PL-OM 2010 Problem 3

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$ . Point $O$ is the circumcenter of the triangle $CPQ$ . Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.