brain

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

The incircle of ABC touches BC, CA, and AB at D, E, and

Let … be three positive integers with ….

Let n be a positive integer that is not a perfect cube. Define

Let N be the number of integral solutions of the equation

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

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

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

Points A, B, C divide the circumcircle Ωof the triangle ABC

A sequence of real numbers u1, u2, u3, . . . is determined by u1

We are given a positive integer … and a rectangular board … with dimensions …, …. The rectangle is divided into a grid…

Let ABC be a triangle, H its orthocenter, O its circumcenter,

Solve the system of equations

The bisectors of angles A, B, C of a triangle ABC meet its cir-

An odd integer n \geq3 is said to be “nice” if there is at least one

Let Z denote the set of all integers. Prove that for any integers

A positive integer is written in each square of an m imesn board.

Let … be the real polynomial function

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

For three points A, B, C in the plane we define m(ABC)

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

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

6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that

Let S be the set of all pairs (m, n) of relatively prime positive

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

Consider the triangle ABC, its circumcircle k with center O

Let O be the center of the circumsphere of a tetrahedron

In a convex quadrangle with area 32 cm2, the sum of the

(GBR 1a)IMO6 For all positive integral n, un+1 = un(u2

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

The triangular array (an,k) of numbers is given by an,1 = 1/n,

In an acute-angled triangle ABC, let AD, BE be altitudes and

5b.(BEL 2)

Let n > m \geq1 be natural numbers such that the groups of

Find all functions f defined on the positive real numbers

Show that the set of positive integers that cannot be repre-

(ROM 1′) Let n be a composite natural number and p a proper divisor

An infinite sequence a0, a1, a2, . . . of real numbers satisfies

II 6 (USS 1) In a certain language words are formed using an alphabet

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

Let Q+ be the set of positive rational numbers. Construct

Let a, b, c, d, m, n be positive integers such that a2+b2+c2+d2 =

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

Let n and k be positive integers such that n/2 < k \leq2n/3.

In the triangle ABC let B′ and C′ be the midpoints of the sides

Prove that for every real number M there exists an infinite

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

Find all functions f from the reals to the reals such that

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

Let n be an integer greater than 1. In a circular arrange-

A circle is called a separator for a set of five points in a plane

Let p and q be integers. Show that there exists an interval I of

A rectangular array of numbers is given. In each row and each

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

Given riangleABC with no side equal to another side, let G, K,

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

Let ABC be a triangle and M an interior point. Prove that

At a round table are 1994 girls, playing a game with a deck

Define a k-clique to be a set of k people such that every pair

A set of 10 positive integers is given such that the decimal

Let P be a set of 7 different prime numbers and C a set of

Find all natural numbers n for which every natural number

Let A, B, and C be three points on the edge of a circular

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

Let f(k) be the number of integers n that satisfy the following

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

In a triangle ABC, let D and E be the intersections of the bisec-

A set S of points in space will be called completely sym-

We consider three distinct half-lines Ox, Oy, Oz in a plane.

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

Let a, b be natural numbers with 1 \leqa \leqb, and M =

In acute triangle ABC with circumcenter O and altitude

Consider pairs of sequences of positive real numbers a1 \geq

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

A rectangular box can be filled completely with unit cubes.

Find the least natural number n such that if the set

Prove that

In a triangle ABC, choose any points K \inBC, L \inAC,

For a triangle ABC, let k be its circumcircle with radius r. The

The numbers from 1 to n2 are randomly arranged in the cells

For a positive integer n, let f(n) denote the number of ways to

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

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

Let AX, BY, CZ be three cevians concurrent at an inte-

Prove that for every natural number k (k \geq2) there exists an

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

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

A circle of radius 1 is located in a right-angled trihedron and

Find a set A of positive integers such that for any infinite

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

The localities P1, P2, . . . , P1983 are served by ten international

Given a tetrahedron ABCD whose all faces are acute-

II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k,

For any integer r \geq1, determine the smallest integer h(r) \geq1

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

Let ABC be a triangle and L the line through C parallel to

Let … be nonnegative real numbers, not all zero.

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

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

Let O be the circumcenter of an acute-angled triangle ABC