Question 1
Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}$. Let $O$ be the circumcenter of the triangle $ABC$ and let $\Gamma$ be the circumcircle of the triangle $BOC$. Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C$. Let $ON$ be a diameter of the circle $\Gamma$. Prove that the quadrilateral $APNQ$ is a parallelogram.