Middle European Mathematical Olympiad 2018 Problem I-3

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let $D$ be the foot of its altitude from $A$. Let $R$ and $Q$ be the centroids of the triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and the points $P, Q, R$ and $D$ are concyclic. Prove that the lines $AP, BQ$ and $CR$ are concurrent.