KöMaL 2012 A554

The circumcenter of the cyclic quadrilateral ABCD is $O$. The second intersection point of the circles ABO and CDO , other than $O$, is $P$, which lies in the interior of the triangle DAO . Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle$ QAP =$\angle$ OBR holds if and only if $\angle$ PDQ =$\angle$ RCO .