RO-NMO 2008 Problem 74

Let $ABCD$ be a rectangle with center $O$ , $AB\neq BC$ . The perpendicular from $O$ to $BD$ cuts the lines $AB$ and $BC$ in $E$ and $F$ respectively. Let $M,N$ be the midpoints of the segments $CD,AD$ respectively. Prove that $FM \perp EN$ .