BALKAN 2008 Problem 1

An acute-angled scalene triangle $A B C$ is given, with $A C>B C$. Let $O$ be its circumcentre, $H$ its orthocentre, and $F$ the foot of the altitude from $C$. Let $P$ be the point (other than $A$ ) on the line $A B$ such that $A F=P F$, and $M$ be the midpoint of $A C$. We denote the intersection of $P H$ and $B C$ by $X$, the intersection of $O M$ and $F X$ by $Y$, and the intersection of $O F$ and $A C$ by $Z$. Prove that the points $F, M, Y$ and $Z$ are concyclic.