Question 4
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $\Gamma$ be a circle with centre on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that $BP \cdot CQ = AP \cdot AQ$. Prove that $\Gamma$ is tangent to the circumcircle of triangle $BOC$.