CN-SMO 2010 Problem 1

$ABC$ is a triangle with a right angle at $C$ . $M_1$ and $M_2$ are two arbitrary points inside $ABC$ , and $M$ is the midpoint of $M_1M_2$ . The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$ Attachments: