TR-MO 2012 Problem 2

Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$ . $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$ . $CP$ and $AD$ intersects at $Q$ , $BP$ and $AD$ intersects at $R$ . Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$ . Show that $RS=RT$