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Let ABC be a triangle with interior angle bisectors AA1,

Let x, y, and z be real numbers satisfying x + y + z = xyz.

Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that

Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1

Solve the equation 28x = 19y + 87z, where x, y, z are integers.

Through a point P within a triangle ABC the lines l, m, and

Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that

Find all pairs of integers a and b for which

Let A1A2, B1B2, C1C2 be three equal segments on the three

Show that for any natural number n there exist two prime

Given a positive integer n, find the greatest integer p with the

Let n be a natural number. If 4n + 2n + 1 is a prime, prove

For every a \inN denote by M(a) the number of elements of

Let an =

Let M be the point inside the right-angled triangle ABC

Consider the number lpha obtained by writing one after another

Let n be a positive integer. Find the maximal number of non-

Describe which natural numbers do not belong to the set

All the irreducible positive rational numbers such that the prod-

Given a triangle, prove that the points of intersection of three

Thirty-four countries participated in a jury session of the IMO,

Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an

(a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the

Let n and k be natural numbers and a1, a2, . . . , an positive real

Consider 37 distinct points in space, all with integer coordi-

Find the number of five-digit numbers with the following

Show that there exists a set S of 15 distinct circles on the

Let a, b, c, d be positive integers such that ab = cd and a + b =

Let t(n), for n = 3, 4, 5, . . ., represent the number of distinct,

Given any triangle ABC and any positive integer n, we say

For a point O inside a triangle ABC, denote by A1, B1, C1

In the coordinate system in the plane we consider a convex

Determine the maximum value of x2y2z2w when x, y, z, w \geq0

The number 0 or 1 is to be assigned to each of the n vertices

Consider the set of grid points (m, n) in the plane, m, n inte-

Given a ring G in the plane bounded by two concentric circles

It is given that x = −2272, y = 103 +102c+10b+a, and z = 1

A regular n-gon A1A2A3 . . . Ak . . . An inscribed in a circle of

Let x = p, y = q, z = r, w = s be the unique solution of the

Solve the following system of linear equations with unknown

Let c be the inscribed circle of the triangle ABC, d a line tan-

The vertices of an (n + 1)-gon are placed on the edges of a

Given a regular convex 2m-sided polygon P, show that there is

Decide whether it is possible to color the 1984 natural numbers

Several segments, which we shall call white, are given, and

Solve the system

Show that the reciprocal of any number of the form 2(m2 +

A circle of radius ho is tangent to the sides AB and AC of the

Let a and b be integers. Is it possible to find integers p and q

Find all square numbers S1 and S2 such that S1 −S2 = 1989.

Find a function f(x) defined for all real values of x such that

Given an integer n \geq2, determine all n-digit numbers

Let f : (0, +\infty) \toR be a function having the property

Determine the range of w(w + x)(w + y)(w + z), where x, y,

Consider the set S of all the different odd positive integers

Let a quadratic polynomial g(x) = ax2 + bx + c be given and

A 2 imes 2 imes 12 box fixed in space is to be filled with twenty-four

Given a polynomial f(x) with integer coefficients whose value

Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real

A circle touches the sides AB, BC, CD, DA of a square at

It is proposed to partition the set of positive integers into two

Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show

Let M be an interior point of tetrahedron V ABC. Denote

In a plane are given n points Pi (i = 1, 2, . . . , n) and two

Let a, b, c, d be the lengths of the sides of a quadrilateral

(a) What is the maximal number of acute angles in a convex

The function F is a one-to-one transformation of the plane into

Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a

A sequence (an)N

Suppose ABCD and A′B′C′D′ are two parallelograms arbi-

If a0 is a positive real number, consider the sequence {an}

Show that for nonnegative real numbers a, b and integers n \geq2,

Prove that there exist 78 lines in the plane such that they have

Find all the functions f : R+ oR satisfying the identity

Two circles touch each other from inside, and an equilateral

Prove that the sequence (an)n\geq0, an = [n

Five distinct numbers are drawn successively and at random

The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one

In what case does the system

Find the integer solutions of the equation

Let n, k \geq1 be natural numbers. Find the number A(n, k) of

In a given country, all inhabitants are knights or knaves. A

The plane is divided into equal squares by parallel lines; i.e.,

Find all numbers x \inZ for which the number

A square hole of depth h whose base is of length a is given.

Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary

Assume that the bisecting plane of the dihedral angle at edge

Given a positive integer k, find the least integer nk for which

Find necessary and sufficient conditions on given positive num-

For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .)

It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2

Given n points A1, A2, . . . , An, no three collinear, show that

Given the expression

Suppose p and q are two different positive integers and x is a

Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges

A sequence {an} of positive integers is defined by

Let 1 \leqk < n. Consider all finite sequences of positive integers

Let n = 2k −1, where k \geq6 is an integer. Let T be the set

Suppose that a triangle whose sides are of integer lengths is

Given two points A, B outside of a given plane P, find the