Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Let $ABC$ be a triangle with circumcicle $\omega$. Then the bisector of $\angle ABC$ meets $AC$ at $D$ and circle $\omega$ at $M \ne B$. The circumcircle of $\triangle BDC$ meets $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets $AC$ at $Q \ne C$. If $PQ = MC$, what is $\angle ABC$?