FKMO 2015 Problem 4

$\triangle ABC$ is an acute triangle and its orthocenter is $H$ . The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$ . Lines $DH$ and $AC$ meets at $P$ , and the circumcenter of $\triangle ADP$ is $Q$ . Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$ .