Question 2
Let $\triangle ABC$ be an acute-angled triangle with $AB \lt AC$, which is inscribed in the circle $\omega_1$ centered at $O$. Denote by $H$ the orthocenter of $\triangle ABC$ and by $M$ the midpoint of side $BC$. Let $\omega_2$ be the circumcircle of triangle $\triangle BHC$, let the line $HO$ meet $\omega_2$ at $K \neq H$, and let the line $MK$ meet $\omega_2$ at $P \neq K$. Prove that the tangent line to $\omega_1$ at $A$ and the tangent line to $\omega_2$ at $P$ meet on the circumcircle of triangle $\triangle APK$.