or equal to n. Show that if the equation n \nu(n) = m has a solution, then so does the equation n \nu(n) = m −1.

IMO 1987 LL GRE31

IMO 1987 LL GRE31 Origin: GRE Problem Construct a triangle ABC given its side a = BC, its circum- radius R (2R \geqa), and the difference 1/k = 1/c−1/b, where c = AB and b = AC.

IMO 1985 LL USS93

IMO 1985 LL USS93 Origin: USS Problem The sphere inscribed in tetrahedron ABCD touches the sides ABD and DBC at points K and M, respectively. Prove that ∡AKB = ∡DMC.

IMO 1977 LL FRG12

IMO 1977 LL FRG12 Origin: FRG Problem Let z be an integer > 1 and let M be the set of all numbers of the form zk = 1 + z + \cdot \cdot \cdot + zk, k = 0, 1, . . . . Determine the set T of divisors of at least one of the numbers zk from M. Solution According to part (a) of the previous problem...

IMO 1985 LL TUR82

IMO 1985 LL TUR82 Origin: TUR Problem Find all cubic polynomials x3 + ax2 + bx + c admitting the rational numbers a, b, and c as roots.

IMO 1974 LL VIE48

IMO 1974 LL VIE48 Origin: VIE Problem Let a be a number different from zero. For all integers n define Sn = an + a−n. Prove that if for some integer k both Sk and Sk+1 are integers, then for each integer n the number Sn is an integer.

IMO 1972 LL SWE41

IMO 1972 LL SWE41 Origin: SWE Problem The ternary expansion x = 0.10101010 . . . is given. Give the binary expansion of x. Alternatively, transform the binary expansion y = 0.110110110 . . . into a ternary expansion.

IMO 1970 LL CZS21

IMO 1970 LL CZS21 Origin: CZS Problem Find necessary and sufficient conditions on given positive num- bers u, v for the following claim to be valid: there exists a right-angled triangle \triangleABC with CD = u, CE = v, where D, E are points of the segments AB such that AD = DE = EB = 1 3AB.

IMO 1969 LL GBR27

IMO 1969 LL GBR27 Origin: GBR Problem The segment AB perpendicularly bisects CD at X. Show that, subject to restrictions, there is a right circular cone whose axis passes through X and on whose surface lie the points A, B, C, D. What are the restrictions?

IMO 1989 LL ISR59

IMO 1989 LL ISR59 Origin: ISR Problem Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1 for 1 \leqr \leq1989. Show that it is possible to find ϵr (1 \leqr \leq1989), each equal to \pm1, such that 1989  r=1 ϵrvr \leq \sqrt 3.

IMO 1988 LL SPA75

IMO 1988 LL SPA75 Origin: SPA Problem Let ABC be a triangle with inradius r and circumradius R. Show that sin A 2 sin B 2 + sin B 2 sin C 2 + sin C 2 sin A 2 \leq5 8 + r 4R.

IMO 1986 LL LUX50

IMO 1986 LL LUX50 Origin: LUX Problem Let D be the point on the side BC of the triangle ABC such that AD is the bisector of \angleCAB. Let I be the incenter of \triangleABC. (a) Construct the points P and Q on the sides AB and AC, respectively, such that PQ is parallel to BC and the perimeter of the triangle APQ is equal to k \cdot BC, where...

IMO 1979 LL ROM57

IMO 1979 LL ROM57 Origin: ROM Problem Let M be a set, and A, B, C given subsets of M. Find a necessary and sufficient condition for the existence of a set X \subsetM for which (X \cupA) \ (X \capB) = C. Describe all such sets X.

IMO 1967 LL MON33

IMO 1967 LL MON33 Origin: MON Problem In what case does the system x + y + mz = a, x + my + z = b, mx + y + z = c, have a solution? Find the conditions under which the unique solution of the above system is an arithmetic progression. Solution If m ̸\in{−2, 1}, the system has the unique solution x = b + a −(1...

IMO 1992 LL FRG18

IMO 1992 LL FRG18 Origin: FRG Problem Fibonacci numbers are defined as follows: F1 = F2 = 1, Fn+2 = Fn+1+Fn, n \geq1. Let an be the number of words that consist of n letters 0 or 1 and contain no two letters 1 at distance two from each other. Express an in terms of Fibonacci numbers.

IMO 1985 LL MON51

IMO 1985 LL MON51 Origin: MON Problem Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers. From f1 one constructs a sequence fk of sequences as follows: if fk = (c1, c2, . . . , cn), then fk+1 = (ci1, ci2, ci3 + 1, ci4 + 1, . . . , cin + 1), where (ci1, ci2, . . ....

IMO 1989 LL BUL6

IMO 1989 LL BUL6 Origin: BUL Problem The circles c1 and c2 are tangent at the point A. A straight line l through A intersects c1 and c2 at points C1 and C2 respectively. A circle c, which contains C1 and C2, meets c1 and c2 at points B1 and B2 respectively. Let \kappa be the circle circumscribed around triangle AB1B2. The circle k tangent to \kappa at the point...

IMO 1969 LL HUN34

IMO 1969 LL HUN34 Origin: HUN Problem Let a and b be arbitrary integers. Prove that if k is an integer not divisible by 3, then (a + b)2k + a2k + b2k is divisible by a2 + ab + b2.

IMO 1984 LL MOR37

IMO 1984 LL MOR37 Origin: MOR Problem Denote by [x] the greatest integer not exceeding x. For all real k > 1, define two sequences: an(k) = [nk] and bn(k) = nk k −1 . If A(k) = {an(k) : n \inN} and B(k) = {bn(k) : n \inN}, prove that A(k) and B(k) form a partition of N if and only if k is irrational.

IMO 1987 LL MOR45

IMO 1987 LL MOR45 Origin: MOR Problem Let us consider a variable polygon with 2n sides (n \inN) in a fixed circle such that 2n −1 of its sides pass through 2n −1 fixed points lying on a straight line ∆. Prove that the last side also passes through a fixed point lying on ∆.

IMO 1967 LL SWE48

IMO 1967 LL SWE48 Origin: SWE Problem Determine all positive roots of the equation xx = 1/ \sqrt 2. Solution Put f(x) = x ln x. The given equation is equivalent to f(x) = f(1/2), which has the solutions x1 = 1/2 and x2 = 1/4. Since the function f is decreasing on (0, 1/e), and increasing on (1/e, +\infty), this equation has no other solutions.

IMO 1982 LL USS51

IMO 1982 LL USS51 Origin: USS Problem Let n numbers x1, x2, . . . , xn be chosen in such a way that 1 \geqx1 \geqx2 \geq\cdot \cdot \cdot \geqxn \geq0. Prove that (1 + x1 + x2 + \cdot \cdot \cdot + xn)\alpha \leq1 + x\alpha 1 + 2\alpha−1x\alpha 2 + \cdot \cdot \cdot + n\alpha−1x\alpha n if 0 \leq\alpha \leq1.

IMO 1992 LL ITA40

IMO 1992 LL ITA40 Origin: ITA Problem The colonizers of a spherical planet have decided to build N towns, each having area 1/1000 of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of N?

IMO 1989 LL TUR102

IMO 1989 LL TUR102 Origin: TUR Problem If in a convex quadrilateral ABCD, E and F are the midpoints of the sides BC and DA respectively. Show that the sum of the areas of the triangles EDA and FBC is equal to the area of the quadrangle.

IMO 1984 LL FRG23

IMO 1984 LL FRG23 Origin: FRG Problem A 2 \times 2 \times 12 box fixed in space is to be filled with twenty-four 1 \times 1 \times 2 bricks. In how many ways can this be done?

IMO 1976 LL GDR22

IMO 1976 LL GDR22 Origin: GDR Problem A regular pentagon A1A2A3A4A5 with side length s is given. At each point Ai a sphere Ki of radius s/2 is constructed. There are two spheres K1′ and K2′ eah of radius s/2 touching all the five spheres Ki. Decide whether K1′ and K2′ intersect each other, touch each other, or have no common points.

IMO 1977 LL ROM38

IMO 1977 LL ROM38 Origin: ROM Problem Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1 \leq\cdot \cdot \cdot \leq bn < c1 \leq\cdot \cdot \cdot \leqcn be real numbers. Prove: ⎡ ⎣ n  j=1 mj(aj + bj + cj) ⎤ ⎦ 3 ⎛ ⎝ n  j=1 mj ⎞ ⎠ ⎡ ⎣ n  j=1 mj(ajbj...

IMO 1987 LL USS67

IMO 1987 LL USS67 Origin: USS Problem If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1, find the maximum of the expression (a + b)4 + (a + c)4 + (a + d)4 + (b + c)4 + (b + d)4 + (c + d)4.

IMO 1979 LL BEL3

IMO 1979 LL BEL3 Origin: BEL Problem Is it possible to partition 3-dimensional Euclidean space into 1979 mutually isometric subsets?

IMO 1966 LL USS55

IMO 1966 LL USS55 Origin: USS Problem Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such that all the angles of the triangle lie in the interval [40◦, 70◦].

IMO 1969 LL SWE59

IMO 1969 LL SWE59 Origin: SWE Problem For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .) construct a continuous function f such that there do not exist x, y with 0 < \lambda < y = x + \lambda \leq1 for which f(x) = f(y).

IMO 1972 LL BUL2

IMO 1972 LL BUL2 Origin: BUL Problem Find all real values of the parameter a for which the system of equations x4 = yz −x2 + a, y4 = zx −y2 + a, z4 = xy −z2 + a, has at most one real solution.

IMO 1992 LL ROM63

IMO 1992 LL ROM63 Origin: ROM Problem Let a and b be integers. Prove that 2a2−1 b2+2 is not an integer.

IMO 1989 LL POR83

IMO 1989 LL POR83 Origin: POR Problem Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth n bourbakis and each silver coin is worth m bourbakis (n and m are positive integers). Using gold and solver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072...

IMO 1984 LL FRA18

IMO 1984 LL FRA18 Origin: FRA Problem Let c be the inscribed circle of the triangle ABC, d a line tan- gent to c which does not pass through the vertices of triangle ABC. Prove the existence of points A1, B1, C1, respectively, on the lines BC, CA, AB satisfying the following two properties: (i) Lines AA1, BB1, and CC1 are parallel. (ii) Lines AA1, BB1, and CC1 meet d...

IMO 1987 LL VIE72

IMO 1987 LL VIE72 Origin: VIE Problem Is it possible to cover a rectangle of dimensions m \times n with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter L) if: (a) m \times n = 1985 \times 1987; (b) m \times n = 1987 \times 1989?

IMO 1984 LL ROM46

IMO 1984 LL ROM46 Origin: ROM Problem Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers such that an+1 = nan + 1, bn+1 = nbn −1 for every n \geq1. Show that these two sequences can have only a finite number of terms in common.

IMO 1978 LL NET32

IMO 1978 LL NET32 Origin: NET Problem Let C be the circumcircle of the square with vertices (0, 0), (0, 1978), (1978, 0), (1978, 1978) in the Cartesian plane. Prove that C con- tains no other point for which both coordinates are integers.

IMO 1966 LL CZS42

IMO 1966 LL CZS42 Origin: CZS Problem Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that there is a subsequence ak1, ak2, . . . , akm, where 1 \leqk1 < k2 < \cdot \cdot \cdot < km \leq n, such that a2 k1 + a2 k2 + \cdot \cdot \cdot + a2 km is divisible by n.

IMO 1966 LL CZS40

IMO 1966 LL CZS40 Origin: CZS Problem For a positive real number p, find all real solutions to the equation x2 + 2px −p2 − x2 −2px −p2 = 1.

IMO 1992 LL ROM64

IMO 1992 LL ROM64 Origin: ROM Problem For any positive integer n consider all representations n = a1 + \cdot \cdot \cdot + ak, where a1 > a2 > \cdot \cdot \cdot > ak > 0 are integers such that for all i \in{1, 2, . . ., k −1}, the number ai is divisible by ai+1. Find the longest such representation of the number 1992.

IMO 1978 LL TUR39

IMO 1978 LL TUR39 Origin: TUR Problem A is a 2m-digit positive integer each of whose digits is 1. B is an m-digit positive integer each of whose digits is 4. Prove that A+B + 1 is a perfect square.

IMO 1967 LL CZS7

IMO 1967 LL CZS7 Origin: CZS Problem Find all real solutions of the system of equations x1 + x2 + \cdot \cdot \cdot + xn = a, x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = a2, . . . . . . . . . . . . . . . . . . xn 1 + xn 2 + \cdot \cdot \cdot + xn...

IMO 1984 LL GDR30

IMO 1984 LL GDR30 Origin: GDR Problem Decide whether it is possible to color the 1984 natural numbers 1, 2, 3, . . ., 1984 using 15 colors so that no geometric sequence of length 3 of the same color exists.

IMO 1976 LL GDR20

IMO 1976 LL GDR20 Origin: GDR Problem Let (an), n = 0, 1, . . ., be a sequence of real numbers such that a0 = 0 and a3 n+1 = 1 2a2 n −1, n = 0, 1, . . .. Prove that there exists a positive number q, q < 1, such that for all n = 1, 2, . . ., |an+1 −an| \leqq|an −an−1|, and give...

IMO 1969 LL NET50

IMO 1969 LL NET50 Origin: NET Problem The bisectors of the exterior angles of a pentagon B1B2B3B4B5 form another pentagon A1A2A3A4A5. Construct B1B2B3B4B5 from the given pentagon A1A2A3A4A5.

IMO 1988 LL KOR53

IMO 1988 LL KOR53 Origin: KOR Problem Let x = p, y = q, z = r, w = s be the unique solution of the system of linear equations x + aiy + a2 i z + a3 i w = a4 i , i = 1, 2, 3, 4. Express the solution of the following system in terms of p, q, r, and s: x + a2 i...

IMO 1988 LL USS87

IMO 1988 LL USS87 Origin: USS Problem All the irreducible positive rational numbers such that the prod- uct of the numerator and the denominator is less than 1988 are written in increasing order. Prove that any two adjacent fractions a/b and c/d, a/b < c/d, satisfy the equation bc −ad = 1.

IMO 1969 LL BEL6

IMO 1969 LL BEL6 Origin: BEL Problem Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two different ways and prove that 10  − 10  1 10  = 24.

IMO 1974 LL BUL4

IMO 1974 LL BUL4 Origin: BUL Problem Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a triangle ABC, touching the interiors of the sides BC, CA, AB at points Ta, Tb, Tc respectively. Prove that the lines OaTa, ObTb, OcTc are concurrent in a point P for which POa = POb = POc = 2R holds, where R denotes the circumradius of ABC. Also prove...

IMO 1976 LL FIN10

IMO 1976 LL FIN10 Origin: FIN Problem Show that the reciprocal of any number of the form 2(m2 + m + 1), where m is a positive integer, can be represented as a sum of consecutive terms in the sequence (aj)\infty j=1, aj = j(j + 1)(j + 2).

IMO 1984 LL SPA51

IMO 1984 LL SPA51 Origin: SPA Problem Two cyclists leave simultaneously a point P in a circular run- way with constant velocities v1, v2 (v1 > v2) and in the same sense. A pedestrian leaves P at the same time, moving with velocity v3 = v1+v2 . If the pedestrian and the cyclists move in opposite directions, the pedes- trian meets the second cyclist 91 seconds after he meets the...

IMO 1989 LL USA104

IMO 1989 LL USA104 Origin: USA Problem For each nonzero complex number z, let arg z be the unique real number t such that −\pi < t \leq\pi and z = |z|(cos t + ı sin t). Given a real number c > 0 and a complex number z ̸= 0 with arg z ̸= \pi, define B(c, z) = {b \inR | |w −z| < b ⇒| arg w...

IMO 1985 LL ISR42

IMO 1985 LL ISR42 Origin: ISR Problem Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.

IMO 1972 LL NET29

IMO 1972 LL NET29 Origin: NET Problem Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a triangle A1B1C1 such that A1A, B1B, C1C are the bisectors of angles of the triangle. We have that AC = BC and A1C1 ̸= B1C1. (a) Prove that C1 lies on the circumcircle of the triangle ABC. (b) Suppose that ∡BAC1 = \pi/6; find the form of triangle ABC.

IMO 1985 LL SPA72

IMO 1985 LL SPA72 Origin: SPA Problem Construct a triangle ABC given the side AB and the distance OH from the circumcenter O to the orthocenter H, assuming that OH and AB are parallel.

IMO 1985 LL POL65

IMO 1985 LL POL65 Origin: POL Problem Define the functions f, F : N \toN, by f(n) = 3 − \sqrt n , F(k) = min{n \inN|f k(n) > 0}, where f k = f ◦\cdot \cdot \cdot ◦f is f iterated n times. Prove that F(k + 2) = 3F(k + 1) −F(k) for all k \inN.

IMO 1978 LL GDR27

IMO 1978 LL GDR27 Origin: GDR Problem Determine the sixth number after the decimal point in the number
\sqrt 1978 + /\sqrt 1978 020.

IMO 1977 LL POL32

IMO 1977 LL POL32 Origin: POL Problem In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted. Solution It is a known result that among six persons there are 3 mutually ac- quainted or 3 mutually unacquainted. By the condition of the problem the last case is excluded. If there is a man in...

IMO 1986 LL TUR71

IMO 1986 LL TUR71 Origin: TUR Problem Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

IMO 1977 LL CZS7

IMO 1977 LL CZS7 Origin: CZS Problem Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real numbers such that (n −1)(c2 1 + c2 2 + \cdot \cdot \cdot + c2 n) = (c1 + c2 + \cdot \cdot \cdot + cn)2, then either all these numbers are nonnegative or all these numbers are nonpositive. Solution Let us suppose that c1 \leqc2 \leq\cdot...

IMO 1988 LL USS88

IMO 1988 LL USS88 Origin: USS Problem There are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, A1, A2, A3, A4, A5, and A6, prove that A1A2 \cdot A3A4 \cdot A5A6 = A2A3 \cdot A4A5 \cdot A6A1.

IMO 1966 LL GDR2

IMO 1966 LL GDR2 Origin: GDR Problem Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an = 1, prove that (1 + a1)(1 + a2) \cdot \cdot \cdot (1 + an) \geq2n.

IMO 1985 LL MON53

IMO 1985 LL MON53 Origin: MON Problem For each P inside the triangle ABC, let A(P), B(P), and C(P) be the points of intersection of the lines AP, BP, and CP with the sides opposite to A, B, and C, respectively. Determine P in such a way that the area of the triangle A(P)B(P)C(P) is as large as possible.

IMO 1987 LL GBR26

IMO 1987 LL GBR26 Origin: GBR Problem Prove that if x, y, z are real numbers such that x2+y2+z2 = 2, then x + y + z \leqxyz + 2.

IMO 1985 LL FRG28

IMO 1985 LL FRG28 Origin: FRG Problem Let M be the set of the lengths of an octahedron whose sides are congruent quadrangles. Prove that M has at most three elements. (FRG 1a) Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.

IMO 1985 LL CZS21

IMO 1985 LL CZS21 Origin: CZS Problem Let A be a set of positive integers such that for any two elements x, y of A, |x−y| \geqxy 25 . Prove that A contains at most nine elements. Give an example of such a set of nine elements.

IMO 1988 LL FRG16

IMO 1988 LL FRG16 Origin: FRG Problem Show that if n runs through all positive integers, f(n) =  n + n/3 + 1/2  runs through all positive integers skipping the terms of the sequence an = 3n2 −2n.

IMO 1969 LL SWE61

IMO 1969 LL SWE61 Origin: SWE Problem Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n. Prove that if n is power of 2, then so is an.

IMO 1992 LL IND27

IMO 1992 LL IND27 Origin: IND Problem Let ABC be an arbitrary scalene triangle. Define \Sigma to be the set of all circles y that have the following properties: (i) y meets each side of \triangleABC in two (possibly coincident) points; (ii) if the points of intersection of y with the sides of the triangle are la- beled by P, Q, R, S, T , U, with the points occurring...

IMO 1988 LL POL69

IMO 1988 LL POL69 Origin: POL Problem For a convex polygon P in the plane let P ′ denote the convex polygon with vertices at the midpoints of the sides of P. Given an integer n \geq3, determine sharp bounds for the ratio area(P ′) area(P) over all convex n-gons P.

IMO 1976 LL CZS7

IMO 1976 LL CZS7 Origin: CZS Problem Let P be a fixed point and T a given triangle that contains the point P. Translate the triangle T by a given vector v and denote by T ′ this new triangle. Let r, R, respectively, be the radii of the smallest disks centered at P that contain the triangles T , T ′, respectively. Prove that r + |v| \leq3R and...

IMO 1985 LL ITA45

IMO 1985 LL ITA45 Origin: ITA Problem Two persons, X and Y , play with a die. X wins a game if the outcome is 1 or 2; Y wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within 5 games. Determine the probabilities of winning in an unlimited number of games. If...

IMO 1989 LL VIE111

IMO 1989 LL VIE111 Origin: VIE Problem Find the greatest number c such that for all natural numbers n, {n \sqrt 2} \geqc n (where {n \sqrt 2} = n \sqrt 2 −[n \sqrt 2]; [x] is the integer part of x). For this number c, find all natural numbers n for which {n \sqrt 2} = c n.

IMO 1983 LL VIE73

IMO 1983 LL VIE73 Origin: VIE Problem Let ABC be a nonequilateral triangle. Prove that there exist two points P and Q in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of ABC, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.

IMO 1989 LL FRA19

IMO 1989 LL FRA19 Origin: FRA Problem Let a1, . . . , an be distinct positive integers that do not contain a 9 in their decimal representations. Prove that a1 \cdot \cdot \cdot + 1 an \leq30.

IMO 1967 LL ITA27

IMO 1967 LL ITA27 Origin: ITA Problem Which regular polygons can be obtained (and how) by cutting a cube with a plane? Solution Regular polygons with 3, 4, and 6 sides can be obtained by cutting a cube with a plane, as shown in the figure. A polygon with more than 6 sides cannot be obtained in such a way, for a cube has 6 faces. Also, if a pentagon...

IMO 1983 LL FRG26

IMO 1983 LL FRG26 Origin: FRG Problem Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) =

IMO 1976 LL SWE34

IMO 1976 LL SWE34 Origin: SWE Problem Let {an}\infty and {bn}\infty be two sequences determined by the recursion formulas an+1 = an + bn, bn+1 = 3an + bn, n = 0, 1, 2, . . ., and the initial values a0 = b0 = 1. Prove that there exists a uniquely determined constant c such that n|can−bn| < 2 for all nonnegative integers n.

IMO 1986 LL ROM62

IMO 1986 LL ROM62 Origin: ROM Problem Determine all pairs of positive integers (x, y) satisfying the equation px −y3 = 1, where p is a given prime number.

IMO 1986 LL GBR35

IMO 1986 LL GBR35 Origin: GBR Problem Establish the maximum and minimum values that the sum |a| + |b| + |c| can have if a, b, c are real numbers such that the maximum value of |ax2 + bx + c| is 1 for −1 \leqx \leq1.

IMO 1969 LL MON45

IMO 1969 LL MON45 Origin: MON Problem Given n points in the plane such that no three of them are collinear, prove that one can find at least
n−3  convex quadrilaterals with their vertices at these points.

IMO 1983 LL ISR37

IMO 1983 LL ISR37 Origin: ISR Problem The points A1, A2, . . . , A1983 are set on the circumference of a circle and each is given one of the values \pm1. Show that if the number of points with the value +1 is greater than 1789, then at least 1207 of the points will have the property that the partial sums that can be formed by taking the...

IMO 1982 LL BRA11

IMO 1982 LL BRA11 Origin: BRA Problem A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers a and b. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between a and b be for this to happen?

IMO 1969 LL SWE58

IMO 1969 LL SWE58 Origin: SWE Problem Six points P1, . . . , P6 are given in 3-dimensional space such that no four of them lie in the same plane. Each of the line segments PjPk is colored black or white. Prove that there exists one triangle PjPkPl whose edges are of the same color.

IMO 1992 LL FIN15

IMO 1992 LL FIN15 Origin: FIN Problem Prove that there exist 78 lines in the plane such that they have exactly 1992 points of intersection.

IMO 1967 LL GDR14

IMO 1967 LL GDR14 Origin: GDR Problem Which fraction p/q, where p, q are positive integers less than 100, is closest to \sqrt 2? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of \sqrt 2 (without using any tables). Solution We have that p q − \sqrt = |p −q \sqrt 2| q = |p2 −2q2|...

IMO 1970 LL GDR28

IMO 1970 LL GDR28 Origin: GDR Problem A set G with elements u, v, w, . . . is a group if the following conditions are fulfilled: (1) There is a binary algebraic operation ◦defined on G such that for all u, v \inG there is a w \inG with u ◦v = w. (2) This operation is associative; i.e., for all u, v, w \inG, (u ◦v) ◦w =...

IMO 1992 LL IRN36

IMO 1992 LL IRN36 Origin: IRN Problem Find all rational solutions of a2 + c2 + 17(b2 + d2) = 21, ab + cd = 2.

IMO 1978 LL CZS10

IMO 1978 LL CZS10 Origin: CZS Problem Show that for any natural number n there exist two prime numbers p and q, p ̸= q, such that n divides their difference.

IMO 1966 LL POL37

IMO 1966 LL POL37 Origin: POL Problem Prove that the perpendiculars drawn from the midpoints of the sides of a cyclic quadrilateral to the opposite sides meet at one point.

IMO 1979 LL VIE73

IMO 1979 LL VIE73 Origin: VIE Problem In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?

IMO 1966 LL CZS16

IMO 1966 LL CZS16 Origin: CZS Problem We are given a circle K with center S and radius 1 and a square Q with center M and side 2. Let XY be the hypotenuse of an isosceles right triangle XY Z. Describe the locus of points Z as X varies along K and Y varies along the boundary of Q.

IMO 1985 LL ROM71

IMO 1985 LL ROM71 Origin: ROM Problem For every integer r > 1 find the smallest integer h(r) > 1 having the following property: For any partition of the set {1, 2, . . ., h(r)} into r classes, there exist integers a \geq0, 1 \leqx \leqy such that the numbers a + x, a + y, a + x + y are contained in the same class of the...

IMO 1971 LL SWE41

IMO 1971 LL SWE41 Origin: SWE Problem Consider the set of grid points (m, n) in the plane, m, n inte- gers. Let \sigma be a finite subset and define S(\sigma) =  (m,n)\in\sigma (100 −|m| −|n|). Find the maximum of S, taken over the set of all such subsets \sigma.

IMO 1967 LL SWE49

IMO 1967 LL SWE49 Origin: SWE Problem Let n and k be positive integers such that 1 \leqn \leqN + 1, 1 \leqk \leqN + 1. Show that min n̸=k | sin n −sin k| < 2 N . Solution Since sin 1, sin 2, . . . , sin(N +1) \in(−1, 1), two of these N +1 numbers have distance less than 2/N. Therefore | sin n −sin k|...

IMO 1970 LL CZS19

IMO 1970 LL CZS19 Origin: CZS Problem Let n > 1 be a natural number, a \geq1 a real number, and x1, x2, . . . , xn numbers such that x1 = 1, xk+1 xk = a+\alphak for k = 1, 2, . . . , n− 1, where \alphak are real numbers with \alphak \leq k(k+1). Prove that n−1\sqrtxn < a + n −1.

IMO 1979 LL GDR30

IMO 1979 LL GDR30 Origin: GDR Problem Let M be a set of points in a plane with at least two elements. Prove that if M has two axes of symmetry g1 and g2 intersecting at an angle \alpha = q\pi, where q is irrational, then M must be infinite.

IMO 1971 LL CUB12

IMO 1971 LL CUB12 Origin: CUB Problem A system of n numbers x1, x2, . . . , xn is given such that x1 = logxn−1 xn, x2 = logxn x1, . . . , xn = logxn−2 xn−1. Prove that $n k=1 xk = 1.

IMO 1972 LL USS43

IMO 1972 LL USS43 Origin: USS Problem A fixed point A inside a circle is given. Consider all chords XY of the circle such that \angleXAY is a right angle, and for all such chords construct the point M symmetric to A with respect to XY . Find the locus of points M.

IMO 1988 LL NET68

IMO 1988 LL NET68 Origin: NET Problem Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}. The distance between two elements {ai} and {bi} of S is defined as 7 i=1 |ai −bi|. Let T be a subset of S in which any two elements have a distance apart greater than or equal to 3. Prove that T contains at most...

IMO 1979 LL POL53

IMO 1979 LL POL53 Origin: POL Problem An infinite increasing sequence of positive integers nj (j = 1, 2, . . . ) has the property that for a certain c, N  nj\leqN nj \leqc, for every N > 0 Prove that there exist finitely many sequences m(i) j (i = 1, 2, . . . , k) such that {n1, n2, . . . } = %k i=1{m(i)...

IMO 1966 LL BUL21

IMO 1966 LL BUL21 Origin: BUL Problem Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality
6V \pi 2 \leq  2S \pi \sqrt . When does equality occur?

IMO 1985 LL ROM67

IMO 1985 LL ROM67 Origin: ROM Problem Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the property n2 | 2n1 −1, n3 | 2n2 −1, . . . , nk | 2nk−1 −1, and n1 | 2nk −1. Show that n1 = n2 = \cdot \cdot \cdot = nk = 1.

IMO 1989 LL IND49

IMO 1989 LL IND49 Origin: IND Problem Let A, B denote two distinct fixed points in space. Let X, P denote variable points (in space), while K, N, n denote positive integers. Call (X, K, N, P) admissible if (N −K)PA + K \cdot PB \geqN \cdot PX. Call (X, K, N) admissible if (X, K, N, P) is admissible for all choices of P. Call (X, N) admissible if...

IMO 1989 LL THA100

IMO 1989 LL THA100 Origin: THA Problem Let A be an n\timesn matrix whose elements are nonnegative real numbers. Assume that A is a nonsingular matrix and all elements of A−1 are nonnegative real numbers. Prove that every row and every column of A has exactly one nonzero element.

IMO 1977 LL POL33

IMO 1977 LL POL33 Origin: POL Problem A circle K centered at (0, 0) is given. Prove that for every vector (a1, a2) there is a positive integer n such that the circle K translated by the vector n(a1, a2) contains a lattice point (i.e., a point both of whose coordinates are integers). Solution Let r be the radius of K and s > \sqrt 2/r an integer. Consider the...

IMO 1992 LL IND30

IMO 1992 LL IND30 Origin: IND Problem Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive integer n. Determine, with proof, the least positive integer m, if it exists, for which Pm is divisible by 3333.

IMO 1974 LL SWE33

IMO 1974 LL SWE33 Origin: SWE Problem Let a be a real number such that 0 < a < 1, and let n be a positive integer. Define the sequence a0, a1, a2, . . . , an recursively by a0 = a; ak+1 = ak + 1 na2 k for k = 0, 1, . . ., n −1. Prove that there exists a real number A, depending on...

IMO 1971 LL GBR14

IMO 1971 LL GBR14 Origin: GBR Problem Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

IMO 1971 LL NET32

IMO 1971 LL NET32 Origin: NET Problem Two half-lines a and b, with the common endpoint O, make an acute angle \alpha. Let A on a and B on b be points such that OA = OB, and let b′ be the line through A parallel to b. Let \beta be the circle with center B and radius BO. We construct a sequence of half-lines c1, c2, c3, . ....

IMO 1966 LL USS50

IMO 1966 LL USS50 Origin: USS Problem Given a quadrangle of sides a, b, c, d and area S, show that S \leq a+c \cdot b+d 2 .

IMO 1978 LL GDR25

IMO 1978 LL GDR25 Origin: GDR Problem Consider a polynomial P(x) = ax2 + bx + c with a > 0 that has two real roots x1, x2. Prove that the absolute values of both roots are less than or equal to 1 if and only if a + b + c \geq0, a −b + c \geq0, and a −c \geq0.

IMO 1979 LL ROM55

IMO 1979 LL ROM55 Origin: ROM Problem Let a, b be coprime integers. Show that the equation ax2 + by2 = z3 has an infinite set of solutions (x, y, z) with x, y, z \inZ and x, y mutually coprime (in each solution).

IMO 1985 LL CZS23

IMO 1985 LL CZS23 Origin: CZS Problem Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s = [nx + y], n \inN}. Prove that if r > 1 is a rational number, there exist real numbers u and v such that S(r, 0) \capS(u, v) = \emptyset, S(r, 0) \cupS(u, v) = N.

IMO 1989 LL IRE55

IMO 1989 LL IRE55 Origin: IRE Problem Let [x] denote the greatest integer less than or equal to x. Let \alpha be the positive root of the equation x2 −1989x −1 = 0. Prove that there exist infinitely many natural numbers n that satisfy the equation [\alphan + 1989\alpha[\alphan]] = 1989n + (19892 + 1)[\alphan].

IMO 1984 LL CAN15

IMO 1984 LL CAN15 Origin: CAN Problem Consider all the sums of the form 1985  k=1 ekk5 = \pm15 \pm 25 \pm \cdot \cdot \cdot \pm 19855, where ek = \pm1. What is the smallest nonnegative value attained by a sum of this type?

IMO 1982 LL POL38

IMO 1982 LL POL38 Origin: POL Problem Numbers un,k (1 \leqk \leqn) are defined as follows: u1,1 = 1, un,k = n k  −  d|n, d|k, d>1 un/d,k/d (the empty sum is defined to be equal to zero). Prove that n | un,k for every natural number n and for every k (1 \leqk \leqn).

IMO 1982 LL CZS20

IMO 1982 LL CZS20 Origin: CZS Problem Consider a cube C and two planes \sigma, \tau, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.

IMO 1969 LL BUL7

IMO 1969 LL BUL7 Origin: BUL Problem Prove that the equation x3 + y3 + z3 = 1969 has no integral solutions.

IMO 1969 LL USS67

IMO 1969 LL USS67 Origin: USS Problem Under the conditions x1, x2 > 0, x1y1 > z2 1, and x2y2 > z2 2, prove the inequality (x1 + x2)(y1 + y2) −(z1 + z2)2 \leq x1y1 −z2 + x2y2 −z2 .

IMO 1972 LL ROM36

IMO 1972 LL ROM36 Origin: ROM Problem A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.

IMO 1966 LL GDR25

IMO 1966 LL GDR25 Origin: GDR Problem Show that tan 7◦30′ = \sqrt 6 + \sqrt 2 − \sqrt 3 −2.

IMO 1979 LL USA65

IMO 1979 LL USA65 Origin: USA Problem Given f(x) \leqx for all real x and f(x + y) \leqf(x) + f(y) for all real x, y, prove that f(x) = x for all x.

IMO 1979 LL BRA8

IMO 1979 LL BRA8 Origin: BRA Problem The sequence (an) of real numbers is defined as follows: a1 = 1, a2 = 2 and an = 3an−1 −an−2, n \geq3. Prove that for n \geq3, an =  a2 n−1 an−2  1, where [x] denotes the integer p such that p \leqx < p + 1.

IMO 1979 LL HUN42

IMO 1979 LL HUN42 Origin: HUN Problem Let a quadratic polynomial g(x) = ax2 + bx + c be given and an integer n \geq1. Prove that there exists at most one polynomial f(x) of nth degree such that f(g(x)) = g(f(x)).

IMO 1989 LL GBR27

IMO 1989 LL GBR27 Origin: GBR Problem Integers cm,n (m \geq0, n \geq0) are defined by cm,0 = 1 for all m \geq0, c0,n = 1 for all n \geq0, and cm,n = cm−1,n −ncm−1,n−1 for all m > 0, n > 0. Prove that cm,n = cn,m for all m \geq0, n \geq0.

IMO 1976 LL USS43

IMO 1976 LL USS43 Origin: USS Problem Prove that if for a polynomial P(x, y) we have P(x −1, y −2x + 1) = P(x, y), then there exists a polynomial \Phi(x) with P(x, y) = \Phi(y −x2).

IMO 1970 LL USS56

IMO 1970 LL USS56 Origin: USS Problem A square hole of depth h whose base is of length a is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length L > \sqrt 2a2 + h2, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide...

IMO 1983 LL VIE74

IMO 1983 LL VIE74 Origin: VIE Problem In a plane we are given two distinct points A, B and two lines a, b passing through B and A respectively (a ∋B, b ∋A) such that the line AB is equally inclined to a and b. Find the locus of points M in the plane such that the product of distances from M to A and a equals the product of...

IMO 1979 LL NET49

IMO 1979 LL NET49 Origin: NET Problem Let there be given two sequences of integers fi(1), fi(2), . . . (i = 1, 2) satisfying: (i) fi(nm) = fi(n)fi(m) if gcd(n, m) = 1; (ii) for every prime P and all k = 2, 3, 4, . . ., fi(P k) = fi(P)fi(P k−1) − P 2f(P k−2). Moreover, for every prime P: (iii) f1(P) = 2P, (iv) f2(P) <...

IMO 1985 LL NOR62

IMO 1985 LL NOR62 Origin: NOR Problem A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed \pi 3 cm per second (relative to the disk). Describe the path of the insect (a) relative to the disk; (b) relative to the wall.

IMO 1989 LL MOR72

IMO 1989 LL MOR72 Origin: MOR Problem Let ABCD be a quadrilateral inscribed in a circle with diam- eter AB such that BC = a, CD = 2a, DA = 3 \sqrt 5−1 a. For each point M on the semicircle AB not containing C and D, denote by h1, h2, h3 the distances from M to the sides BC, CD, and DA. Find the maximum of h1 + h2...

IMO 1987 LL GRE30

IMO 1987 LL GRE30 Origin: GRE Problem Consider the regular 1987-gon A1A2 . . . A1987 with center O. Show that the sum of vectors belonging to any proper subset of M = {OAj | j = 1, 2, . . . , 1987} is nonzero.

IMO 1974 LL CUB7

IMO 1974 LL CUB7 Origin: CUB Problem Let P be a prime number and n a natural number. Prove that the product N = pn2 2n−1 i=1; 2∤i ((p −1)i)! p2i pi  is a natural number that is not divisible by p.

IMO 1967 LL USS57

IMO 1967 LL USS57 Origin: USS Problem Consider the sequence (cn): c1 = a1 + a2 + \cdot \cdot \cdot + a8, c2 = a2 1 + a2 2 + \cdot \cdot \cdot + a2 8, . . . . . . . . . . . . cn = an 1 + an 2 + \cdot \cdot \cdot + an 8, . . . . . . . ....

IMO 1967 LL GDR13

IMO 1967 LL GDR13 Origin: GDR Problem Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius r there exists one (or more) with maximal area. If so, determine their shape and area. Solution The maximum area is 3 \sqrt 3r2/4 (where r is the radius of the semicircle) and is attained in the case of a trapezoid with two vertices at the endpoints of the diameter...

IMO 1967 LL CZS11

IMO 1967 LL CZS11 Origin: CZS Problem Let n be a positive integer. Find the maximal number of non- congruent triangles whose side lengths are integers less than or equal to n. Solution We have to find the number pn of triples of positive integers (a, b, c) satisfying a \leqb \leqc \leqn and a + b > c. Let us denote by pn(k) the number of such triples with...

IMO 1988 LL KOR58

IMO 1988 LL KOR58 Origin: KOR Problem For each pair of positive integers k and n, let Sk(n) be the base-k digit sum of n. Prove that there are at most two primes p less than 20,000 for which S31(p) is a composite number.

IMO 1966 LL USS54

IMO 1966 LL USS54 Origin: USS Problem Find the last two digits of a sum of eighth powers of 100 consecutive integers.

IMO 1986 LL MON54

IMO 1986 LL MON54 Origin: MON Problem Find the least integer n with the following property: For any set V of 8 points in the plane, no three lying on a line, and for any set E of n line segments with endpoints in V , one can find a straight line intersecting at least 4 segments in E in interior points.

IMO 1974 LL NET21

IMO 1974 LL NET21 Origin: NET Problem Let M be a nonempty subset of Z+ such that for every element x in M, the numbers 4x and [\sqrtx] also belong to M. Prove that M = Z+.

IMO 1966 LL POL36

IMO 1966 LL POL36 Origin: POL Problem Let ABCD be a cyclic quadrilateral. Show that the centroids of the triangles ABC, CDA, BCD, DAB lie on a circle.

IMO 1985 LL ISR43

IMO 1985 LL ISR43 Origin: ISR Problem Suppose that 1985 points are given inside a unit cube. Show that one can always choose 32 of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than 8 \sqrt 3.

IMO 1970 LL BUL16

IMO 1970 LL BUL16 Origin: BUL Problem Show that the equation 2 −x2 + 3 −x3 = 0 has no real roots.

IMO 1970 LL BEL10

IMO 1970 LL BEL10 Origin: BEL Problem Let A, B, C be angles of a triangle. Prove that 1 < cos A + cos B + cos C \leq3 2.

IMO 1966 LL GDR27

IMO 1966 LL GDR27 Origin: GDR Problem We are given a circle K and a point P lying on a line g. Construct a circle that passes through P and touches K and g.

IMO 1972 LL MON23

IMO 1972 LL MON23 Origin: MON Problem Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an arbitrary n) for which the following equality holds: a2n . . . a1 = (an . . . a1)2?

IMO 1977 LL USS50

IMO 1977 LL USS50 Origin: USS Problem Determine all positive integers n for which there exists a poly- nomial Pn(x) of degree n with integer coefficients that is equal to n at n different integer points and that equals zero at zero. Solution Suppose that Pn(x) = n for x \in{x1, x2, . . . , xn}. Then Pn(x) = (x −x1)(x −x2) \cdot \cdot \cdot (x −xn) + n....

IMO 1966 LL GDR10

IMO 1966 LL GDR10 Origin: GDR Problem How many real solutions are there to the equation x = 1964 sinx −189?

IMO 1989 LL INA43

IMO 1989 LL INA43 Origin: INA Problem Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num- bers. Find all values of a, b, and c such that the following three conditions are satisfied simultaneously: (i) f(x) = 381 if sin x = 1/2. (ii) The absolute maximum of f(x) is 444. (iii) The absolute minimum of f(x) is 364.

IMO 1992 LL SPA66

IMO 1992 LL SPA66 Origin: SPA Problem A circle of radius \rho is tangent to the sides AB and AC of the triangle ABC, and its center K is at a distance p from BC. (a) Prove that a(p −\rho) = 2s(r −\rho), where r is the inradius and 2s the perimeter of ABC. (b) Prove that if the circle intersect BC at D and E, then DE = 4...

IMO 1967 LL HUN23

IMO 1967 LL HUN23 Origin: HUN Problem Prove that for an arbitrary pair of vectors f and g in the plane, the inequality af 2 + bfg + cg2 \geq0 holds if and only if the following conditions are fulfilled: a \geq0, c \geq0, 4ac \geqb2. Solution Suppose that a \geq0, c \geq0, 4ac \geqb2. If a = 0, then b = 0, and the inequality reduces to the obvious...

IMO 1978 LL FRA15

IMO 1978 LL FRA15 Origin: FRA Problem Prove that for every positive integer n coprime to 10 there exists a multiple of n that does not contain the digit 1 in its decimal representation.

IMO 1970 LL ROM47

IMO 1970 LL ROM47 Origin: ROM Problem Given a polynomial P(x) = ab(a −c)x3 + (a3 −a2c + 2ab2 −b2c + abc)x2 +(2a2b + b2c + a2c + b3 −abc)x + ab(b + c), where a, b, c ̸= 0, prove that P(x) is divisible by Q(x) = abx2 + (a2 + b2)x + ab and conclude that P(x0) is divisible by (a + b)3 for x0 = (a +...

IMO 1970 LL FRA23

IMO 1970 LL FRA23 Origin: FRA Problem Let E be a finite set, PE the family of its subsets, and f a mapping from PE to the set of nonnegative real numbers such that for any two disjoint subsets A, B of E, f(A \cupB) = f(A) + f(B). Prove that there exists a subset F of E such that if with each A \subsetE we associate a subset A′...

IMO 1985 LL SWE79

IMO 1985 LL SWE79 Origin: SWE Problem Let a, b, and c be real numbers such that bc −a2 + ca −b2 + ab −c2 = 0. Prove that a (bc −a2)2 + b (ca −b2)2 + c (ab −c2)2 = 0.

IMO 1971 LL NET30

IMO 1971 LL NET30 Origin: NET Problem Prove that the system of equations 2yz + x −y −z = a, 2xz −x + y −z = a, 2xy −x −y + z = a, a being a parameter, cannot have five distinct solutions. For what values of a does this system have four distinct integer solutions?

IMO 1979 LL CZS14

IMO 1979 LL CZS14 Origin: CZS Problem Let S be a set of n2 + 1 closed intervals (n a positive integer). Prove that at least one of the following assertions holds: (i) There exists a subset S′ of n + 1 intervals from S such that the inter- section of the intervals in S′ is nonempty. (ii) There exists a subset S′′ of n + 1 intervals from S...

IMO 1986 LL MOR56

IMO 1986 LL MOR56 Origin: MOR Problem Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with center O. Consider the circular arc with endpoints A1, A6 not containing

IMO 1986 LL IRE46

IMO 1986 LL IRE46 Origin: IRE Problem We wish to construct a matrix with 19 rows and 86 columns, with entries xij \in{0, 1, 2} (1 \leqi \leq19, 1 \leqj \leq86), such that: (i) in each column there are exactly k terms equal to 0; (ii) for any distinct j, k \in{1, . . . , 86} there is i \in{1, . . . , 19} with xij + xik...

IMO 1985 LL BEL6

IMO 1985 LL BEL6 Origin: BEL Problem On a one-way street, an unending sequence of cars of width a, length b passes with velocity v. The cars are separated by the distance c. A pedestrian crosses the street perpendicularly with velocity w, without paying attention to the cars. (a) What is the probability that the pedestrian crosses the street unin- jured? (b) Can he improve this probability by crossing the...

IMO 1971 LL HUN25

IMO 1971 LL HUN25 Origin: HUN Problem Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments A2B1, B2C1, C2A1 by P, Q, R in this order. Prove that the triangle PQR is equilateral.

IMO 1966 LL POL24

IMO 1966 LL POL24 Origin: POL Problem There are n \geq2 people in a room. Prove that there exist two among them having equal numbers of friends in that room. (Friendship is always mutual.)

IMO 1988 LL KOR56

IMO 1988 LL KOR56 Origin: KOR Problem The Fibonacci sequence is defined by an+1 = an + an−1 (n \geq1), a0 = 0, a1 = a2 = 1. Find the greatest common divisor of the 1960th and 1988th terms of the Fibonacci sequence.

IMO 1971 LL YUG53

IMO 1971 LL YUG53 Origin: YUG Problem Denote by xn(p) the multiplicity of the prime p in the canonical representation of the number n! as a product of primes. Prove that xn(p) n < p−1 and limn\to\infty xn(p) n p−1.

IMO 1977 LL HUN25

IMO 1977 LL HUN25 Origin: HUN Problem Prove the identity (z + a)n = zn + a n  k=1 n k  (a −kb)k−1(z + kb)n−k. Solution Let fn(z) = zn + a n  k=1 n k  (a −kb)k−1(z + kb)n−k. We shall prove by induction on n that fn(z) = (z + a)n. This is trivial for n = 1. Suppose that the statement is true...

IMO 1992 LL KOR47

IMO 1992 LL KOR47 Origin: KOR Problem Find the largest integer not exceeding $1992 n=1 3n+2 3n+1.

IMO 1974 LL CZS12

IMO 1974 LL CZS12 Origin: CZS Problem A circle K with radius r, a point D on K, and a convex angle with vertex S and rays a and b are given in the plane. Construct a parallelogram ABCD such that A and B lie on a and b respectively, SA + SB = r, and C lies on K.

IMO 1992 LL TUR74

IMO 1992 LL TUR74 Origin: TUR Problem Let S = \pin 1992m | n, m \inZ . Show that every real number x \geq0 is an accumulation point of S.

IMO 1987 LL FRA16

IMO 1987 LL FRA16 Origin: FRA Problem Let ABC be a triangle. For every point M belonging to segment BC we denote by B′ and c′ the orthogonal projections of M on the straight lines AC and BC. Find points M for which the length of segment B′C′ is a minimum.

IMO 1969 LL CZS17

IMO 1969 LL CZS17 Origin: CZS Problem Let d and p be two real numbers. Find the first term of an arith- metic progression a1, a2, a3, . . . with difference d such that a1a2a3a4 = p. Find the number of solutions in terms of d and p.

IMO 1992 LL THA70

IMO 1992 LL THA70 Origin: THA Problem Let two circles A and B with unequal radii r and R, respec- tively, be tangent internally at the point A0. If there exists a sequence of distinct circles (Cn) such that each circle is tangent to both A and B, and each circle Cn+1 touches circle Cn at the point An, prove that \infty  n=1 |An+1An| < 4\piRr R + r...

IMO 1992 LL AUS3

IMO 1992 LL AUS3 Origin: AUS Problem Let ABC be a triangle, O its circumcenter, S its centroid, and H its orthocenter. Denote by A1, B1, and C1 the centers of the circles circumscribed about the triangles CHB, CHA, and AHB, respectively. Prove that the triangle ABC is congruent to the triangle A1B1C1 and that the nine-point circle of \triangleABC is also the nine-point circle of \triangleA1B1C1.

IMO 1970 LL AUT4

IMO 1970 LL AUT4 Origin: AUT Problem Solve the system of equations x2 + xy = a2 + ab y2 + xy = a2 −ab, a, b real, a ̸= 0.

IMO 1987 LL ROM55

IMO 1987 LL ROM55 Origin: ROM Problem Two moving bodies M1, M2 are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines M1M2.

IMO 1976 LL USA39

IMO 1976 LL USA39 Origin: USA Problem In \triangleABC, the inscribed circle is tangent to side BC at X. Segment AX is drawn. Prove that the line joining the midpoint of segment AX to the midpoint of side BC passes through the center I of the inscribed circle.

IMO 1969 LL POL52

IMO 1969 LL POL52 Origin: POL Problem Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

IMO 1985 LL MOR55

IMO 1985 LL MOR55 Origin: MOR Problem The points A, B, C are in this order on line D, and AB = 4BC. Let M be a variable point on the perpendicular to D through C. Let MT1 and MT2 be tangents to the circle with center A and radius AB. Determine the locus of the orthocenter of the triangle MT1T2.

IMO 1966 LL ROM38

IMO 1966 LL ROM38 Origin: ROM Problem Two concentric circles have radii R and r respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between 2 \cdot \sqrt R+\sqrtr \sqrt R−\sqrtr −1 and 63 20 \cdot R+r R−r.

IMO 1983 LL SWE61

IMO 1983 LL SWE61 Origin: SWE Problem Let a and b be integers. Is it possible to find integers p and q such that the integers p + na and q + nb have no common prime factor no matter how the integer n is chosen.

IMO 1989 LL AUS1

IMO 1989 LL AUS1 Origin: AUS Problem In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined with the following properties: (i) c = a ∗b is in Sn for any a \inSn, b \inSn. (ii) If the ordinary product a\cdotb is less than or equal to n, then a∗b = a\cdotb. (iii) The ordinary rules of multiplication hold for ∗, i.e., (1)...

IMO 1987 LL FIN13

IMO 1987 LL FIN13 Origin: FIN Problem A be an infinite set of positive integers such that every n \inA is the product of at most 1987 prime numbers. Prove that there is an infinite set B \subsetA and a number p such that the greatest common divisor of any two distinct numbers in B is b.

IMO 1977 LL BUL1

IMO 1977 LL BUL1 Origin: BUL Problem A pentagon ABCDE inscribed in a circle for which BC < CD and AB < DE is the base of a pyramid with vertex S. If AS is the longest edge starting from S, prove that BS > CS. Solution Let P be the projection of S onto the plane ABCDE. Obviously BS > CS is equivalent to BP > CP. The conditions...

IMO 1979 LL POL51

IMO 1979 LL POL51 Origin: POL Problem Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be circles satisfying the following conditions: S1 is tangent to CA and AB, S2 is tangent to S1, AB, and BC, S3 is tangent to S2, BC, and CA, \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot S7...

IMO 1967 LL BUL2

IMO 1967 LL BUL2 Origin: BUL Problem Prove that 1 3n2 + 1 2n + 1 6 \geq(n!)2/n (n is a positive integer) and that equality is possible only in the case n = 1. Solution (n!)2/n = ((1 \cdot 2 \cdot \cdot \cdot n)1/n)2 \leq
1+2+\cdot\cdot\cdot+n n 2 =
n+1 2 \leq1 3n2 + 1 2n + 1 6.

IMO 1970 LL SWE50

IMO 1970 LL SWE50 Origin: SWE Problem The area of a triangle is S and the sum of the lengths of its sides is L. Prove that 36S \leqL2\sqrt 3 and give a necessary and sufficient condition for equality.

IMO 1967 LL ITA29

IMO 1967 LL ITA29 Origin: ITA Problem The triangles A0B0C0 and A′B′C′ have all their angles acute. Describe how to construct one of the triangles ABC, similar to A′B′C′ and circumscribing A0B0C0 (so that A, B, C correspond to A′, B′, C′, and AB passes through C0, BC through A0, and CA through B0). Among these triangles ABC, describe, and prove, how to construct the triangle with the maximum area....

IMO 1966 LL USS57

IMO 1966 LL USS57 Origin: USS Problem Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

IMO 1987 LL SPA59

IMO 1987 LL SPA59 Origin: SPA Problem It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2 are complex numbers, and that a11a22 = a12a12 (where a12 is the conju- gate of a12). We consider the following system in x1, x2: x1(a11x1 + a12x2) = b1, x2(a12x1 + a22x2) = b2. (a) Give one condition to make the system consistent. (b) Give one condition to...

IMO 1988 LL USS86

IMO 1988 LL USS86 Origin: USS Problem Let a, b, c be integers different from zero. It is known that the equation ax2 + by2 + cz2 = 0 has a solution (x, y, z) in integers different from the solution x = y = z = 0. Prove that the equation ax2+by2+cz2 = 1 has a solution in rational numbers.

IMO 1978 LL USA42

IMO 1978 LL USA42 Origin: USA Problem A, B, C, D, E are points on a circle O with radius equal to r. Chords AB and DE are parallel to each other and have length equal to x. Diagonals AC, AD, BE, CE are drawn. If segment XY on O meets AC at X and EC at Y , prove that lines BX and DY meet at Z on the...

IMO 1974 LL USS46

IMO 1974 LL USS46 Origin: USS Problem Outside an arbitrary triangle ABC, triangles ADB and BCE are constructed such that \angleADB = \angleBEC = 90◦and \angleDAB = \angleEBC = 30◦. On the segment AC the point F with AF = 3FC is chosen. Prove that \angleDFE = 90◦ and \angleFDE = 30◦.

IMO 1984 LL SPA55

IMO 1984 LL SPA55 Origin: SPA Problem Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30), p > q being two given positive integers. (a) Prove that k = a3 + b3 + c3 is not a prime number. (b) Prove that if a \cdot b \cdot c is maximum, then 1984 divides k.

IMO 1987 LL USA65

IMO 1987 LL USA65 Origin: USA Problem The runs of a decimal number are its increasing or decreasing blocks of digits. Thus 024379 has three runs: 024, 43, and 379. Determine the average number of runs for a decimal number in the set {d1d2 . . . dn | dk ̸= dk+1, k = 1, 2, . . ., n −1}, where n \geq2.

IMO 1979 LL GRE34

IMO 1979 LL GRE34 Origin: GRE Problem Notice that in the fraction 16 64 we can perform a simplification as 1̸6 ̸64 = 1 4 obtaining a correct equality. Find all fractions whose numer- ators and denominators are two-digit positive integers for which such a simplification is correct.

IMO 1988 LL POL70

IMO 1988 LL POL70 Origin: POL Problem In 3-dimensional space a point O is given and a finite set A of segments with the sum of the lengths equal to 1988. Prove that there exists a plane disjoint from A such that the distance from it to O does not exceed 574.

IMO 1983 LL BUL13

IMO 1983 LL BUL13 Origin: BUL Problem Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat- ural numbers less than or equal to p. Prove that for each natural number r less than or equal to p, there exist two numbers (perhaps equal) ai and aj such that p \equivaiaj(mod r).

IMO 1988 LL HUN38

IMO 1988 LL HUN38 Origin: HUN Problem In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any 3 of them there was a question that the three students answered differently. What is the maximal possible number of students tested?

IMO 1970 LL SWE53

IMO 1970 LL SWE53 Origin: SWE Problem A square ABCD is divided into (n −1)2 congruent squares, with sides parallel to the sides of the given square. Consider the grid of all n2 corners obtained in this manner. Determine all integers n for which it is possible to construct a nondegenerate parabola with its axis parallel to one side of the square and that passes through exactly n points of...

IMO 1977 LL GDR15

IMO 1977 LL GDR15 Origin: GDR Problem Let n be an integer greater than 1. In the Cartesian coordinate system we consider all squares with integer vertices (x, y) such that 1 \leq x, y \leqn. Denote by pk (k = 0, 1, 2, . . . ) the number of pairs of points that are vertices of exactly k such squares. Prove that  k(k −1)pk = 0. Solution...

IMO 1992 LL POL57

IMO 1992 LL POL57 Origin: POL Problem For positive numbers a, b, c define A = (a + b + c)/3, G = (abc)1/3, H = 3/(a−1 + b−1 + c−1). Prove that A G 3 \geq1 4 + 3 4 \cdot A H , for every a, b, c > 0.

IMO 1985 LL CYP18

IMO 1985 LL CYP18 Origin: CYP Problem The circles (R, r) and (P, \rho), where r > \rho, touch externally at A. Their direct common tangent touches (R, r) at B and (P, \rho) at C. The line RP meets the circle (P, \rho) again at D and the line BC at E. If |BC| = 6|DE|, prove that: (a) the lengths of the sides of the triangle RBE are...

IMO 1977 LL USS48

IMO 1977 LL USS48 Origin: USS Problem The intersection of a plane with a regular tetrahedron with edge a is a quadrilateral with perimeter P. Prove that 2a \leqP \leq3a. Solution Let a plane cut the edges AB, BC, CD, DA at points K, L, M, N respec- tively. Let D′, A′, B′ be distinct points in the plane ABC such that the triangles BCD′, CD′A′, D′A′B′ are equilateral, and...

IMO 1972 LL ROM33

IMO 1972 LL ROM33 Origin: ROM Problem A rectangle ABCD is given whose sides have lengths 3 and 2n, where n is a natural number. Denote by U(n) the number of ways in which one can cut the rectangle into rectangles of side lengths 1 and 2. (a) Prove that U(n + 1) + U(n −1) = 4U(n); (b) Prove that U(n) = \sqrt 3[( \sqrt 3 + 1)(2 +...

IMO 1967 LL BUL3

IMO 1967 LL BUL3 Origin: BUL Problem Prove the trigonometric inequality cos x < 1 −x2 2 + x4 16, where x \in(0, \pi/2). Solution Consider the function f : [0, \pi/2] \toR defined by f(x) = 1 −x2/2 + x4/16 −cos x. It is easy to calculate that f ′(0) = f ′′(0) = f ′′′(0) = 0 and f ′′′′(x) = 3/2 −cos x. Since f ′′′′(x) >...

IMO 1985 LL CAN14

IMO 1985 LL CAN14 Origin: CAN Problem Let k be a positive integer. Define u0 = 0, u1 = 1, and un = kun−1 −un−2, n \geq2. Show that for each integer n, the number u3 1 + u3 2 + \cdot \cdot \cdot + u3 n is a multiple of u1 + u2 + \cdot \cdot \cdot + un.

IMO 1970 LL BEL11

IMO 1970 LL BEL11 Origin: BEL Problem Let ABCD and A′B′C′D′ be two squares in the same plane and oriented in the same direction. Let A′′, B′′, C′′, and D′′ be the midpoints of AA′, BB′, CC′, and DD′. Prove that A′′B′′C′′D′′ is also a square.

IMO 1985 LL ITA46

IMO 1985 LL ITA46 Origin: ITA Problem Let C be the curve determined by the equation y = x3 in the rectangular coordinate system. Let t be the tangent to C at a point P of C; t intersects C at another point Q. Find the equation of the set L of the midpoints M of PQ as P describes C. Is the correspondence associating P and M a bijection...

IMO 1979 LL HUN40

IMO 1979 LL HUN40 Origin: HUN Problem A polynomial P(x) has degree at most 2k, where k = 0, 1, 2, . . . . Given that for an integer i, the inequality −k \leqi \leqk implies |P(i)| \leq1, prove that for all real numbers x, with −k \leqx \leqk, the following inequality holds: |P(x)| < (2k + 1) 2k k  .

IMO 1966 LL USS51

IMO 1966 LL USS51 Origin: USS Problem In a school, n children numbered 1 to n are initially arranged in the order 1, 2, . . . , n. At a command, every child can either exchange its position with any other child or not move. Can they rearrange into the order n, 1, 2, . . . , n −1 after two commands?

IMO 1984 LL BEL6

IMO 1984 LL BEL6 Origin: BEL Problem Let P, Q, R be the polynomials with real or complex coefficients such that at least one of them is not constant. If P n +Qn +Rn = 0, prove that n < 3.

IMO 1989 LL POL79

IMO 1989 LL POL79 Origin: POL Problem To each pair (x, y) of distinct elements of a finite set X a number f(x, y) equal to 0 or 1 is assigned in such a way that f(x, y) ̸= f(y, x) for all x, y (x ̸= y). Prove that exactly one of the following situations occurs: (i) X is the union of two disjoint nonempty subsets U, V such...

IMO 1987 LL MON42

IMO 1987 LL MON42 Origin: MON Problem Find the integer solutions of the equation \sqrt 2 m   (2 + \sqrt 2)n  .

IMO 1992 LL MON50

IMO 1992 LL MON50 Origin: MON Problem Let N be a point inside the triangle ABC. Through the mid- points of the segments AN, BN, and CN the lines parallel to the opposite sides of \triangleABC are constructed. Let AN, BN, and CN be the intersection points of these lines. If N is the orthocenter of the triangle ABC, prove that the nine-point circles of \triangleABC and \triangleANBNCN coincide. Remark....

IMO 1970 LL AUT5

IMO 1970 LL AUT5 Origin: AUT Problem Prove that n n+1 + n+1 + \cdot \cdot \cdot + n n+1 \geq1 for n \geq2.

IMO 1967 LL USS55

IMO 1967 LL USS55 Origin: USS Problem Find all x for which for all n, sin x + sin 2x + sin 3x + \cdot \cdot \cdot + sin nx \leq \sqrt 2 . Solution It is enough to find all x from (0, 2\pi] such that the given inequality holds for all n. Suppose 0 < x < 2\pi/3. If n is the maximum integer for which nx \leq...

IMO 1976 LL GBR17

IMO 1976 LL GBR17 Origin: GBR Problem Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are \sqrt 3 : \sqrt 3 : 2.

IMO 1967 LL BUL4

IMO 1967 LL BUL4 Origin: BUL Problem Suppose medians ma and mb of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled triangle. (b) The inequality 5(a2 + b2 −c2) \geq8ab is valid, where a, b, and c are side lengths of the given triangle. Solution (a) Let ABCD be a parallelogram, and K, L the midpoints of segments BC...

IMO 1989 LL HUN38

IMO 1989 LL HUN38 Origin: HUN Problem Connecting the vertices of a regular n-gon we obtain a closed (not necessarily convex) n-gon. Show that if n is even, then there are two parallel segments among the connecting segments and if n is odd then there cannot be exactly two parallel segments.

IMO 1966 LL USS6

IMO 1966 LL USS6 Origin: USS Problem A convex planar polygon M with perimeter l and area S is given. Let M(R) be the set of all points in space that lie a distance at most R from a point of M. Show that the volume V (R) of this set equals V (R) = 4 3\piR3 + \pi 2 lR2 + 2SR.

IMO 1969 LL BEL1

IMO 1969 LL BEL1 Origin: BEL Problem A parabola P1 with equation x2 −2py = 0 and parabola P2 with equation x2 + 2py = 0, p > 0, are given. A line t is tangent to P2. Find the locus of pole M of the line t with respect to P1.

IMO 1978 LL VIE50

IMO 1978 LL VIE50 Origin: VIE Problem A variable tetrahedron ABCD has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal (BC = DA, CA = DB, AB = DC); and the vertices A, B, C lie respectively on three fixed spheres with the same center P and radii 3, 4, 12. What is the maximal length of PD?

IMO 1983 LL KUW39

IMO 1983 LL KUW39 Origin: KUW Problem If \alpha is the real root of the equation E(x) = x3 −5x −50 = 0 such that xn+1 = (5xn + 50)1/3 and x1 = 5, where n is a positive integer, prove that: (a) x3 n+1 −\alpha3 = 5(xn −\alpha) (b) \alpha < xn+1 < xn

IMO 1983 LL CUB22

IMO 1983 LL CUB22 Origin: CUB Problem Does there exist an infinite number of sets C consisting of 1983 consecutive natural numbers such that each of the numbers is divisible by some number of the form a1983, with a \inN, a ̸= 1?

IMO 1988 LL INA45

IMO 1988 LL INA45 Origin: INA Problem (a) Consider a circle K with diameter AB, a circle L tangent to AB and to K, and a circle M tangent to circle K, circle L, and AB. Calculate the ratio of the area of circle K to the area of circle M. (b) In triangle ABC, AB = AC and ∡CAB = 80◦. If points D, E, and F lie on...

IMO 1970 LL BEL7

IMO 1970 LL BEL7 Origin: BEL Problem Let ABCD be any quadrilateral. A square is constructed on each side of the quadrilateral, all in the same manner (i.e., outward or inward). Denote the centers of the squares by M1, M2, M3, and M4. Prove: (a) M1M3 = M2M4; (b) M1M3 is perpendicular to M2M4.

IMO 1977 LL GDR17

IMO 1977 LL GDR17 Origin: GDR Problem A ball K of radius r is touched from the outside by mutually equal balls of radius R. Two of these balls are tangent to each other. Moreover, for two balls K1 and K2 tangent to K and tangent to each other there exist two other balls tangent to K1, K2 and also to K. How many balls are tangent to K? For...

IMO 1982 LL AUS2

IMO 1982 LL AUS2 Origin: AUS Problem Given a finite number of angular regions A1, . . . , Ak in a plane, each Ai being bounded by two half-lines meeting at a vertex and provided with a + or −sign, we assign to each point P of the plane and not on a bounding half-line the number k −l, where k is the number of + regions and l...

IMO 1984 LL MON36

IMO 1984 LL MON36 Origin: MON Problem The set {1, 2, . . ., 49} is divided into three subsets. Prove that at least one of these subsets contains three different numbers a, b, c such that a + b = c.

IMO 1971 LL HUN23

IMO 1971 LL HUN23 Origin: HUN Problem Find all integer solutions of the equation x2 + y2 = (x −y)3.

IMO 1967 LL MON35

IMO 1967 LL MON35 Origin: MON Problem Prove the identity n  k=0 n k   tan x 2k 1 + 2k
1 −tan2(x/2) k = sec2n x 2 + secn x. Solution The given sum can be rewritten as n  k=0 n k   tan2 x k + n  k=0 n k   2 tan2 x 1 −tan2 x k . Since 2...

IMO 1992 LL FRG20

IMO 1992 LL FRG20 Origin: FRG Problem Let X and Y be two sets of points in the plane and M be a set of segments connecting points from X and Y . Let k be a natural number. Prove that the segments from M can be painted using k colors in such a way that for any point x \inX \cupY and two colors \alpha and \beta (\alpha ̸=...

IMO 1986 LL MON55

IMO 1986 LL MON55 Origin: MON Problem Given an integer n \geq2, determine all n-digit numbers M0 = a1a2 . . . an (ai ̸= 0, i = 1, 2, . . ., n) divisible by the numbers M1 = a2a3 . . . ana1, M2 = a3a4 . . . ana1a2, . . . , Mn−1 = ana1a2 . . . an−1.

IMO 1979 LL HUN38

IMO 1979 LL HUN38 Origin: HUN Problem Prove the following statement: If a polynomial f(x) with real coefficients takes only nonnegative values, then there exists a positive integer n and polynomials g1(x), g2(x), . . . , gn(x) such that f(x) = g1(x)2 + g2(x)2 + \cdot \cdot \cdot + gn(x)2.

IMO 1976 LL USS46

IMO 1976 LL USS46 Origin: USS Problem For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality a4 + b4 + c4 + d4 + 2abcd \geqa2b2 + a2c2 + a2d2 + b2c2 + b2d2 + c2d2.

IMO 1988 LL INA42

IMO 1988 LL INA42 Origin: INA Problem (a) Four balls of radius 1 are mutually tangent, three resting an the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length s, is circumscribed around the balls. Find the value of s. (b) Suppose that ABCD and EFGH are opposite faces of a rectangu- lar solid, with \angleDHC = 45◦and \angleFHB = 60◦. Find the...

IMO 1972 LL ROM35

IMO 1972 LL ROM35 Origin: ROM Problem (a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b = −cm + d = m, (i) \sqrt a2 + b2 + \sqrt c2 + d2 + (a −c)2 + (b −d)2 \geq 4m2 1+m2 , and (ii) 2 \leq 4m2 1+m2 < 4. (b) Express a, b, c, d as functions of m so that there...

IMO 1989 LL POR84

IMO 1989 LL POR84 Origin: POR Problem Let a, b, c, r, and s be real numbers. Show that if r is a root of ax2 +bx+c = 0 and s is a root of −ax2 +bx+c = 0, then a 2x2 +bx+c = 0 has a root between r and s.

IMO 1969 LL NET47

IMO 1969 LL NET47 Origin: NET Problem Let A and B be points on the circle \gamma. A point C, different from A and B, is on the circle \gamma. Let D be the projection of the point C onto the line AB. Consider three other circles \gamma1, \gamma2, and \gamma3 with the common tangent AB: \gamma1 inscribed in the triangle ABC, and \gamma2 and \gamma3 tangent to both (the...

IMO 1979 LL HUN39

IMO 1979 LL HUN39 Origin: HUN Problem A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of n days of walking from the camp, under the following conditions: (i) Each member of the expedition can pick up at most 3 liters of water. (ii) Each member must drink one liter...

IMO 1983 LL FRG25

IMO 1983 LL FRG25 Origin: FRG Problem How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange ai and ai+1 whenever ai > ai+1 for i running from 1 up to n −1?

IMO 1966 LL HUN19

IMO 1966 LL HUN19 Origin: HUN Problem Construct a triangle given the three exradii.

IMO 1988 LL HKG30

IMO 1988 LL HKG30 Origin: HKG Problem Find the total number of different integers that the function f(x) = [x] + [2x] + 5x [3x] + [4x] takes for 0 \leqx \leq100.

IMO 1988 LL IRE49

IMO 1988 LL IRE49 Origin: IRE Problem Let −1 < x < 1. Show that  k=0 1 −x2 1 −2x cos(2\pik/7) + x2 = 7(1 + x7) 1 −x7 . Deduce that csc2 \pi 7 + csc2 2\pi 7 + csc2 3\pi 7 = 8.

IMO 1966 LL ROM9

IMO 1966 LL ROM9 Origin: ROM Problem Find x such that sin 3x cos(60◦−4x) + 1 sin(60◦−7x) −cos(30◦+ x) + m = 0, where m is a fixed real number.

IMO 1985 LL SPA74

IMO 1985 LL SPA74 Origin: SPA Problem Find the triples of positive integers x, y, z satisfying x + 1 y + 1 z = 4 5.

IMO 1985 LL FRA24

IMO 1985 LL FRA24 Origin: FRA Problem Let d \geq1 be an integer that is not the square of an integer. Prove that for every integer n \geq1, (n \sqrt d + 1)| sin(n\pi \sqrt d)| \geq1.

IMO 1986 LL FIN20

IMO 1986 LL FIN20 Origin: FIN Problem For any angle \alpha with 0 < \alpha < 180◦, we call a closed convex planar set an \alpha-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is \alpha. Given a (closed) triangle T , find the greatest \alpha such that any two points in T are contained in an \alpha-set S...

IMO 1979 LL HUN41

IMO 1979 LL HUN41 Origin: HUN Problem Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

IMO 1988 LL INA44

IMO 1988 LL INA44 Origin: INA Problem (a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the polynomial g(x12) is divided by the polynomial g(x)? (b) If k is a positive integer and f is a function such that for every positive number x, f(x2 +1) \sqrtx = k, find the value of f  9+y2 y2 \sqrt 12/y...

IMO 1988 LL VIE93

IMO 1988 LL VIE93 Origin: VIE Problem Given a natural number n, find all polynomials P(x) of degree less than n satisfying the following condition: n  i=0 P(i)(−1)i n i  = 0.

IMO 1976 LL VIE49

IMO 1976 LL VIE49 Origin: VIE Problem Determine whether there exist 1976 nonsimilar triangles with angles \alpha, \beta, \gamma, each of them satisfying the relations sin \alpha + sin \beta + sin \gamma cos \alpha + cos \beta + cos \gamma = 12 and sin \alpha sin \beta sin \gamma = 12 25.

IMO 1969 LL GBR25

IMO 1969 LL GBR25 Origin: GBR Problem Let a, b, x, y be positive integers such that a and b have no common divisor greater than 1. Prove that the largest number not ex- pressible in the form ax + by is ab −a −b. If N(k) is the largest number not expressible in the form ax + by in only k ways, find N(k).

IMO 1989 LL GBR26

IMO 1989 LL GBR26 Origin: GBR Problem Let a, b, c, d be positive integers such that ab = cd and a + b = c −d. Prove that there exists a right-angled triangle the measures of whose sides (in some unit) are integers and whose area measure is ab square units.

IMO 1979 LL FRA23

IMO 1979 LL FRA23 Origin: FRA Problem Consider the set E consisting of pairs of integers (a, b), with a \geq 1 and b \geq1, that satisfy in the decimal system the following properties: (i) b is written with three digits, as \alpha2\alpha1\alpha0, \alpha2 ̸= 0; (ii) a is written as \betap . . . \beta1\beta0 for some p; (iii) (a + b)2 is written as \betap . . ....

IMO 1978 LL FIN12

IMO 1978 LL FIN12 Origin: FIN Problem The equation x3 + ax2 + bx + c = 0 has three (not necessarily distinct) real roots t, u, v. For which a, b, c do the numbers t3, u3, v3 satisfy the equation x3 + a3x2 + b3x + c3 = 0?

IMO 1983 LL COL21

IMO 1983 LL COL21 Origin: COL Problem Prove that there are infinitely many positive integers n for which it is possible for a knight, starting at one of the squares of an n \times n chessboard, to go through each of the squares exactly once.

IMO 1989 LL HKG35

IMO 1989 LL HKG35 Origin: HKG Problem Find all square numbers S1 and S2 such that S1 −S2 = 1989.

IMO 1983 LL USA65

IMO 1983 LL USA65 Origin: USA Problem Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect in a point P. Prove that AP PC = cot \angleBAC + cot \angleDAC cot \angleBCA + cot \angleDCA.

IMO 1982 LL USA48

IMO 1982 LL USA48 Origin: USA Problem Given a finite sequence of complex numbers c1, c2, . . . , cn, show that there exists an integer k (1 \leqk \leqn) such that for every finite sequence a1, a2, . . . , an of real numbers with 1 \geqa1 \geqa2 \geq\cdot \cdot \cdot \geqan \geq0, the following inequality holds: n  m=1 amcmn \leq n  m=1 cm .

IMO 1969 LL HUN38

IMO 1969 LL HUN38 Origin: HUN Problem Let r and m (r \leqm) be natural numbers and Ak = 2k−1 2m \pi. Evaluate m2 m  k=1 m  l=1 sin(rAk) sin(rAl) cos(rAk −rAl).

IMO 1982 LL USA47

IMO 1982 LL USA47 Origin: USA Problem Evaluate sec′′ \pi 4 +sec′′ 3\pi 4 +sec′′ 5\pi 4 +sec′′ 7\pi 4 . (Here sec′′ means the second derivative of sec.)

IMO 1988 LL SWE80

IMO 1988 LL SWE80 Origin: SWE Problem Let S be an infinite set of integers containing zero and such that the distance between successive numbers never exceeds a given fixed number. Consider the following procedure: Given a set X of integers, we construct a new set consisting of all numbers x \pm s, where x belongs to X and s belongs to S. Starting from S0 = {0} we successively...

IMO 1983 LL BEL5

IMO 1983 LL BEL5 Origin: BEL Problem Consider the set Q2 of points in R2, both of whose coordinates are rational. (a) Prove that the union of segments with vertices from Q2 is the entire set R2. (b) Is the convex hull of Q2 (i.e., the smallest convex set in R2 that con- tains Q2) equal to R2?

IMO 1976 LL USS42

IMO 1976 LL USS42 Origin: USS Problem For a point O inside a triangle ABC, denote by A1, B1, C1 the respective intersection points of AO, BO, CO with the corresponding sides. Let n1 = AO A1O, n2 = BO B1O, n3 = CO C1O . What possible values of n1, n2, n3 can all be positive integers?

IMO 1977 LL ROM36

IMO 1977 LL ROM36 Origin: ROM Problem Consider a sequence of numbers (a1, a2, . . . , a2n). Define the operation S((a1, a2, . . . , a2n)) = (a1a2, a2a3, . . . , a2n−1a2n, a2na1). Prove that whatever the sequence (a1, a2, . . . , a2n) is, with ai \in{−1, 1} for i = 1, 2, . . . , 2n, after finitely many applications of...

IMO 1969 LL FRA21

IMO 1969 LL FRA21 Origin: FRA Problem A right-angled triangle OAB has its right angle at the point B. An arbitrary circle with center on the line OB is tangent to the line OA. Let AT be the tangent to the circle different from OA (T is the point of tangency). Prove that the median from B of the triangle OAB intersects AT at a point M such that MB...

IMO 1985 LL USA88

IMO 1985 LL USA88 Origin: USA Problem Determine the range of w(w + x)(w + y)(w + z), where x, y, z, and w are real numbers such that x + y + z + w = x7 + y7 + z7 + w7 = 0.

IMO 1987 LL GRE32

IMO 1987 LL GRE32 Origin: GRE Problem Solve the equation 28x = 19y + 87z, where x, y, z are integers.

IMO 1984 LL USA61

IMO 1984 LL USA61 Origin: USA Problem A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

IMO 1983 LL COL20

IMO 1983 LL COL20 Origin: COL Problem Let f and g be functions from the set A to the same set A. We define f to be a functional nth root of g (n is a positive integer) if f n(x) = g(x), where f n(x) = f n−1(f(x)). (a) Prove that the function g : R \toR, g(x) = 1/x has an infinite number of nth functional roots for...

IMO 1979 LL VIE72

IMO 1979 LL VIE72 Origin: VIE Problem Let f(x) be a polynomial with integer coefficients. Prove that if f(x) equals 1979 for four different integer values of x, then f(x) cannot be equal to 2 \times 1979 for any integral value of x.

IMO 1986 LL MON52

IMO 1986 LL MON52 Origin: MON Problem Solve the system of equations tan x1 + cot x1 = 3 tan x2, tan x2 + cot x2 = 3 tan x3, \cdot \cdot \cdot \cdot \cdot \cdot tan xn + cot xn = 3 tan x1.

IMO 1967 LL POL38

IMO 1967 LL POL38 Origin: POL Problem Does there exist an integer such that its cube is equal to 3n2 + 3n + 7, where n is integer? Solution Suppose that there exist integers n and m such that m3 = 3n2 + 3n + 7. Then from m3 \equiv1 (mod 3) it follows that m = 3k + 1 for some k \inZ. Substituting into the initial equation we...

IMO 1987 LL MON41

IMO 1987 LL MON41 Origin: MON Problem Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

IMO 1976 LL BUL2

IMO 1976 LL BUL2 Origin: BUL Problem Let P be a set of n points and S a set of l segments. It is known that: (i) No four points of P are coplanar. (ii) Any segment from S has its endpoints at P. (iii) There is a point, say g, in P that is the endpoint of a maximal number of segments from S and that is not a...

IMO 1988 LL CUB5

IMO 1988 LL CUB5 Origin: CUB Problem Let k be a positive integer and Mk the set of all the integers that are between 2k2 + k and 2k2 + 3k, both included. Is it possible to partition Mk into two subsets A and B such that  x\inA x2 =  x\inB x2?

IMO 1969 LL FRA18

IMO 1969 LL FRA18 Origin: FRA Problem Let a and b be two nonnegative integers. Denote by H(a, b) the set of numbers n of the form n = pa + qb, where p and q are positive integers. Determine H(a) = H(a, a). Prove that if a ̸= b, it is enough to know all the sets H(a, b) for coprime numbers a, b in order to know all...

IMO 1992 LL NET52

IMO 1992 LL NET52 Origin: NET Problem Let n be an integer > 1. In a circular arrangement of n lamps L0, . . . , Ln−1, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, Step0, Step1, . . . . If Lj−1 (j is taken mod n) is ON,...

IMO 1992 LL USA81

IMO 1992 LL USA81 Origin: USA Problem Suppose that points X, Y, Z are located on sides BC, CA, and AB, respectively, of \triangleABC in such a way that \triangleXY Z is similar to \triangleABC. Prove that the orthocenter of \triangleXY Z is the circumcenter of \triangleABC.

IMO 1989 LL MON70

IMO 1989 LL MON70 Origin: MON Problem Three mutually nonparallel lines li (i = 1, 2, 3) are given in a plane. The lines li determine a triangle and reflections fi with axes on lines li. Prove that for every point of the plane, there exists a finite composition of the reflections fi that maps that point to a point interior to the triangle.

IMO 1988 LL VIE91

IMO 1988 LL VIE91 Origin: VIE Problem A regular 14-gon with side length a is inscribed in a circle of radius one. Prove that 2 −a 2a  3 cos \pi 7 .

IMO 1979 LL BRA7

IMO 1979 LL BRA7 Origin: BRA Problem M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four. Given that: (i) for each i = 1, 2, 3, 4 and for each k = 5, 6, 7, ai,k = ai,k−4; Pi = a1,i + a2,i+1 + a3,i+2 + a4,i+3; Si = a4,i + a3,i+1 + a2,i+2 + a1,i+3; Li = ai,1 + ai,2 +...

IMO 1977 LL FIN46

IMO 1977 LL FIN46 Origin: FIN Problem Let f be a strictly increasing function defined on the set of real numbers. For x real and t positive, set g(x, t) = f(x + t) −f(x) f(x) −f(x −t). Assume that the inequalities 2−1 < g(x, t) < 2 hold for all positive t if x = 0, and for all t \leq|x| otherwise. Show that 14−1 < g(x, t) <...

IMO 1970 LL GDR31

IMO 1970 LL GDR31 Origin: GDR Problem Prove that for any triangle with sides a, b, c and area P the following inequality holds: P \leq \sqrt 4 (abc)2/3. Find all triangles for which equality holds.

IMO 1987 LL ROM54

IMO 1987 LL ROM54 Origin: ROM Problem Let n be a natural number. Solve in integers the equation xn + yn = (x −y)n+1.

IMO 1972 LL BUL5

IMO 1972 LL BUL5 Origin: BUL Problem Given a pyramid whose base is an n-gon inscribable in a circle, let H be the projection of the top vertex of the pyramid to its base. Prove that the projections of H to the lateral edges of the pyramid lie on a circle.

IMO 1982 LL USA49

IMO 1982 LL USA49 Origin: USA Problem Simplify n  k=0 (2n)! (k!)2((n −k)!)2 .

IMO 1976 LL CZS6

IMO 1976 LL CZS6 Origin: CZS Problem For each point X of a given polytope, denote by f(X) the sum of the distances of the point X from all the planes of the faces of the polytope. Prove that if f attains its maximum at an interior point of the polytope, then f is constant.

IMO 1989 LL KOR60

IMO 1989 LL KOR60 Origin: KOR Problem A real-valued function f on Q satisfies the following conditions for arbitrary \alpha, \beta \inQ: (i) f(0) = 0, (ii) f(\alpha) > 0 if \alpha ̸= 0, (iii) f(\alpha\beta) = f(\alpha)f(\beta), (iv) f(\alpha + \beta) \leqf(\alpha) + f(\beta), (v) f(m) \leq1989 for all m \inZ. Prove that f(\alpha + \beta) = max{f(\alpha), f(\beta)} if f(\alpha) ̸= f(\beta). Here, Z, Q denote the sets...

IMO 1970 LL FRA27

IMO 1970 LL FRA27 Origin: FRA Problem Find a natural number n such that for all prime numbers p, n is divisible by p if and only if n is divisible by p −1.

IMO 1966 LL ROM30

IMO 1966 LL ROM30 Origin: ROM Problem If n is a natural number, prove that (a) log10(n + 1) > 10n + log10 n; (b) log n! > 3n
1 2 + 1 3 + \cdot \cdot \cdot + 1 n −1  .

IMO 1989 LL ROM90

IMO 1989 LL ROM90 Origin: ROM Problem Prove that the sequence (an)n\geq0, an = [n \sqrt 2], contains an infinite number of perfect squares.

IMO 1983 LL LUX40

IMO 1983 LL LUX40 Origin: LUX Problem Four faces of tetrahedron ABCD are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are a < b < c, determine the radius of the sphere circumscribed about the tetrahedron as a function on a, b, and c. What is the ratio c/a if R = a?

IMO 1977 LL FRG11

IMO 1977 LL FRG11 Origin: FRG Problem Let n and z be integers greater than 1 and (n, z) = 1. Prove: (a) At least one of the numbers zi = 1+z+z2+\cdot \cdot \cdot+zi, i = 0, 1, . . . , n−1, is divisible by n. (b) If (z−1, n) = 1, then at least one of the numbers zi, i = 0, 1, . . ., n−2, is...

IMO 1978 LL CUB5

IMO 1978 LL CUB5 Origin: CUB Problem Prove that for any triangle ABC there exists a point P in the plane of the triangle and three points A′, B′, and C′ on the lines BC, AC, and AB respectively such that AB \cdot PC′ = AC \cdot PB′ = BC \cdot PA′ = 0.3M 2, where M = max{AB, AC, BC}.

IMO 1985 LL MON50

IMO 1985 LL MON50 Origin: MON Problem From each of the vertices of a regular n-gon a car starts to move with constant speed along the perimeter of the n-gon in the same direction. Prove that if all the cars end up at a vertex A at the same time, then they never again meet at any other vertex of the n-gon. Can they meet again at A?

IMO 1969 LL YUG70

IMO 1969 LL YUG70 Origin: YUG Problem A park has the shape of a convex pentagon of area 5 \sqrt 3 ha (= 50000 \sqrt 3 m2). A man standing at an interior point O of the park notices that he stands at a distance of at most 200 m from each vertex of the pentagon. Prove that he stands at a distance of at least 100 m from each...

IMO 1989 LL FIN17

IMO 1989 LL FIN17 Origin: FIN Problem Let a, 0 < a < 1, be a real number and f a continuous function on [0, 1] satisfying f(0) = 0, f(1) = 1, and f x + y  = (1 −a)f(x) + af(y) for all x, y \in[0, 1] with x \leqy. Determine f(1/7).

IMO 1983 LL USA67

IMO 1983 LL USA67 Origin: USA Problem The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.

IMO 1989 LL BUL5

IMO 1989 LL BUL5 Origin: BUL Problem The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal- ities a0 = \sqrt 2 , an+1 = \sqrt 1 − 1 −a2n, n = 0, 1, 2, . . . and b0 = 1, bn+1 = 1 + b2n −1 bn , n = 0, 1, 2, . . .. Prove the inequalities 2n+2an...

IMO 1970 LL ROM46

IMO 1970 LL ROM46 Origin: ROM Problem Given a triangle ABC and a plane \pi having no common points with the triangle, find a point M such that the triangle determined by the points of intersection of the lines MA, MB, MC with \pi is congruent to the triangle ABC.

IMO 1989 LL INA46

IMO 1989 LL INA46 Origin: INA Problem Given two distinct numbers b1 and b2, their product can be formed in two ways: b1 \times b2 and b2 \times b1. Given three distinct numbers, b1, b2, b3, their product can be formed in twelve ways: b1 \times(b2 \timesb3); (b1 \times b2) \times b3; b1 \times (b3 \times b2); (b1 \times b3) \times b2; b2 \times (b1 \times b3); (b2 \times b1)...

IMO 1986 LL FRG30

IMO 1986 LL FRG30 Origin: FRG Problem Prove that a convex polyhedron all of whose faces are equilat- eral triangles has at most 30 edges.

IMO 1967 LL POL36

IMO 1967 LL POL36 Origin: POL Problem Prove that the center of the sphere circumscribed around a tetrahedron ABCD coincides with the center of a sphere inscribed in that tetrahedron if and only if AB = CD, AC = BD, and AD = BC. Solution Suppose that the skew edges of the tetrahedron ABCD are equal. Let K, L, M, P, Q, R be the midpoints of edges AB, AC,...

IMO 1989 LL MON67

IMO 1989 LL MON67 Origin: MON Problem A family of sets A1, A2, . . . , An has the following properties: (i) Each Ai contains 30 elements. (ii) Ai \capAj contains exactly one element for all i, j, 1 \leqi < j \leq30. Find the largest possible n if the intersection of all these sets is empty.

IMO 1971 LL BUL9

IMO 1971 LL BUL9 Origin: BUL Problem The base of an inclined prism is a triangle ABC. The per- pendicular projection of B1, one of the top vertices, is the midpoint of BC. The dihedral angle between the lateral faces through BC and AB is \alpha, and the lateral edges of the prism make an angle \beta with the base. If r1, r2, r3 are exradii of a perpendicular section...

IMO 1983 LL ROM55

IMO 1983 LL ROM55 Origin: ROM Problem For every a \inN denote by M(a) the number of elements of the set {b \inN | a + b is a divisor of ab}. Find maxa\leq1983 M(a).

IMO 1989 LL CUB13

IMO 1989 LL CUB13 Origin: CUB Problem Let n be a natural number not greater than 44. Prove that for any function f defined over N2 whose images are in the set {1, 2, . . ., n}, there are four ordered pairs (i, j), (i, k), (l, j), and (l, k) such that f(i, j) = f(i, k) = f(l, j) = f(l, k), where i, j, k, l...

IMO 1971 LL SWE40

IMO 1971 LL SWE40 Origin: SWE Problem Prove that  1 −1   1 −1   1 −1  \cdot \cdot \cdot  1 −1 n3  1 2, n = 2, 3, . . ..

IMO 1986 LL FIN19

IMO 1986 LL FIN19 Origin: FIN Problem Let f : [0, 1] \to[0, 1] satisfy f(0) = 0, f(1) = 1 and f(x + y) −f(x) = f(x) −f(x −y) for all x, y \geq0 with x −y, x + y \in[0, 1]. Prove that f(x) = x for all x \in[0, 1].

IMO 1988 LL KOR55

IMO 1988 LL KOR55 Origin: KOR Problem Find all positive integers x such that the product of all digits of x is given by x2 −10x −22.

IMO 1971 LL CUB13

IMO 1971 LL CUB13 Origin: CUB Problem One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: Martian : I have spent 1/12 of my life on Pluton. Human : I also have. Venusian : Me too. Martian : But Venusian and I have spend much more time here than you, Human. Human : That is true. However, Venusian and I are of...

IMO 1992 LL PRK61

IMO 1992 LL PRK61 Origin: PRK Problem There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange...

IMO 1982 LL BRA9

IMO 1982 LL BRA9 Origin: BRA Problem Let n be a natural number, n \geq2, and let \varphi be Euler’s function; i.e., \varphi(n) is the number of positive integers not exceeding n and coprime to n. Given any two real numbers \alpha and \beta, 0 \leq\alpha < \beta \leq1, prove that there exists a natural number m such that \alpha < \varphi(m) m < \beta.

IMO 1967 LL ROM47

IMO 1967 LL ROM47 Origin: ROM Problem Prove the inequality x1x2 \cdot \cdot \cdot xk
xn−1 xn−1 \cdot \cdot \cdot + xn−1 k  \leqxn+k−1 +xn+k−1 +\cdot \cdot \cdot+xn+k−1 k , where xi > 0 (i = 1, 2, . . ., k), k \inN, n \inN. Solution Using the A–G mean inequality we get (n + k −1)xn 1x2 \cdot \cdot \cdot xk \leqnxn+k−1 xn+k−1 \cdot \cdot \cdot...

IMO 1985 LL USS90

IMO 1985 LL USS90 Origin: USS Problem Decompose the number 51985−1 into a product of three integers, each of which is larger than 5100.

IMO 1982 LL CZS21

IMO 1982 LL CZS21 Origin: CZS Problem All edges and all diagonals of regular hexagon A1A2A3A4A5A6 are colored blue or red such that each triangle AjAkAm, 1 \leqj < k < m \leq6 has at least one red edge. Let Rk be the number of red segments AkAj, (j ̸= k). Prove the inequality  k=1 (2Rk −7)2 \leq54.

IMO 1978 LL CUB6

IMO 1978 LL CUB6 Origin: CUB Problem Prove that for all X > 1 there exists a triangle whose sides have lengths P1(X) = X4+X3+2X2+X+1, P2(X) = 2X3+X2+2X+1, and P3(X) = X4−1. Prove that all these triangles have the same greatest angle and calculate it.

IMO 1982 LL POL40

IMO 1982 LL POL40 Origin: POL Problem We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a...

IMO 1989 LL GBR28

IMO 1989 LL GBR28 Origin: GBR Problem Let b1, b2, . . . , b1989 be positive real numbers such that the equations xr−1 −2xr + xr+1 + brxr = 0 (1 \leqr \leq1989) have a solution with x0 = x1990 = 0 but not all of x1, . . . , x1989 are equal to zero. Prove that b1 + b2 + \cdot \cdot \cdot + b1989 \geq 995.

IMO 1984 LL AUS3

IMO 1984 LL AUS3 Origin: AUS Problem The opposite sides of the reentrant hexagon AFBDCE in- tersect at the points K, L, M (as shown in the figure). It is given that AL = AM = a, BM = BK = b, CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f. (a) Given length a and the three angles...

IMO 1966 LL USS5

IMO 1966 LL USS5 Origin: USS Problem Prove the inequality tan \pi sin x 4 sin \alpha + tan \pi cos x 4 cos \alpha > 1 for any x, \alpha with 0 \leqx \leq\pi/2 and \pi/6 < y < \pi/3.

IMO 1969 LL FRA23

IMO 1969 LL FRA23 Origin: FRA Problem Consider the integer d = ab−1 c , where a, b, and c are positive integers and c \leqa. Prove that the set G of integers that are between 1 and d and relatively prime to d (the number of such integers is denoted by ϕ(d)) can be partitioned into n subsets, each of which consists of b elements. What can be said...

IMO 1967 LL HUN24

IMO 1967 LL HUN24 Origin: HUN Problem Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer...

IMO 1971 LL CUB11

IMO 1971 LL CUB11 Origin: CUB Problem Prove that n! cannot be the square of any natural number.

IMO 1982 LL POL42

IMO 1982 LL POL42 Origin: POL Problem Let F be the family of all k-element subsets of the set {1, 2, . . ., 2k + 1}. Prove that there exists a bijective function f : F \toF such that for every A \inF, the sets A and f(A) are disjoint.

IMO 1971 LL POL38

IMO 1971 LL POL38 Origin: POL Problem Let A, B, C be three points with integer coordinates in the plane and K a circle with radius R passing through A, B, C. Show that AB\cdotBC\cdotCA \geq2R, and if the center of K is in the origin of the coordinates, show that AB \cdot BC \cdot CA \geq4R.

IMO 1986 LL ROM64

IMO 1986 LL ROM64 Origin: ROM Problem Let (an)n\inN be the sequence of integers defined recursively by a1 = a2 = 1, an+2 = 7an+1 −an −2 for n \geq1. Prove that an is a perfect square for every n.

IMO 1979 LL BEL6

IMO 1979 LL BEL6 Origin: BEL Problem Prove that 1 4 sin2 36◦−1 = cos 72◦.

IMO 1967 LL ROM46

IMO 1967 LL ROM46 Origin: ROM Problem If x, y, z are real numbers satisfying the relations x+y+z = 1 and arctanx + arctan y + arctan z = \pi/4, prove that x2n+1 + y2n+1 + z2n+1 = 1 for all positive integers n. Solution Let us set arctan x = a, arctan y = b, arctanz = c. Then tan(a+b) = x+y 1−xy and tan(a + b + c)...

IMO 1983 LL ROM56

IMO 1983 LL ROM56 Origin: ROM Problem Consider the expansion (1 + x + x2 + x3 + x4)496 = a0 + a1x + \cdot \cdot \cdot + a1984x1984. (a) Determine the greatest common divisor of the coefficients a3, a8, a13, . . . , a1983. (b) Prove that 10340 < a992 < 10347.

IMO 1972 LL SWE38

IMO 1972 LL SWE38 Origin: SWE Problem Congruent rectangles with sides m (cm) and n (cm) are given (m, n positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)

IMO 1977 LL USS51

IMO 1977 LL USS51 Origin: USS Problem Several segments, which we shall call white, are given, and the sum of their lengths is 1. Several other segments, which we shall call black, are given, and the sum of their lengths is 1. Prove that every such system of segments can be distributed on the segment that is 1.51 long in the following way: Segments of the same color are disjoint,...

IMO 1978 LL TUR36

IMO 1978 LL TUR36 Origin: TUR Problem The integers 1 through 1000 are located on the circumference of a circle in natural order. Starting with 1, every fifteenth number (i.e., 1, 16, 31, . . .) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?

IMO 1967 LL SWE50

IMO 1967 LL SWE50 Origin: SWE Problem The function ϕ(x, y, z), defined for all triples (x, y, z) of real numbers, is such that there are two functions f and g defined for all pairs of real numbers such that ϕ(x, y, z) = f(x + y, z) = g(x, y + z) for all real x, y, and z. Show that there is a function h of one...

IMO 1989 LL CZS15

IMO 1989 LL CZS15 Origin: CZS Problem A sequence a1, a2, a3, . . . is defined recursively by a1 = 1 and a2k+j = −aj (j = 1, 2, . . . , 2k). Prove that this sequence is not periodic.

IMO 1992 LL FRG19

IMO 1992 LL FRG19 Origin: FRG Problem Denote by an the greatest number that is not divisible by 3 and that divides n. Consider the sequence s0 = 0, sn = a1 + a2 + \cdot \cdot \cdot+ an, n \inN. Denote by A(n) the number of all sums sk (0 \leqk \leq3n, k \inN0) that are divisible by 3. Prove the formula A(n) = 3n−1 + 2 \cdot 3(n/2)−1...

IMO 1989 LL HKG34

IMO 1989 LL HKG34 Origin: HKG Problem Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.

IMO 1974 LL CUB6

IMO 1974 LL CUB6 Origin: CUB Problem Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.

IMO 1976 LL USA37

IMO 1976 LL USA37 Origin: USA Problem From a square board 11 squares long and 11 squares wide, the central square is removed. Prove that the remaining 120 squares cannot be covered by 15 strips each 8 units long and one unit wide.

IMO 1978 LL USA43

IMO 1978 LL USA43 Origin: USA Problem If p is a prime greater than 3, show that at least one of the numbers p2 , 4 p2 , . . . , p−2 p2 is expressible in the form 1 x + 1 y, where x and y are positive integers.

IMO 1989 LL POR86

IMO 1989 LL POR86 Origin: POR Problem Given two natural numbers w and n, the tower of n w’s is the natural number Tn(w) defined by Tn(w) = ww...w , with n w’s on the right side. More precisely, T1(w) = w and Tn+1(w) = wTn(w). For example, T3(2) = 222 = 16, T4(2) = 216 = 65536, and T2(3) = 33 = 27. Find the smallest tower of 3’s...

IMO 1966 LL POL15

IMO 1966 LL POL15 Origin: POL Problem Points A, B, C, D lie on a circle such that AB is a diameter and CD is not. If the tangents at C and D meet at P while AC and BD meet at Q, show that PQ is perpendicular to AB.

IMO 1983 LL ISR36

IMO 1983 LL ISR36 Origin: ISR Problem The set X has 1983 members. There exists a family of subsets {S1, S2, . . . , Sk} such that: (i) the union of any three of these subsets is the entire set X, while (ii) the union of any two of them contains at most 1979 members. What is the largest possible value of k?

IMO 1969 LL BUL9

IMO 1969 LL BUL9 Origin: BUL Problem One hundred convex polygons are placed on a square with edge of length 38 cm. The area of each of the polygons is smaller than \pi cm2, and the perimeter of each of the polygons is smaller than 2\pi cm. Prove that there exists a disk with radius 1 in the square that does not intersect any of the polygons.

IMO 1979 LL CZS16

IMO 1979 LL CZS16 Origin: CZS Problem Let Q be a square with side length 6. Find the smallest integer n such that in Q there exists a set S of n points with the property that any square with side 1 completely contained in Q contains in its interior at least one point from S.

IMO 1985 LL AUS2

IMO 1985 LL AUS2 Origin: AUS Problem We are given a triangle ABC and three rectangles R1, R2, R3 with sides parallel to two fixed perpendicular directions and such that their union covers the sides AB, BC, and CA; i.e., each point on the perimeter of ABC is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the...

IMO 1992 LL AUS1

IMO 1992 LL AUS1 Origin: AUS Problem Points D and E are chosen on the sides AB and AC of the triangle ABC in such a way that if F is the intersection point of BE and CD, then AE + EF = AD + DF. Prove that AC + CF = AB + BF.

IMO 1986 LL FRA23

IMO 1986 LL FRA23 Origin: FRA Problem Let I and J be the centers of the incircle and the excircle in the angle BAC of the triangle ABC. For any point M in the plane of the triangle, not on the line BC, denote by IM and JM the centers of the incircle and the excircle (touching BC) of the triangle BCM. Find the locus of points M for which...

IMO 1979 LL BUL11

IMO 1979 LL BUL11 Origin: BUL Problem Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges and equal space angles between adjacent lateral walls is regular. Variant. Prove that a pyramid A1 . . . A2k+1S with equal space angles between adjacent lateral walls is regular if there exists a sphere tangent to all its edges.

IMO 1966 LL USS7

IMO 1966 LL USS7 Origin: USS Problem For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

IMO 1992 LL COL12

IMO 1992 LL COL12 Origin: COL Problem Given a triangle ABC such that the circumcenter is in the interior of the incircle, prove that the triangle ABC is acute-angled.

IMO 1986 LL SWE66

IMO 1986 LL SWE66 Origin: SWE Problem One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.

IMO 1989 LL IRE56

IMO 1989 LL IRE56 Origin: IRE Problem Let n = 2k −1, where k \geq6 is an integer. Let T be the set of all n-tuples (x1, x2, . . . , xn) where xi is 0 or 1 (i = 1, 2, . . ., n). For x = (x1, . . . , xn) and y = (y1, . . . , yn) in T , let d(x,...

IMO 1966 LL BUL33

IMO 1966 LL BUL33 Origin: BUL Problem Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle one draws segments tangent to the smaller circle. Prove that the length of one of these segments equals the sum of the lengths of the other two.

IMO 1986 LL FRG27

IMO 1986 LL FRG27 Origin: FRG Problem In an urn there are n balls numbered 1, 2, . . . , n. They are drawn at random one by one one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.

IMO 1970 LL SWE51

IMO 1970 LL SWE51 Origin: SWE Problem Let p be a prime number. A rational number x, with 0 < x < 1, is written in lowest terms. The rational number obtained from x by adding p to both the numerator and the denominator differs from x by 1/p2. Determine all rational numbers x with this property.

IMO 1992 LL JAP41

IMO 1992 LL JAP41 Origin: JAP Problem Let S be a set of positive integers n1, n2, . . . , n6 and let n(f) denote the number n1nf(1) + n2nf(2) + \cdot \cdot \cdot + n6nf(6), where f is a permu- tation of {1, 2, . . ., 6}. Let Ω= {n(f) | f is a permutation of {1, 2, . . ., 6}}. Give an example of positive...

IMO 1977 LL ROM35

IMO 1977 LL ROM35 Origin: ROM Problem Find all numbers N = a1a2 . . . an for which 9 \times a1a2 . . . an = an . . . a2a1 such that at most one of the digits a1, a2, . . . , an is zero. Solution The solutions are 0 and Nk = 10 99 . . .9    k 89, where k =...

IMO 1976 LL GDR19

IMO 1976 LL GDR19 Origin: GDR Problem For a positive integer n, let 6(n) be the natural number whose decimal representation consists of n digits 6. Let us define, for all natural numbers m, k with 1 \leqk \leqm, m k = 6(m) \cdot 6(m−1) \cdot \cdot \cdot 6(m−k+1) 6(1) \cdot 6(2) \cdot \cdot \cdot 6(k) . Prove that for all m, k, m k is a natural number whose...

IMO 1987 LL AUS3

IMO 1987 LL AUS3 Origin: AUS Problem A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop X to bus stop Y ̸= X, then we shall say Y can be reached from X. We shall use the phrase Y comes after X when...

IMO 1977 LL NET30

IMO 1977 LL NET30 Origin: NET Problem A triangle ABC with \angleA = 30◦and \angleC = 54◦is given. On BC a point D is chosen such that \angleCAD = 12◦. On AB a point E is chosen such that \angleACE = 6◦. Let S be the point of intersection of AD and CE. Prove that BS = BC. Solution Suppose \angleSBA = x. By the trigonometric form of Ceva’s theorem...

IMO 1984 LL POL45

IMO 1984 LL POL45 Origin: POL Problem Let X be an arbitrary nonempty set contained in the plane and let sets A1, A2, . . . , Am and B1, B2, . . . , Bn be its images under parallel translations. Let us suppose that A1 \cupA2 \cup\cdot \cdot \cdot \cupAm \subsetB1 \cupB2 \cup\cdot \cdot \cdot \cupBn and that the sets A1, A2, . . . , Am are...

IMO 1988 LL NET67

IMO 1988 LL NET67 Origin: NET Problem Given a set of 1988 points in the plane, no three points of the set collinear, the points of a subset with 1788 points are colored blue, and the remaining 200 are colored red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100...

IMO 1985 LL SPA75

IMO 1985 LL SPA75 Origin: SPA Problem Let ABCD be a rectangle, AB = a, BC = b. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being d) that are at an the angle \varphi, 0 \leq\varphi \leq90◦, with respect to AB. Let L be the sum of the lengths of all the segments intersecting the rectangle. Find: (a) how L varies, (b)...

IMO 1967 LL BUL6

IMO 1967 LL BUL6 Origin: BUL Problem Solve the system |x + y| + |1 −x| = 6, |x + y + 1| + |1 −y| = 4. Solution The given system has two solutions: (−2, −1) and (−14/3, 13/3).

IMO 1966 LL ROM47

IMO 1966 LL ROM47 Origin: ROM Problem Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the triangle. Compute this minimum length in terms of the sides a, b, c of the triangle.

IMO 1967 LL MON32

IMO 1967 LL MON32 Origin: MON Problem Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are \alpha, \beta, \gamma. Solution Let us denote by V the volume of the given body, and by Va, Vb, Vc the volumes of the parts of the given ball that lie inside the...

IMO 1972 LL BUL3

IMO 1972 LL BUL3 Origin: BUL Problem On a line a set of segments is given of total length less than n. Prove that every set of n points of the line can be translated in some direction along the line for a distance smaller than n/2 so that none of the points remain on the segments.

IMO 1970 LL AUT1

IMO 1970 LL AUT1 Origin: AUT Problem Prove that bc b + c + ca c + a + ab a + b \leq1 2(a + b + c) (a, b, c > 0).

IMO 1982 LL POL39

IMO 1982 LL POL39 Origin: POL Problem Let S be the unit circle with center O and let P1, P2, . . . , Pn be points of S such that the sum of vectors vi = −−\to OPi is the zero vector. Prove that the inequality n i=1 XPi \geqn holds for every point X.

IMO 1988 LL MON63

IMO 1988 LL MON63 Origin: MON Problem Let ABCD be a quadrilateral. Let A′BCD′ be the reflection of ABCD in BC, while A′′B′CD′ is the reflection of A′BCD′ in CD′ and A′′B′′C′D′ is the reflection of A′′B′CD′ in D′A′′. Show that if the lines AA′′ and BB′′ are parallel, then ABCD is a cyclic quadrilateral.

IMO 1969 LL CZS15

IMO 1969 LL CZS15 Origin: CZS Problem Let K1, . . . , Kn be nonnegative integers. Prove that K1!K2! \cdot \cdot \cdot Kn! \geq[K/n]!n, where K = K1 + \cdot \cdot \cdot + Kn.

IMO 1985 LL BUL11

IMO 1985 LL BUL11 Origin: BUL Problem Let a and b be integers and n a positive integer. Prove that bn−1a(a + b)(a + 2b) \cdot \cdot \cdot (a + (n −1)b) n! is an integer.

IMO 1988 LL FRG17

IMO 1988 LL FRG17 Origin: FRG Problem Show that if n runs through all positive integers, f(n) = / n + \sqrt 3n + 1/2 runs through all positive integers skipping the terms of the sequence an =  n2+2n  .

IMO 1987 LL BEL9

IMO 1987 LL BEL9 Origin: BEL Problem In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z} we have made a random sequence of 28 throws. What is the probability that the sequence CUBA JULY 1987 appears in this order in the sequence already thrown?

IMO 1969 LL MON41

IMO 1969 LL MON41 Origin: MON Problem Given two numbers x0 and x1, let \alpha and \beta be coefficients of the equation 1 −\alphay −\betay2 = 0. Under the given conditions, find an expression for the solution of the system xn+2 −\alphaxn+1 −\betaxn = 0, n = 0, 1, 2, . . . .

IMO 1987 LL AUS1