IT-CESENATICO 2021 Problem 5

Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$ -altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$ .

Show that the circumcircles of the two triangles $ACM$ , $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$ .