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Let ABC be a triangle such that ngleA = 90◦and ngleB < ngleC. The

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

Consider two concentric circles of radii R and r (R > r)

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

Let ABC be a triangle with semiperimeter s and inradius

If a, b, and c are sides of a triangle, prove that

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

Let p be an odd prime. Determine positive integers x and

Let N be the number of integral solutions of the equation

The quadrilateral ABCD has the following properties:

Let ABCD be a cyclic quadrilateral. Let P, Q, R be the

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

Let S be a convex quadrilateral ABCD and O a point inside

B4 (BRA 1) A box contains p white balls and q black balls. Beside the

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

The positive integers … and … are such that the numbers

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

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

Given k parallel lines and a few points on each of them, find

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

Prove that for all positive real numbers a, b, c,

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

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

For which positive integers n does there exist a positive

Unit cubes are made into beads by drilling a hole through

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

Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn)

We call a positive integer alternate if its decimal digits

C7 (CZS 3)

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

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

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

Let p \geq5 be a prime number. Prove that there exists an

Assume that the set of all positive integers is decomposed into

Let n and k be positive integers. There are given n circles

Determine the maximum value of the sum

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

Prove that there exists a four-coloring of the set M =

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

Prove for each triangle ABC the inequality

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

Prove that there exist two strictly increasing sequences (an)

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

Let S1 and S2 be circles meeting at the points A and B. A

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

Let … be an arbitrary triangle and … a point inside it. Let … be the distances from … to sides …; … the lengths of the…

Does there exist a set M with the following properties?

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

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

Let c be a positive integer. The sequence {fn} is defined as

The circle S has center O, and BC is a diameter of S.

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

Let f(n) be a function defined on the set of all positive integers

Let P be a polynomial of degree n satisfying

The function F is defined on the set of nonnegative integers

The tangents at B and A to the circumcircle of an acute-

Consider a sequence of polynomials P0(x), P1(x), P2(x), . . . ,

Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by

Let the sides of two rectangles be … and … with

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

Does there exist a finite set M of points in space, not all in

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

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

Given an integer n > 1, denote by Pn the product of all

Let p be a prime number and let A be a set of positive integers

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

For each finite set U of nonzero vectors in the plane we define

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

We are given 100 points in the plane, no three of which are

Let P(z) and Q(z) be complex-variable polynomials, with degree

A nonempty set A of real numbers is called a B3-set if the

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

Find all solutions of the following system of n equations in n

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

Consider the sequences (an), (bn) defined by

Knowing that the system

Does there exist a positive integer n such that n has

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

Three persons A, B, C, are playing the following game: A k-

The function \psi from the set N of positive integers into itself

Let R be the set of real numbers. Does there exist a function

Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove

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

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

Let f : N oN be a function that satisfies the inequality

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

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

Let n be an integer greater than 1. Define

Prove that N = 5125−1

For every integer d \geq1, let Md be the set of all positive

What is the smallest positive integer t such that there exist

A polyhedron has 12 faces and is such that:

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

Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that

A particle moves from (0, 0) to (n, n) directed by a fair coin.

A circle whose center is on the side ED of the cyclic

Find all natural numbers n for which 28 + 211 + 2n is a perfect

Ten points such that no three of them lie on a line are marked in

Determine whether there exist distinct real numbers a, b, c, t

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