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

In a test, 3n students participate, who are located in three

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

(a) If a 5 imes n rectangle can be tiled using n pieces like those

In a plane two different points O and A are given. For

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

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

For which positive integers n does there exist a positive

Circles S1 and S2 intersect at points P and Q. Distinct points

Assume that the set of all positive integers is decomposed into

II 2 (NET 3)IMO5 If a, b, c, d are arbitrary positive real numbers, find all

On the sides of a square ABCD one constructs inwardly

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

Let x1, x2, . . . , xn be real numbers satisfying the conditions

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

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

2b.(VIE 1)

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

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

Let … be integers such that …. Determine the maximum size of a subset … of the set … such that no … distinct elements…

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

A4 (BUL 2) Determine all real values of the parameter a for which the

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

Does there exist a positive integer n such that n has

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

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

In a triangle ABC we have AB = AC. A circle is tangent

Let ABC be a triangle with centroid G. Determine, with

Let n \geq4 be a fixed positive integer. Given a set S =

In the plane we have n rectangles with parallel sides. The

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

Let n > 1 be an integer and let f(x) = xn + 5xn−1 + 3.

Two players A and B play a game in which they choose

Determine all m imes n rectangles that can be covered with

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

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

In a certain city, age is reckoned in terms of real numbers

Is it possible to plot 1975 points on a circle with radius 1 so

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

For all rational x satisfying 0 \leqx < 1, f is defined by

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

C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri-

Let n be an even positive integer. We say that two dif-

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

A positive integer is called a double number if its decimal rep-

The lock on a safe consists of three wheels, each of which may

Let f(0) = f(1) = 0 and

II 4 (FIN 3)IMO2 Let riangleABC be a triangle. Prove that there exists a

Let … be an acute-angled triangle with circumcenter … and circumradius …. Let … meet the circle … again in …, let ……

Let a, b, c, d, e be real numbers. Prove that the expression

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

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

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

Let r1, r2, . . . , rn be real numbers greater than or equal to 1.

Let … be a convex quadrilateral, and let …, …, …, and … denote the circumradii of the triangles …, …, …, and ……

There are 2n words of length n over the alphabet {0, 1}. Prove

Let n, k be positive integers such that n is not divisible by

For which positive integers n do there exist two polynomials f

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

Given a, \theta \inR, m \inN, and P(x) = x2m −2|a|mxm cos \theta+a2m,

Let A1 be the center of the square inscribed in acute triangle

Let a > b > c > d be positive integers and suppose

Let ABCD be a convex quadrilateral such that AC =

Find, with proof, all functions f defined on the nonnegative

Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers

Let a1, a2, . . . , an be positive numbers and q a given real

A pile of n pebbles is placed in a vertical column. This

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

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

Let x0 = 5 and xn+1 = xn +

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

Let n be an even positive integer. Show that there is a

An infinite arithmetic progression whose terms are positive in-

A wobbly number is a positive integer whose digits in base 10

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

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

6a.(SWE 3)IMO6 The sequence f1, f2, . . . , fn, . . . of functions is defined

Let n be an integer greater than 1. Define

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

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

For all integers n, n \geq0, there exist uniquely determined

Let T denote the set of all ordered triples (p, q, r) of nonneg-

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

The diagonals of a quadrilateral ABCD are perpendicular:

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

Let A, B, and C be noncollinear points. Prove that there is

The circle \Gamma and the line ℓdo not intersect. Let AB be the

Let a0, a1, . . . , an, an+1 be a sequence of real numbers satisfying

Ten localities are served by two international airlines such

Let …, …, and … be positive real numbers such that ….

Let n be a positive integer. Each point (x, y) in the plane,

Let A be a 101-element subset of the set S = {1, 2, . . . ,

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

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

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

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

Let ABC be an acute-angled triangle. Three lines LA, LB,

C5 (USS 5) The right triangles ABC and AB1C1 are similar and have

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

Let I be the incenter of triangle ABC. Let K, L, and M