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Let S be a set of n points in the plane. No three points of

Let E be the set of 19833 points of the space R3 all three

Let n be a positive integer. Find the number of odd coefficients

For a given triangle ABC, let X be a variable point on

Let f and g be two integer-valued functions defined on the set

(FIN 2′) Let f : [0, 1] oR be continuous and satisfy:

Given a convex polygon A1A2 . . . An with area S, and a point

Find all polynomials f(x) with real coefficients for which

Let a and b be the bases of two number systems and let

(a) Decide whether the fields of the 8 imes 8 chessboard can be numbered

Four different points A, B, C, D are chosen on a circle \Gamma such

Prove that the set of positive integers cannot be partitioned

Given a convex polyhedron P1 with 9 vertices A1, . . . , A9,

4a.(BUL 1)

Show that 20

Determine the minimum of a2 + b2 if a and b are real

Let p and q be relatively prime positive integers. A subset

A finite sequence of integers … is called quadratic if for each … we have the equality ….

Let A1A2 . . . An be a convex polygon, n \geq4. Prove that

Let x1, x2, . . . , xn be real numbers satisfying x1+x2+\cdot \cdot \cdot+xn =

Two circles Ω1 and Ω2 touch internally the circle Ωin

Prove that there exist infinitely many positive integers n

The triangle ABC is acute-angled. Let L be any line in the

Two students A and B are playing the following game: Each

Let p1, p2, . . . , pn be distinct primes greater than 3. Show

Let A be a nonempty set of positive integers. Suppose that

Let A be a set of N residues (mod N 2). Prove that there

Let ABC be a triangle and P an exterior point in the plane

Let n be an integer greater than 2. A positive integer is said to be

Each positive integer a undergoes the following procedure in

Let M be a point in the interior of triangle ABC. Let A′ lie

I 4 (USS 4) The sum of the squares of five real numbers a1, a2, a3, a4, a5

Define a sequence ⟨f(n)⟩\infty

Peter has three accounts in a bank, each with an integral

Given n countries with three representatives each, m commit-

Let p > 3 be a prime number. For each nonempty subset T of

Given the integer n > 1 and the real number a > 0 determine

Let n \geqk \geq0 be integers. The numbers c(n, k) are defined as

If … and … are arbitrary positive real numbers and … an integer, prove that

Two circles G1 and G2 intersect at M and N. Let AB

A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: at each…

Let P be a polynomial of degree n satisfying

Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality

Find all polynomials P(x) with real coefficients that

Let P be a cubic polynomial with rational coefficients, and let

Given … (…) points in space such that every three of them form a triangle with one angle greater than or equal to …,…

Prove that for any positive integer m there exist an infinite

Given a nonequilateral triangle ABC, the vertices listed coun-

Consider the n imes n array of nonnegative integers

B5 (CAN 3)

In a convex quadrilateral ABCD the diagonal BD does

Let … have orthocenter …, and let … be a point on its circumcircle, distinct from …, …, …. Let … be the foot of the…

Given a point … and lengths …, prove that there exists an equilateral triangle … for which …, …, …, if and only if …,…

For any positive integer n, let au(n) denote the number of its

The polynomial 1976(x+x2+\cdot \cdot \cdot+xn) is decomposed into a sum

Determine all triples of positive integers (a, m, n) such that

Prove that for every real number M there exists an infinite

Find the least natural number n such that if the set

Prove that for all n \inN the following is true:

Prove that there are exactly

Prove the following assertion: The four altitudes of a tetrahe-

Let p be an odd prime. Find the number of p-element

(IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let

Let {f(n)} be a strictly increasing sequence of positive

Let p be an odd prime and n a positive integer. In the

Let …, …, … be real numbers. Prove that the system of equations

Find four positive integers each not exceeding 70000 and each

(a) Let (m, k) = 1. Prove that there exist integers a1, a2, . . . , am

Let n be a positive integer. Let \sigma(n) be the sum of the natural

C3 (CAN 5) Show that

Let a and b be nonnegative integers such that ab \geqc2,

The set S = {2, 5, 13} has the property that for every

Prove that on the coordinate plane it is impossible to draw a

A brick staircase with three steps of width 2 is made of twelve

Decide whether there exists a set M of natural numbers satis-

The matrix

Let O be an interior point of acute triangle ABC. Let A1

Let ABCD be a regular tetrahedron and M, N distinct points

An n imes n matrix with entries from {1, 2, . . ., 2n −1} is called

(a) A plane \pi passes through the vertex O of the regular

Let n be a positive integer and let a, b be given real numbers.

Consider two segments of length … (…) and a segment of length ….

Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 =

Let N be a positive integer. Two players A and B, taking

In the tetrahedron SABC the angle BSC is a right angle,

Real constants a, b, c are such that there is exactly one square

Let ABCD be a cyclic quadrilateral. Let E and F be variable

Find the sphere of maximal radius that can be placed inside

The nonnegative real numbers x1, x2, x3, x4, x5, a satisfy the

M is a subset of {1, 2, 3, . . ., 15} such that the product of

Let n be a natural number and a1, a2, . . . , a2n mutually distinct

Let ABC be a triangle with semiperimeter s and inradius

Consider in a plane 
i the points O, A1, A2, A3, A4 such that

The circle inscribed in a triangle ABC touches the sides

Let f be a function that satisfies the following conditions:

Given a set S in the plane containing n points and satis-

A tetrahedron ABCD is given such that AD = BC = a;

Let a and b be natural numbers and let q and r be the

Determine the smallest integer n \geq4 for which one can choose

Let A = (a1, a2, . . . , a2001) be a sequence of positive integers.