HMMT 2015 T4

(Convex) quadrilateral $A B C D$ with $B C=C D$ is inscribed in circle $\Omega$; the diagonals of $A B C D$ meet at $X$. Suppose $A D<A B$, the circumcircle of triangle $B C X$ intersects segment $A B$ at a point $Y \neq B$, and ray $\overrightarrow{C Y}$ meets $\Omega$ again at a point $Z \neq C$. Prove that ray $\overrightarrow{D Y}$ bisects angle $Z D B$. (We have only included the conditions $A D<A B$ and that $Z$ lies on ray $\overrightarrow{C Y}$ for everyone's convenience. With slight modifications, the problem holds in general. But we will only grade your team's solution in this special case.)