# 2007 BMO Problems

Problems of the 2007 Balkan Mathematical Olympiad.

## Problem 1

Let be a convex quadrilateral with , not equal to , and let be the intersection point of its diagonals. Prove that if and only if .

## Problem 2

Find all functions such that

, for any .

## Problem 3

Find all positive integers such that there exists a permutation on the set for which

is a rational number.

**Note**: A permutation of the set is a one-to-one function of this set to itself.

## Problem 4

For a given positive integer , let be the boundaries of three convex -gons in the plane such that , , are finite. Find the maximum number of points in the set .