CMO 2008 Problem 1

$A B C D$ is a convex quadrilateral in which $A B$ is the longest side. Points $M$ and $N$ are located on sides $A B$ and $B C$ respectively, so that each of the segments $A N$ and $C M$ divides the quadrilateral into two parts of equal area. Prove that the segment $M N$ bisects the diagonal $B D$.