OMM 2018 Problem 1

Let $A$ and $B$ be two points on a line $\ell$ , $M$ the midpoint of $AB$ , and $X$ a point on segment $AB$ other than $M$ . Let $\Omega$ be a semicircle with diameter $AB$ . Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$ . Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$ . The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$ . Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$ . Show that $MX > XY$