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Let x1, . . . , xn and y1, . . . , yn be real numbers. Let A =

(a) Let n be a positive integer. Prove that there exist distinct

In the triangle ABC, let D, E be points on the side BC such

A point M is chosen on the side AC of the triangle ABC in

Let Ox, Oy, Oz be three rays, and G a point inside the trihe-

Prove that the sequence 2n −3 (n > 1) contains a subse-

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

Let f(x) = x2 + x + p, p \inN. Prove that if the numbers

Determine the least possible value of f(1998), where f is a

Let R denote the set of all real numbers and R+ the subset

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

Show that the solution set of the inequality

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

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

Solve the system of equations

The altitudes through the vertices A, B, C of an acute-angled

Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be

Ali Barber, the carpet merchant, has a rectangular piece of

Let p be a prime number and let f(x) be a polynomial of degree

Given two congruent triangles … and … …, prove that there exists a plane such that the orthogonal projections of these…

Let N = {1, 2, 3, . . .}. Determine whether there exists a

Prove that the sum of an odd number of unit vectors passing

Let P(x) be a polynomial with real coefficients such that P(x) >

Let {fn} be the Fibonacci sequence {1, 1, 2, 3, 5, . . .}.

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

The quadrilateral ABCD has the following properties:

The incenter of the triangle ABC is K. The midpoint of AB

Let F be a nonempty set of functions f : R oR of the

Prove that it is possible to place 2n(2n + 1) parallelepipedic

Prove that the functional equations

Determine all pairs (m, n) of positive integers such that

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

I 6 (ROM 4)IMO3 Does there exist a natural number n for which the

The vertices D, E, F of an equilateral triangle lie on the sides

Let n \geq2 be a positive integer, with divisors 1 = d1 <

Prove that every partition of 3-dimensional space into three

Several equal spherical planets are given in outer space. On the

A pentagonal prism A1A2 . . . A5B1B2 . . . B5 is given. The

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

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

A function f defined on the positive integers (and taking

Let an be the last nonzero digit in the decimal representation

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

Let n \geq2 be a natural number. Find a way to assign nat-

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

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

Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three

Let au(n) denote the number of positive divisors of the positive

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

Let … denote the set of nonnegative integers. Find all functions … such that

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

S3 (POL) For an integer x \geq1, let p(x) be the least prime that does not

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

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

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

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

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

Prove that for every integer n > 1 the equation

Find all positive integers n such that

A plane cuts a right circular cone into two parts. The plane is

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

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

A set of three nonnegative integers {x, y, z} with x < y < z

Let f(x) = xn where n is a fixed positive integer and x =

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

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

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

For a given positive integer k denote the square of the sum of

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

Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a…

On the sides of an arbitrary triangle ABC, triangles BPC,

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

Find the integer represented by

Consider a regular 2n-gon and the n diagonals of it that

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

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

Let xn =

The following operation is allowed on a finite graph: Choose

Prove that N = 5125−1

Does there exist a set M with the following properties?

Find all positive integers … for which …,

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

The positive integers … and … are such that the numbers

Determine whether or not there exist two disjoint infinite sets … and … of points in the plane satisfying the following…

Given any point P in the interior of a sphere with ra-

The incircle Ωof the acute-angled triangle ABC is tangent

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

Two identically oriented equilateral triangles, ABC with center

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

A circle \omega is tangent to two parallel lines l1 and l2. A second

Prove that for each positive integer n, there exists a positive

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

Let B be a set of k sequences each having n terms equal to 1 or

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

From a bag containing 5 pairs of socks, each pair a different

Prove that

Suppose G is a connected graph with n edges. Prove that

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

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

Prove that every integer k greater than 1 has a multiple that is