# 2007 IMO Problems

## Problem 1

Real numbers are given. For each () define and let

(a) Prove that, for any real numbers ,

(b) Show that there are real numbers such that equality holds in (*)

## Problem 2

Consider five points , and such that is a parallelogram and is a cyclic quadrilateral. Let be a line passing through . Suppose that intersects the interior of the segment at and intersects line at . Suppose also that . Prove that is the bisector of .

## Problem 3

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

## Problem 4

In the bisector of intersects the circumcircle again at , the perpendicular bisector of at , and the perpendicular bisector of at . The midpoint of is and the midpoint of is . Prove that the triangles and have the same area.

## Problem 5

(*Kevin Buzzard and Edward Crane, United Kingdom*)
Let and be positive integers. Show that if divides , then .

## Problem 6

Let be a positive integer. Consider as a set of points in three-dimensional space. Determine the smallest possible number of planes, the union of which contain but does not include .