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Let a and b be natural numbers and let q and r be the

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

Consider the triangle ABC, its circumcircle k with center O

The vertex A of the acute triangle ABC is equidistant from

I 2 (POL 1) Prove that the squares with sides 1/1, 1/2, 1/3, . . . may be

A natural number n is said to have the property P if whenever

In the plane we are given two circles intersecting at X and Y .

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

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

Let D1, . . . , Dn be closed disks in the plane. (A closed disk

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

(POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given

Let … be real numbers such that

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

Let d be the sum of the lengths of all diagonals of a convex

Prove that for all n \inN the following is true:

Let n \geq2 be a positive integer and \lambda a positive real

An integer sequence is defined by

Prove for each triangle ABC the inequality

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

A cube is assembled with 27 white cubes. The larger cube is then

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

Let ABC be a triangle and M an interior point in ABC.

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

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

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

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

The triangle ABC is inscribed in a circle. The interior bi-

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

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

Determine the maximum value of m2 + n2 where m and n

Let 1 = a0 \leqa1 \leqa2 \leq\cdot \cdot \cdot \leqan \leq\cdot \cdot \cdot be a sequence of

Three equal circles touch the sides of a triangle and have

Let … be a finite set and let …, … be bijective functions from … onto itself. Let

A tetrahedron ABCD is given such that AD = BC = a;

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

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

Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in

Prove that the volume of a tetrahedron inscribed in a right

The set {a0, a1, . . . , an} of real numbers satisfies the following

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

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

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

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

Find all natural numbers n for which every natural number

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

A finite set of unit circles is given in a plane such that the area

An eccentric mathematician has a ladder with n rungs that he

A finite set of (distinct) positive integers is called a “DS-set”

On the sides of a square ABCD one constructs inwardly

Determine all m imes n rectangles that can be covered with

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

A rectangular box can be filled completely with unit cubes.

Cards numbered 1 to 9 are arranged at random in a row. In a

Let ABCD be a regular tetrahedron and M, N distinct points

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

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

If an acute-angled triangle ABC is given, construct an equilat-

Let f(x) = x2+1

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

Let n \geq2 be an integer. Find the maximal cardinality of a set

Let a, b, A, B be given constant real numbers and

We are given 3n points A1, A2, . . . , A3n in the plane, no three

Consider the n imes n array of nonnegative integers

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

The function f(x, y) is a homogeneous polynomial of the nth

Let ABCD be a convex quadrilateral such that AC =

Given a point M on the side AB of the triangle ABC, let

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

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

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

Determine all the triples (a, b, c) of positive real numbers such

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

Consider in a plane 
i the points O, A1, A2, A3, A4 such that

Let D be the interior of the circle C and let A \inC. Show

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

Let n be an integer greater than 2. A positive integer is said to be

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

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

Let lpha be the positive root of the equation x2 = 1991x + 1. For

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

Let S be the set of real numbers greater than −1. Find

Let a be a positive integer and let {an} be defined by a0 = 0

Let r \geq2 be a fixed positive integer, and let F be an infinite

An infinite arithmetic progression whose terms are positive in-

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

Is there a 1990-gon with the following properties (i) and

Consider the two square matrices

In a right-angled triangle ABC, let AD be the altitude

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

Show that the solution set of the inequality

Show that for any n ̸quiv0 (mod 10) there exists a multiple of

For any set S of five points in the plane, no three of which

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

Prove or disprove: Given a finite set of points with integer

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

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

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

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

The incircle Ωof the acute-angled triangle ABC is tangent