RMM 2025 Problem 5

Let $A B C$ be an acute triangle with $A B< A C$ , and let $H$ and $o$ be its orthocentre and circumcentre, respectively. Let $\Gamma$ be the circumcircle of triangle $B O C$ . Circle $\Gamma$ intersects line $A O$ at points $o$ and $A^{\prime}$ , and $\Gamma$ intersects the circle of radius $A O$ with centre $A$ at points $o$ and $F$ . Prove that the circle which has diameter $A A^{\prime}$ , the circumcircle of triangle $A F H$ and $\Gamma$ pass through a common point.