BALKAN 2010 Problem 2

Let $A B C$ be an acute-angled triangle with orthocenter $H$ and let $M$ be the midpoint of $A C$. The foot of the altitude from $C$ is $C_{1}$. Point $H_{1}$ is symmetric to $H$ in $A B$. The projections of $C_{1}$ on lines $A H_{1}, A C$ and $B C$ are $P, Q$ and $R$ respectively. If $M_{1}$ is the circumcenter of triangle $P Q R$, prove that the point symmetric to $M$ with respect to $M_{1}$ lies on line $B H_{1}$. (Serbia)