2014 OIM Problems/Problem 5

Problem

Let $ABC$ be an acute triangle and $H$ be the point of intersection of the heights. The height from $A$ cuts $BC$ in $D$ . Let $M$ and $N$ be the midpoints of $BH$ and $CH$ , respectively. $DM$ and $DN$ intersect $AB$ and $AC$ at $X$ and $Y$ , respectively. If $XY$ intersects $BH$ at $P$ and $CH$ in $Q$ , show that $H, P, D$ , and $Q$ are on the same circle.