ELMO 2009 Problem 2

Let $ABC$ be a triangle such that $AB < AC$ . Let $P$ lie on a line through $A$ parallel to line $BC$ such that $C$ and $P$ are on the same side of line $AB$ . Let $M$ be the midpoint of segment $BC$ . Define $D$ on segment $BC$ such that $\angle BAD = \angle CAM$ , and define $T$ on the extension of ray $CB$ beyond $B$ so that $\angle BAT = \angle CAP$ . Given that lines $PC$ and $AD$ intersect at $Q$ , that lines $PD$ and $AB$ intersect at $R$ , and that $S$ is the midpoint of segment $DT$ , prove that if $A$ , $P$ , $Q$ , and $R$ lie on a circle, then $Q$ , $R$ , and $S$ are collinear.

David Rush