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Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot <

Let a0, a1, a2, . . . be an arbitrary infinite sequence of positive

Let a, b, and c be positive real numbers such that abc = 1.

I 4 (USS 4) The sum of the squares of five real numbers a1, a2, a3, a4, a5

Let …, …, … be real numbers. Prove that the system of equations

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

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

Let S1 and S2 be two spheres with distinct radii that touch

Let x, y, and z be positive real numbers such that xyz = 1. Prove

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

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

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

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

Find all the functions f : R oR that satisfy

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

Let S be a set of n points in the plane. No three points of

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

Let ϕ : {1, 2, 3, . . .} … be injective. Prove that

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

Let m and n be nonnegative integers. Prove that m!n!(m+

A circle O with center O on base BC of an isosceles triangle

Let a, b, c, d be odd positive integers such that a < b < c <

Real constants a, b, c are such that there is exactly one square

Knowing that the system

Determine all integers n > 3 such that there are n points

On an infinite square grid, two players alternately mark sym-

Let a set of p equations be given,

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

Define a sequence ⟨f(n)⟩\infty

Let R+ be the set of all positive real numbers. Find all

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

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

Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime

The following operation is allowed on a finite graph: Choose

Let E be the set of 19833 points of the space R3 all three

We are given two mutually tangent circles in the plane, with

The triangle ABC is acute-angled. Let L be any line in the

C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3

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

For a positive integer n, let d(n) be the number of all positive

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

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

Find, with proof, the point P in the interior of an acute-angled

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

Let ABC be a triangle with semiperimeter s and inradius

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

B2 (POL 4) A convex, closed figure lies inside a given circle. The figure

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

Show that there exists a finite set A \subsetR2 such that for

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

Let triangle ABC be such that its circumradius R is equal to

The prolongation of the bisector AL (L \inBC) in the acute-

The point M inside the convex quadrilateral ABCD is such

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

Find all functions f : R oR satisfying the equation

Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an

Let ABCD be a tetrahedron having each sum of opposite sides

Let ABC be a triangle and let P be a point in its interior.

Decide whether there exists a set M of natural numbers satis-

Prove that from x + y = 1 (x, y \inR) it follows that

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

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

Given seven points in the plane, some of them are connected

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

ABCD is a quadrilateral with BC parallel to AD. M is the

C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the

Given that 1 −1

(a) Show that the set Q+ of all positive rational numbers can be par-

For a positive integer n define a sequence of zeros and ones

Let ABC be an acute triangle. Let DAC, EAB, and FBC

Prove that for every integer n > 1 the equation

Prove that for every real number M there exists an infinite

Suppose that every integer has been given one of the colors

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

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

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

Each positive integer a undergoes the following procedure in

The polynomial 1976(x+x2+\cdot \cdot \cdot+xn) is decomposed into a sum

Let P be a polynomial of degree n satisfying

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

Let x1 and x2 be relatively prime positive integers. For n \geq2,

Determine the minimum of a2 + b2 if a and b are real

2b.(VIE 1)

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

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

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

Let P1(x) = x2 −2, Pj(x) = P1(Pj−1(x)), j = 2, 3, . . . .

Let … be a function from the set of real numbers … into itself such that for all …, we have … and

(a) A plane \pi passes through the vertex O of the regular

Let O be the circumcenter and H the orthocenter of an acute

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

Denote by S the set of all primes p such that the decimal

Is it possible to put 1987 points in the Euclidean plane

Let f(x) = x2+1

Prove that the functional equations

A cyclic quadrilateral ABCD is given. The lines AD and

(a) For which values of n > 2 is there a set of n consecutive

Is there a positive integer m such that the equation

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

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