# 2001 IMO Problems

## Problem 1

Consider an acute triangle . Let be the foot of the altitude of triangle issuing from the vertex , and let be the circumcenter of triangle . Assume that . Prove that .

## Problem 2

Let be positive real numbers. Prove that .

## Problem 3

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

## Problem 4

Let be integers where is odd. Let denote a permutation of the integers . Let . Show that for some distinct permutations , the difference is a multiple of .

## Problem 5

is a triangle. lies on and bisects angle . lies on and bisects angle . Angle is . . Find all possible values for angle .

## Problem 6

are positive integers such that . Prove that is not prime.