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Let ABCD be a convex quadrilateral for which the circle

Let R be a set of exactly 6 elements. A set F of subsets of R

Let S be the set of real numbers greater than −1. Find

Let c be a positive integer. The sequence {fn} is defined as

A line l does not meet a circle \omega with center O. E is the

Let n \geq3 be a positive integer. Let C1, C2, C3, . . . , Cn

Prove that there exist infinitely many positive integers n such

For what natural numbers n can the product of some of

The function F is defined on the set of nonnegative integers

Find all complex numbers m such that polynomial

Find all positive integers n having the property that 2n+1

Given a set M of 1985 positive integers, none of which

Let R+ be the set of all nonnegative real numbers. Given two

Let n be an even positive integer. Let A1, A2, . . . , An+1 be

Let A, B, C, and D be distinct points on a line, in that

Let M and N be points inside triangle ABC such that

Does there exist a function f : N oN, such that f(f(n)) =

Let … be an acute-angled triangle with …. Let … be the circumcenter, … its orthocenter, and … the foot of its altitude…

A biologist watches a chameleon. The chameleon catches

3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in

Let O be the circumcenter and H the orthocenter of an acute

Let a be a rational number with 0 < a < 1 and suppose that

Let k be a fixed integer greater than 1, and let m = 4k2 −5.

Prove that

Let A1A2A3 be a nonisosceles triangle with incenter I. Let Ci,

Let a, b, c, d be nonnegative real numbers such that ab + bc +

Let … be a point inside … such that

Let p be a prime and A = {a1, . . . , ap−1} an arbitrary subset

Let P be a point inside a regular tetrahedron T of unit volume.

Let N = {1, 2, . . ., n}, n \geq2. A collection F = {A1, . . . , At}

A finite set of (distinct) positive integers is called a “DS-set”

Prove that the set {1, 2, . . ., 1989} can be expressed as the

Given an oriented line … and a fixed point … on it, consider all trapezoids … one of whose bases … lies on …, in the…

Let V be a finite subset of Euclidean space consisting of

Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn,

Let Tk = k −1 for k = 1, 2, 3, 4 and

A number of signal lights are equally spaced along a one-way

Let n be a positive integer and let p be a prime number, p > 3.

Let a, b, n be positive integers, b > 1 and bn −1 | a. Show

In town A, there are n girls and n boys, and each girl knows each

Let m and n be nonnegative integers. Prove that m!n!(m+

In a right-angled triangle ABC, let AD be the altitude

Given seven points in the plane, some of them are connected

Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta + \gamma < \pi,

Let the sides of two rectangles be … and … with

At n distinct points of a circular race course there are n cars

I 2 (POL 1) Prove that the squares with sides 1/1, 1/2, 1/3, . . . may be

Consider on the first quadrant of the trigonometric circle the

In a given tetrahedron ABCD let K and L be the centers of

An integer n is said to be good if |n| is not the square of

Let a, b, c be positive real numbers with product 1. Prove

Is it possible to put 1987 points in the Euclidean plane

We are given two mutually tangent circles in the plane, with

Determine whether there exist distinct real numbers a, b, c, t

Each pair of opposite sides of a convex hexagon has the

Let lpha(n) be the number of digits equal to one in the binary

Find all solutions in positive real numbers xi (i =

Given a circle K, find the locus of vertices A of parallelograms

A bicentric quadrilateral is one that is both inscribable in

Let ABCDEF be a convex hexagon such that ngleB +ngleD +ngleF =

On the sides of the triangle ABC, three similar isosceles tri-

Let a, b, c, d be odd positive integers such that a < b < c <

Let b, m, n be positive integers such that b > 1 and m ̸= n. Prove

Let ABCD be a convex quadrilateral whose vertices do not

Find the number of partitions of the set {1, 2, . . ., n} into three

In the plane are given a point … and a polygon … (not necessarily convex). Let … denote the perimeter of …, … the sum…

The prolongation of the bisector AL (L \inBC) in the acute-

A polyhedron has 12 faces and is such that:

Let … be a function from the set of real numbers … into itself such that for all …, we have … and

Given real numbers x1, x2, . . . , xn (n \geq2), with xi \geq1/n

Let n be an integer greater than 2. Define V = {1 + kn |

Let P be a convex polygon. Prove that there is a convex

Prove that a convex pentagon (a five-sided polygon) ABCDE

A function f : I oR, defined on an interval I, is called

Let ABCDEF be a convex hexagon with AB = BC =

Let a1, a2, a3, . . . be any infinite increasing sequence of pos-

Ten points such that no three of them lie on a line are marked in

Let a0 = 1994 and an+1 =

A1A2A3 is an acute-angled triangle. The foot of the

Find the point P inside the triangle ABC for which

In a permutation (x1, x2, . . . , xn) of the set 1, 2, . . . , n we call

A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its

A line in the plane of a triangle … intersects the sides … and … respectively at points … and … such that …. Find the…

S1 (UKR) Does there exist a sequence F(1), F(2), F(3), . . . of nonneg-

The triangle ABC is inscribed in a circle. The interior bi-

The tangents at B and A to the circumcircle of an acute-

Inside an equilateral triangle ABC one constructs points P,

A cube is assembled with 27 white cubes. The larger cube is then

Let u1, u2, . . . , un, v1, v2, . . . , vn be real numbers. Prove that

A circle with center O passes through points A and C and

Let a, b, c be given integers a > 0, ac −b2 = P = P1 \cdot \cdot \cdot Pm

At a party attended by n married couples, each person talks

For a triangle T = ABC we take the point X on the side

In the plane we are given a set E of 1991 points, and certain

For a positive integer n, let d(n) be the number of all positive

Consider the two square matrices

Prove that for each n \geq4 every cyclic quadrilateral can

S5 (FIN) For positive integers n, the numbers f(n) are defined induc-

Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove

II 5 (BUL 1)IMO4 Consider a partition of an 8 imes 8 chessboard into p