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Prove that for any positive integers x, y, z with xy−z2 = 1 one

Let R be a rectangle that is the union of a finite number of

S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 =

Let … be real numbers such that

Find all integer triples (p, q, r) such that 1 < p < q < r

There are 10001 students at a university. Some students join

Let m, n \geq2 be positive integers, and let a1, a2, . . . , an

Let f(x) be a monic polynomial of degree 1991 with integer

There are six ports on a lake. Is it possible to organize a series

Let a, b, c be integers and p an odd prime number. Prove that

The tangents at B and C to the circumcircle of the acute-angled

Let ABC be any triangle and P any point in its interior. Let

The expression contains the denominator

We seek all three-digit integers whose quotient upon division by $11$ equals the sum of the squares of their digits.

Define

We seek all real numbers $x$ satisfying

The previous construction failed because adding a small geometric perturbation to $a_k$ cannot repair arbitrarily bad ratios.

The soldier moves inside an equilateral triangle $ABC$ of side length $a$.

Let $G$ be a set of non-constant affine functions of the real variable $x$ of the form

A one-person game with two possible outcomes is played as

Let a, b, c, r, and s be real numbers. Show that if r is a root of

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

Find, with argument, the integer solutions of the equation

Prove that the product of two natural numbers with their sum

Let f be a real-valued function defined on I = (0, +\infty) and

A solid right circular cylinder with height h and base-radius

Let −1 < x < 1. Show that

Let f be a strictly increasing function defined on the set of real

Do there exist two sequences of real numbers {ai}, {bi}, i \in

Let a, b, c be positive real numbers and let [x] denote the

A straight cone is given inside a rectangular parallelepiped

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

In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined

Let Fn be the nth Fibonacci number, defined by F1 = F2 = 1

Let l denote the length of the smallest diagonal of all rectangles

Show that for no integers a \geq1, n \geq1 is the sum

Given a triangle ABC such that the circumcenter is in the

A town has a road network that consists entirely of one-way

Let O be the midpoint of the axis of a right circular cylinder.

A regular triangular prism has height h and a base of side length

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

Given a natural number n, prove that the number M(n) of

Let x, y, z be nonnegative real numbers satisfying

The altitude from a vertex of a given tetrahedron intersects

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

Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with

Connecting the vertices of a regular n-gon we obtain a closed

Find a necessary and sufficient condition on the natural num-

Show that the sequence {an}n\geq1 defined by an = [n

Given two distinct numbers b1 and b2, their product can be

Two concentric circles have radii R and r respectively. Determine

Given a triangle ABC, three equilateral triangles AEB, BFC,

Let I, H, O be the incenter, centroid, and circumcenter of the

We call a coloring f of the elements in the set M = {(x, y) |

Let x = \sqrta +

Prove that the two last digits of 999 and 9999

Let M be the set of the lengths of an octahedron whose sides

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

Numbers 1, 2, . . . , 16 are written in a 4 imes4 square matrix so that

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

Let m be a positive integer and define f(m) to be the number

Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with

A circle of radius 1 rolls around a circle of radius

Let C be a cube with edges of length 2. Construct a solid with

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

A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed

Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an

A smooth solid consists of a right circular cylinder of height

Compute the largest number of regions into which one can divide

The fraction

A set G with elements u, v, w, . . . is a group if the following

Find the sum of the fiftieth powers of all sides and diagonals of

Let four points Ai (i = 1, 2, 3, 4) in the plane determine four

Among all triangles with a given perimeter, find the one with

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

Let S =

Prove the identity

Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) =

Express the number 1988 as the sum of some positive integers

If p and q are distinct prime numbers, then there are integers

Let A be a set of polynomials with real coefficients and let

Let n points be given on the surface of a sphere. Show that the

A system of n numbers x1, x2, . . . , xn is given such that

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

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

Find the average of the quantity

For each nonnegative integer n, Fn(x) is a polynomial in x of

Let X be a bounded, nonempty set of points in the Cartesian

A rectangle ABCD is given whose sides have lengths 3 and

Let M be the circumcenter of a triangle ABC. The line through

Given two natural numbers w and n, the tower of n w’s is the

From point P on arc BC of the circumcircle about triangle

For a convex polygon P in the plane let P ′ denote the convex

For a matrix (pij) of the format m imes n with real entries, set

In the plane a circle C of unit radius is given. For any line l

Find x such that

Let a be a number different from zero. For all integers n define

Let C be a class of functions f : N oN that contains the

Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When

In the plane of a given triangle A1A2A3 determine (with proof)