RO-NMO 2003 Problem 74

In triangle $ABC$ , $P$ is the midpoint of side $BC$ . Let $M\in(AB)$ , $N\in(AC)$ be such that $MN\parallel BC$ and $\{Q\}$ be the common point of $MP$ and $BN$ . The perpendicular from $Q$ on $AC$ intersects $AC$ in $R$ and the parallel from $B$ to $AC$ in $T$ . Prove that: (a) $TP\parallel MR$ ; (b) $\angle MRQ=\angle PRQ$ .

Mircea Fianu