HMIC 2024 Problem 5

Let $A B C$ be an acute, scalene triangle with circumcenter $O$ and symmedian point $K$. Let $X$ be the point on the circumcircle of triangle $B O C$ such that $\angle A X O=90^{\circ}$. Assume that $X \neq K$. The hyperbola passing through $B, C, O, K$, and $X$ intersects the circumcircle of triangle $A B C$ at points $U$ and $V$, distinct from $B$ and $C$. Prove that $U V$ is the perpendicular bisector of $A X$. The symmedian point of triangle $A B C$ is the intersection of the reflections of $B$-median and $C$-median across the angle bisectors of $\angle A B C$ and $\angle A C B$, respectively.