Middle European Mathematical Olympiad 2016 Problem I-3

Let $ABC$ be an acute-angled triangle with $\measuredangle BAC > 45°$ and with circumcentre $O$. The point $P$ lies in its interior such that the points $A, P, O, B$ lie on a circle and $BP$ is perpendicular to $CP$. The point $Q$ lies on the segment $BP$ such that $AQ$ is parallel to $PO$.

Prove that $\measuredangle QCB = \measuredangle PCO$.