RU-ALLRUSSIAN 2015 Problem 117

A scalene triangle $ABC$ is inscribed within circle $\omega$ . The tangent to the circle at point $C$ intersects line $AB$ at point $D$ . Let $I$ be the center of the circle inscribed within $\triangle ABC$ . Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$ , respectively. Let $M$ be the midpoint of $QP$ . Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$ .