RO-NMO 2014 Problem 93

Let $P,Q$ be the midpoints of the diagonals $BD,$ respectively, $AC,$ of the quadrilateral $ABCD,$ and points $M,N,R,S$ on the segments $BC,CD,PQ,$ respectively $AC,$ except their extremities, such that $$\frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} .$$ Show that the center of mass of the triangle $AMN$ is situated on the segment $RS.$