RO-TST 2015 Problem 17

Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$ ; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$ . The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$ , and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$ , situated inside triangle $P_1Q_1R$ . Finally, let $M$ be the midpoint of the side $AC$ . Prove that the angles $P_1RS$ and $Q_1RM$ are equal.