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A subset S of the set of integers 0, . . . , 99 is said to have

A rectangular pool table has a hole at each of three of its

Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle.

To every natural number k, k \geq2, there corresponds a sequence

Given n points in the plane such that no three of them

In connection with a convex pentagon ABCDE we consider

The lengths of the sides of a rectangle are given to be odd

Let n > 1 be a fixed integer. Define functions f0(x) = 0,

Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that

Suppose that n > m \geq1 are integers such that the string of

Find all solutions (x, y) \inZ2 of the equation

We are given a circle K and a point P lying on a line g. Construct

Let A1A2, B1B2, C1C2 be three equal segments on the three

The circles (R, r) and (P, ho), where r > ho, touch externally

Let n numbers x1, x2, . . . , xn be chosen in such a way that

Given a triangle ABC such that the circumcenter is in the

The points A, B, C are in this order on line D, and AB = 4BC.

The colonizers of a spherical planet have decided to build N

Integers cm,n (m \geq0, n \geq0) are defined by cm,0 = 1 for all

A cylindrical container has height 6 cm and radius 4 cm. It

If 0 \leqa \leqb \leqc \leqd, prove that

Let D be the point on the side BC of the triangle ABC such

Two families of parallel lines are given in the plane, consisting

Find for every value of n a set of numbers p for which the fol-

Prove that the sequence (an)n\geq0, an = [n

Find the total number of different integers that the function

Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let

Prove that

If a0 is a positive real number, consider the sequence {an}

Let {an}\infty

Find all positive integers x such that the product of all digits

Find all possible finite sequences {n0, n1, n2, . . . , nk} of integers

Given a graph with n vertices and a positive integer m that is

A pack of 2n cards contains n different pairs of cards. Each

Let S be a set of n2 + 1 closed intervals (n a positive integer).

Construct the circle that is tangent to three given circles.

Show that the equation

Let ABCD be a cyclic quadrilateral. Show that the centroids of

Let M be a finite set and P = {M1, M2, . . . , Mk} a partition

Let f and g be functions from the set A to the same set A.

We are given n > 3 points in the plane, no three of which lie on

Find all numbers x \inZ for which the number

The circles c1 and c2 are tangent at the point A. A straight

For a positive integer n, let 6(n) be the natural number whose

In a group of n people each one knows exactly three others. They

Consider the number lpha obtained by writing one after another

A curve determined by

Given a unit cube, find the locus of the centroids of all tetra-

A collection of 2n letters contains 2 each of n different letters.

A straight cone is given inside a rectangular parallelepiped

The plane is divided into equal squares by parallel lines; i.e.,

Find a function f(x) defined for all real values of x such that

If in a convex quadrilateral ABCD, E and F are the midpoints

Find all integer solutions of the equation

Given a natural number n, find all polynomials P(x) of degree

Prove that 1

Find all natural numbers n < 1978 with the following property:

Compute the largest number of regions into which one can divide

Suppose ABCD and A′B′C′D′ are two parallelograms arbi-

Let A, B denote two distinct fixed points in space. Let X, P

We are given n points in space. Some pairs of these points

Let an =

The number 0 or 1 is to be assigned to each of the n vertices

The incenter of a triangle is the midpoint of the line seg-

Prove that there exist infinitely many natural numbers a

An (n2 +n+1) imes(n2 +n+1) matrix of zeros and ones is given.

Given a polynomial f(x) with integer coefficients whose value

Let lpha(n) be the number of pairs (x, y) of integers such that

It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2

Solve the equation

The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where

There are n \geq3 job openings at a factory, ranked 1 to n in

In how many different ways can three knights be placed on a

Denote by xn(p) the multiplicity of the prime p in the canonical

It is well known that the binomial coefficients

Let M be the point inside the right-angled triangle ABC

A sequence (an)\infty

Prove that if P(x) = (x−a)kQ(x), where k is a positive integer,

Determine the least possible value of the natural number n

In a given country, all inhabitants are knights or knaves. A

Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary

If x, y, z are real numbers satisfying the relations x+y+z = 1

Let ABC and A′B′C′ be any two coplanar triangles. Let L be

Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove

For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n}

A circle K centered at (0, 0) is given. Prove that for every vector

For a positive real number p, find all real solutions to the equation

An arithmetic function is a real-valued function whose do-

Consider the binomial coefficients

(a) Show that the set N of all natural numbers can be parti-

Given an equilateral triangle ABC of side a in a plane, let

A function f has the following property: If k > 1, j > 1,

Prove the inequality

Let f : (0, +\infty) \toR be a function having the property

Let ABC be a triangle with inradius r and circumradius R.

A variable tetrahedron ABCD has the following properties:

A square ABCD is divided into (n −1)2 congruent squares,

A polynomial P(x) has degree at most 2k, where k = 0, 1,

Let ABC be a nonequilateral triangle. Prove that there exist

Let n be a positive integer. Find the maximal number of non-