CAPS Match 2024

Let $M$ be the midpoint of $XY$. Note that $PR$ is midline in triangles $XBD$ and $XBY$, hence $M$ lies on $PR$. Analogously $M$ lies on $QS$.

Let ${\omega }_1$ be a circle with center $B$ and radius $AB=BC$ and ${\omega }_2$ be a circle with center $C$ and radius $BC=CD$. Distance of $X$ from center of ${\omega }_1$ is the same as distance of $Y$ from center of ${\omega }_2$ and also ${\omega }_1$ and ${\omega }_2$ have radius of same size, hence power of $X$ with respect to ${\omega }_1$ is the same as power of $Y$ with respect to ${\omega }_2$, so $$XA\cdot XC=YD\cdot YB.$$ Using homotheties centered at $X,Y$ we get that $MS\cdot MQ=MR\cdot MP$ and thus points $P,Q,R,S$ lie on a circle.