ELMO SL 2013 G13

In $\triangle ABC$ , $AB<AC$ . $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$ , respectively. $M$ is the midpoint of segment $BC$ , and $\omega$ is the circumcircle of $\triangle APD$ . Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$ . $QB$ meets $\omega$ again at $R$ , and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$ . Prove $SD$ is tangent to the circumcircle of $\triangle QDM$ .

Proposed by Ray Li