UA-UMO 2005 Problem 113

In an acute-angled triangle $ABC$ , $\omega$ is the circumcircle and $O$ its center, $\omega _1$ the circumcircle of triangle $AOC$ , and $OQ$ the diameter of $\omega _1$ . Let $M$ and $N$ be points on the lines $AQ$ and $AC$ respectively such that the quadrilateral $AMBN$ is a parallelogram. Prove that the lines $MN$ and $BQ$ intersect on $\omega _1$ .