Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)