HMMT 2011 T11

Let $A B C$ be a non-isosceles, non-right triangle, let $\omega$ be its circumcircle, and let $O$ be its circumcenter. Let $M$ be the midpoint of segment $B C$. Let the circumcircle of triangle $A O M$ intersect $\omega$ again at $D$. If $H$ is the orthocenter of triangle $A B C$, prove that $\angle D A H=\angle M A O$.