brain

Let … be a real number and … a real function defined on all of …, satisfying for all …,

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

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

Determine all pairs (a, b) of real numbers such that a\lfloorbn\rfloor= b\lflooran\rfloor

For what natural numbers n can the product of some of

A magician has one hundred cards numbered 1 to 100.

Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers

Let u1, u2, . . . , um be m vectors in the plane, each of length

Prove that the sum of an odd number of unit vectors passing

If … … are distinct non-zero real numbers, prove that the equation

On a circle, 2n −1 (n \geq3) different points are given. Find

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

B1 (CAN 2)

Prove that for each positive integer n, there exists a positive

For a given positive integer k denote the square of the sum of

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

In the plane, let there be given a circle C, a line l tangent

Consider the system

Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this…

Let a set of p equations be given,

S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 =

For a polynomial P of degree 2000 with distinct real co-

Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+

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

How many words with n digits can be formed from the alphabet

Let f, g, and a be polynomials with real coefficients, f and g

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

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

Find all pairs of integers x, y \geq1 satisfying the equation

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

Let B be a point on a circle S1, and let A be a point distinct

Let A1A2A3A4 be a tetrahedron, G its centroid, and

Let S be a set of n elements. We denote the number of all

Let m and n be positive integers. The set A = {a1, a2, . . . ,

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

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

Let the sequence …, …, be generated as follows:

Let dn be the last nonzero digit of the decimal representation

A solitaire game is played on an m imes n rectangular board, using

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

Find all positive integers k for which the following statement is

Let f and ϕ be real functions defined on the set R satisfying

Is it possible to put 1987 points in the Euclidean plane

B6 (FIN 3) Four distinct circles C, C1, C2, C3 and a line L are given in

An n \times n chessboard (n \geq2) is numbered by the numbers

A flock of 155 birds sit down on a circle C. Two birds Pi, Pj are

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

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

A convex quadrilateral ABCD has perpendicular diagonals.

An integer sequence is defined by

Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot <

Determine all pairs (a, b) of positive real numbers with a ̸= 1

The sequence a0, a1, a2, . . . is defined as follows:

Determine all positive integers n \geq2 that satisfy the following

For which positive integers n do there exist two polynomials f

For any positive integer k, Ak is the subset of {k+1, k+

A1 (GBR 3)IMO1 The function f(n) is defined for all positive integers

Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime

Let K denote the set {a, b, c, d, e}. F is a collection of 16 different

Inside triangle ABC there are three circles k1, k2, k3 each of

The sequence {an} of integers is defined by a1 = 2, a2 = 7,

Let S be the set of all the odd positive integers that are not

On the sides of a square ABCD one constructs inwardly

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

Find all functions f : R oR satisfying the equation

Let a, b, c be positive integers satisfying the conditions b > 2a

Find all pairs of positive integers (x, p) such that p is

Find the point P inside the triangle ABC for which

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

Let n be a positive integer. How many integer solutions

Let a, b, and c be given positive real numbers. Determine all

Let S and F be two opposite vertices of a regular octagon.

Find all functions f : R oR such that

Find all positive integers a1, a2, . . . , an such that

We call a set S on the real line R superinvariant if for any

Find all positive integer solutions x, y, z of the equation 3x +

For points A1, . . . , A5 on the sphere of radius 1, what is the

(a) Do there exist functions f : R oR and g : R oR such that

Let n be an odd integer greater than 1 and let c1, c2, . . . ,

Let K be one of the two intersection points of the circles W1

Let T1 be a triangle having a, b, c as lengths of its sides and let

Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive

Prove or disprove: From the interval [1, . . . , 30000] one

Is there a positive integer m such that the equation

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

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

In a contest, there are m candidates and n judges, where

On an infinite square grid, two players alternately mark sym-

A point M is chosen on the side AC of the triangle ABC in

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

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

Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a…

Find all positive integers x and y such that x+y2+z3 = xyz,

Let ABC be a triangle with semiperimeter s and inradius

Let p be a prime number and let f(x) be a polynomial of degree

For what values of n does there exist an n imes n array of entries

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

Let b be an integer greater than 5. For each positive integer

An international society has its members in 6 different

The matrix