# 2000 IMO Problems

## Day 1

## Problem 1

Two circles and intersect at two points and . Let be the line tangent to these circles at and , respectively, so that lies closer to than . Let be the line parallel to and passing through the point , with on and on . Lines and meet at ; lines and meet at ; lines and meet at . Show that .

## Problem 2

Let be positive real numbers with . Show that

## Problem 3

Let be a positive integer and a positive real number. Initially there are fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points and to the left of , and letting the flea from jump over the flea from to the point so that .

Determine all values of such that, for any point on the line and for any initial position of the fleas, there exists a sequence of moves that will take them all to the position right of .

## Day 2

## Problem 4

A magician has one hundred cards numbered to . He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.

A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)

## Problem 5

Does there exist a positive integer such that has exactly 2000 prime divisors and divides ? Solution

## Problem 6

Let , , and be the altitudes of an acute triangle . The incircle of triangle touches the sides , , and at , , and , respectively. Consider the reflections of the lines , , and with respect to the lines , , and . Prove that these images form a triangle whose vertices line on .