US-TST 2009 Problem 4

Let $ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ABC$ . Let $M$ be the midpoint of segment $AP$ . Given that $\angle PAB = \angle CQB = 45^\circ$ , $\angle ABP = \angle QBC = 75^\circ$ , $\angle RAC = 105^\circ$ , and $RQ^2 = 6CM^2$ , compute $AC^2/AR^2$ .

Zuming Feng.