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Prove that the product of two sides of a triangle is always

Let k, m, and n be positive integers such that m+k + 1 is

(a) Solve the equation

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

Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7.

Find the greatest number c such that for all natural numbers

Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A

(a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the

Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a

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

Let n \geq2 be an integer. Find the minimum k for which there

The integers 1 through 1000 are located on the circumference

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

Let f(x) be a polynomial with integer coefficients. Prove that

Let N be a point inside the triangle ABC. Through the mid-

Two mirror walls are placed to form an angle of measure lpha. There

Three mutually nonparallel lines li (i = 1, 2, 3) are given

In a plane, three pairwise intersecting circles C1, C2, C3 with

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

Let (an), n = 0, 1, . . ., be a sequence of real numbers such that

On a line a set of segments is given of total length less than

Prove that for every positive integer n coprime to 10 there

Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the

Suppose that 1985 points are given inside a unit cube. Show

We call a tetrahedron right-faced if each of its faces is a right-

A balance has a left pan, a right pan, and a pointer that moves

Let k be a positive integer and Mk the set of all the integers

T is a given triangle with vertices P1, P2, P3. Consider an arbi-

Given two numbers x0 and x1, let lpha and eta be coefficients

Through a point O on the diagonal BD of a parallelogram

Two moving bodies M1, M2 are displaced uniformly on two

Show that the set S of natural numbers n for which 3/n

Show that the reciprocal of any number of the form 2(m2 +

Prove that

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

Let T be the set of all lattice points (i.e., all points with

A fox stands in the center of the field which has the form of an

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

Given the expression

In a triangle ABC, the incircle touches the sides BC, CA, AB

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

Find the positions of three points A, B, C on the boundary of

Let n and z be integers greater than 1 and (n, z) = 1. Prove:

Show that there do not exist more than 27 half-lines (or rays)

Consider the set S of all the different odd positive integers

Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1,

Let a and b be integers. Is it possible to find integers p and q

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

Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb

Let b \geq2 be a positive integer.

Let Z be a set of points in the plane. Suppose that there exists

Let M be the set of all functions f with the following proper-

Let ABCD and A′B′C′D′ be two squares in the same plane and

If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n,

We are given an n imes n board, where n is an odd number. In

A figure of area 1 is cut out from a sheet of paper and divided

Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1

Let P be a fixed point and T a given triangle that contains the

For any angle lpha with 0 < lpha < 180◦, we call a closed convex

Seventeen cities are served by four airlines. It is noted that

Let f : R oR be a continuous function. Suppose that the

We define a binary operation ⋆in the plane as follows: Given

For a real number x, let [x] denote the greatest integer not

Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot +

Let n and p be two integers such that 2p \leqn. Prove the

Given a triangle ABC and external points X, Y , and Z such

A regular tetrahedron A1B1C1D1 is inscribed in a regular

Let [x] denote the greatest integer less than or equal to x. Let lpha

Let S be the point of intersection of the two lines l1 : 7x−5y +

Notice that in the fraction 16

Find the maximum value of

The function f(n) is defined on the nonnegative integers n by:

Find all pairs of integers a and b for which

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

There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of

Find the number of positive integers n satisfying arphi(n) | n such

A wheel consists of a fixed circular disk and a mobile circular

Let Sn = {1, . . . , n} and let f be a function that maps every

In the system of base n2 + 1 find a number N with n different

If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1,

If a, b, c, d are integers such that ad is odd and bc is even, prove

The triangle ABC has a right angle at C. The point P is

Show that for nonnegative real numbers a, b and integers n \geq2,

Prove the identity

Poldavia is a strange kingdom. Its currency unit is the bourbaki

Through a point P within a triangle ABC the lines l, m, and

Let a1, . . . , an be distinct positive integers that do not contain

We are given three equal rectangles with the same center in

A rhombus with its incircle is given. At each vertex of the

Prove the following statement: There does not exist a pyramid

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

Consider the sequence (cn):

Construct a scalene triangle such that

Under the conditions x1, x2 > 0, x1y1 > z2

Find all triples (x, y, z) of integers such that

Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive

Let O be an interior point of a tetrahedron A1A2A3A4. Let

Let E be the set of all triangles whose only points with integer

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

Prove that