brain

Is there a positive integer m such that the equation

Let a1 \geqa2 \geqa3 be given positive integers and let N(a1, a2, a3)

Three equal circles touch the sides of a triangle and have

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

Let … be an even positive integer. Prove that there exists a positive integer … such that

Let n \geq1 be an integer. A path from (0, 0) to (n, n) in the

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

Let ABC be a triangle. The bisector of angle A meets

The vertex A of the acute triangle ABC is equidistant from

For which integers n \geq3 does there exist a regular n-gon in the

For n an odd positive integer, the unit squares of an n imes n

Suppose x1, x2, . . . , xn are real numbers with x2

A3 (USS 4)IMO3 Consider the infinite sequences {xn} of positive real

An integer sequence is defined by

Determine all pairs (a, b) of positive integers such that

Prove that the sum of the face angles at each vertex of a tetra-

Let a1, a2, . . . , an, . . . be a sequence of real numbers such that

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

Let a set of p equations be given,

Among a group of 120 people, some pairs are friends. A weak

Ali Barber, the carpet merchant, has a rectangular piece of

We are given a fixed point on the circle of radius …, and going from this point along the circumference in the positive…

Let a0, a1, a2, . . . be an increasing sequence of nonnegative inte-

Let n, k be positive integers with k \leqn and let S be a set

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

Let E be a set of n points in the plane (n \geq3) whose co-

Circles G, G1, G2 are three circles related to each other as

Let M be the set of all positive integers that do not contain the

Prove that for any integer n \geq2, there exists a set of 2n−1

Assume that f(x, y) is defined for all positive integers x and

Find all pairs of positive integers m, n \geq3 for which

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

Let O be the center of the circumsphere of a tetrahedron

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

Find all the functions f : R oR that satisfy

Determine for which positive integers k the set

Does there exist a function s: Q … such that if x

Consider a matrix of size n imesn whose entries are real numbers

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

Let f(x) be a continuous function defined on the closed interval

To each vertex Pi (i = 1, . . . , 5) of a pentagon an integer

Let a1 = 1111, a2 = 1212, a3 = 1313, and

Every point with integer coordinates in the plane is the

Let S be any point on the circumscribed circle of rianglePQR. Then

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

Find all functions f : R oR satisfying

Let a, b be integers that are not perfect squares. Prove that if

Let ABC be an isosceles triangle with AC = BC, whose

Let … be the real polynomial function

Does there exist a set M in usual Euclidean space such that

Let n \geq2 be a fixed integer. Find the least constant C

Two ships sail on the sea with constant speeds and fixed directions. It is known that at … the distance between them…

Consider the system

Let k be a positive integer. Prove that there are infinitely

Let a1, a2, . . . , an be positive real numbers such that a1 + a2 +

Prove that for every natural number m \geq1 there exists a

S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which

Let a1, a2, . . . be an infinite sequence of real numbers for

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

Let a be the greatest positive root of the equation x3−3x2+1 =

Describe all closed bounded figures 
hi in the plane any two

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

We say that a set E of points of the Euclidian plane is

A game is played by n girls (n \geq2), everybody having a ball.

For any positive integer x0, three sequences {xn}, {yn}, and

Prove that there exists a unique triangle whose side

Let ABCDEF be a convex hexagon such that AB = BC, CD =

At a meeting of 12k people, each person exchanges greetings

Let ABC be an acute-angled triangle with AB ̸= AC.

Given a point P in a given plane \pi and also a given point

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

Let f(x) be a polynomial with rational coefficients and lpha be

The integer 9 can be written as a sum of two consecutive

5b.(BEL 2)

For a finite graph G, let f(G) be the number of triangles

Let P, Q, R be polynomials and let S(x) = P(x3) + xQ(x3) +

Let n \geq3 be an integer and t1, t2, . . . , tn positive real

There are two circles in the plane. Let a point A be one

Let U = {1, 2, . . ., n}, where n \geq3. A subset S of U is said to be

I 3 (SWE 3)IMO6 Let P(x) be a polynomial with integer coefficients. If

Let p be the product of two consecutive integers greater than

1b.(TUR 5) Find the smallest positive integer n such that

Solve the following system of equations, in which a is a given

I 5 (GBR 3) Let Ar, Br, Cr be points on the circumference of a given

C1 (NET 1)IMO2 A scalene triangle A1A2A3 is given with sides a1, a2, a3

A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahe-

C2 (AUS 4)

Let m be a positive odd integer, m \geq2. Find the smallest

In the triangle ABC, with ∡A = 60◦, a parallel IF to AC

A sequence (an) is defined by means of the recursion

Determine the maximum value of the sum

Find all positive integers k for which the following statement is

Let … be the apothem (distance from the center to one of the sides) of a regular …-gon (…) inscribed in a circle of…

Each of the numbers in the set N = {1, 2, 3, . . ., n −1},

Determine all positive integers n for which there exists an integer

Find two positive integers a, b such that none of the num-

For which digits a do exist integers n \geq4 such that each digit

Let n \geq2 be a natural number and let the real numbers

Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between

Let ABC be a triangle. A circle passing through B and C in-