# 1993 IMO Problems

## Problem 1

Let , where is an integer. Prove that cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

## Problem 2

Let be a point inside acute triangle such that and .

(a) Compute the ratio

(b) Prove that the tangents at to the circumcircles of and are perpendicular.

## Problem 3

On an infinite chessboard, a game is played as follows. At the start, pieces are arranged on the chessboard in an by block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of for which the game can end with only one piece remaining on the board.

## Problem 4

For three points in the plane, we define as the minimum length of the three altitudes of . (If the points are collinear, we set .)

Prove that for points in the plane,

## Problem 5

Does there exist a function such that for all and for all ?

## Problem 6

There are lamps in a circle (), where we denote . (A lamp at all times is either on or off.) Perform steps as follows: at step , if is lit, switch from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that:

(a) There is a positive integer such that after steps all the lamps are on again;

(b) If , we can take ;

(c) If , we can take