BALTIC-WAY 2019 Problem 11

Let $ABC$ be a triangle with $AB = AC$ . Let $M$ be the midpoint of $BC$ . Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$ . Let $MP$ intersect $AB$ at $Q$ . Let $R$ be a point on $AP$ such that $QR \parallel BP$ . Prove that $CP$ bisects $\angle RCB$ .