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Determine all real functions f(x) that are defined and contin-

Let b1, b2, . . . , b1989 be positive real numbers such that the

For each pair of positive integers k and n, let Sk(n) be the

Let S be a k-element set.

Determine all positive roots of the equation xx = 1/

Given the equation

The area of a triangle is S and the sum of the lengths of its

Prove that all numbers in the sequence

We are given 2n natural numbers

Let M be a nonempty subset of Z+ such that for every element

Given a set of 1988 points in the plane, no three points of the

Let T be a triangle with inscribed circle C. A square with sides

The vertices of a given square are clockwise lettered A, B, C, D.

Let Q be a square with side length 6. Find the smallest integer

Let a, b, and c denote the three sides of a billiard table in the

The ternary expansion x = 0.10101010 . . . is given. Give the

A pentagon ABCDE inscribed in a circle for which BC < CD

Show that if the sides a, b, c of a triangle satisfy the equation

Prove that the set {1, 2, . . ., 1986} can be partitioned into 27

If Cp

A lampshade is part of the surface of a right circular cone

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

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

Let a and b be coprime integers, greater than or equal to 1.

Let O be a point outside a given circle. Two lines OAB, OCD

For k = 1, 2, . . . consider the k-tuples (a1, a2, . . . , ak) of positive

Prove that the equation

Given an acute triangle find a point inside the triangle such

Given n points X1, X2, . . . , Xn in the interval 0 \leqXi \leq1,

Let a, b, and c be real numbers such that

Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1,

In a triangle ABC, let H be its orthocenter, O its circumcenter,

Prove that

Two straight lines perpendicular to each other meet each side

Prove that

Given the side a and the corresponding altitude ha of a triangle

Show that for every natural number n, n

Let p be a prime number. A rational number x, with 0 < x < 1,

One hundred red points and one hundred blue points are

Let l be tangent to the circle k at B. Let A be a point on k

Show that the numbers tan(r\pi/15), where r is a positive integer

Let p and q be two prime numbers greater than 3. Prove that

Find the last eight digits of the binary development of 271986.

Suppose x0, x1, . . . , xn are integers and x0 > x1 > \cdot \cdot \cdot > xn.

Suppose tan lpha = p/q, where p and q are integers and q ̸= 0.

Let there be given a circle with center S and radius 1 in the plane,

Let Zm,n be the set of all ordered pairs (i, j) with i \in

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

In a plane a circle with radius R and center w and a line 
ambda

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

Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two

One hundred convex polygons are placed on a square with edge

Let g(x) be a fixed polynomial and define f(x) by f(x) =

Let P, Q, R be polynomials with real coefficients, satisfying

Find all rational solutions of

Prove the identity

Call a four-digit number (xyzt)B in the number system with

There are some boys and girls sitting in an n imes n quadratic

Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers.

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

Two half-lines a and b, with the common endpoint O, make an

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

Let n be a positive integer, X = {1, 2, . . ., n}, and k a positive

Let k and s be positive integers. For sets of real numbers

The perpendicular line issued from the center of the circum-

With the medians of an acute-angled triangle another triangle is

Let n be an integer that is not divisible by any square greater

Let O be the center of a circle. Let OU, OV be perpendicular

Let L denote the set of all lattice points of the plane (points

Let the polynomials

Let P1(x, y) and P2(x, y) be two relatively prime polynomials

Solve the system of equations

A boy has a set of trains and pieces of railroad track. Each

Points A, B, C, D lie on a circle such that AB is a diameter and

Prove that for a, b \inN, a!b! divides (a + b)!.

Prove that for any triangle ABC there exists a point P in the

If r > s > 0 and a > b > c, prove that

Exactly one side of a tetrahedron is of length greater than

An alphabet consists of n letters. What is the maximal length

Does there exist a number lpha (0 < lpha < 1) such that there is an

A polygon (not necessarily convex) with vertices in the lattice

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

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

Let ABCD be an arbitrary quadrilateral. Let squares ABB1A2,

Let ABCD be a convex quadrilateral whose diagonals intersect

Find the maximum value of

(a) Four balls of radius 1 are mutually tangent, three resting an the floor

For two given triangles A1A2A3 and B1B2B3 with areas ∆A

A park has the shape of a convex pentagon of area 5

We have p players participating in a tournament, each player

Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q

Prove that for a natural number n > 2,

Let ABCD be a convex quadrilateral whose diagonals AC and

Given an isosceles triangle ABC with a right angle at C,

Given a pyramid whose base is an n-gon inscribable in a circle,

Let ABC be a triangle. For every point M belonging to segment

Prove that 1

Let M, N, P be the midpoints of the sides BC, CA, AB of a

Suppose that points X, Y, Z are located on sides BC, CA,

Determine all pairs of positive integers (x, y) satisfying the equa-