TSTST 2012 Problem 7

Triangle $A B C$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $B C$ and $\Omega$ at $D$ and $L$ (other than $A$ ), respectively. Let $M$ be the midpoint of side $B C$. The circumcircle of triangle $A D M$ intersects sides $A B$ and $A C$ again at $Q$ and $P$ (other than $A$ ), respectively. Let $N$ be the midpoint of segment $P Q$, and let $H$ be the foot of the perpendicular from $L$ to line $N D$. Prove that line $M L$ is tangent to the circumcircle of triangle $H M N$.