Question 8
Let $ABCD$ be a convex quadrilateral (that is, all its interior angles are less than $180^\circ$), such that there exist a point $M$ on segment $AB$ and a point $N$ on segment $BC$ for which $AN$ divides the quadrilateral into two equal-area pieces, and $CM$ does so as well. Prove that $MN$ bisects diagonal $BD$.