Question 1
Let $\Gamma$ be the circumcircle of triangle $ABC$, isosceles at $C$. Let $M$ lie in the interior of side $BC$. Suppose $N$ lies on ray $AM$, with $M$ between $A,N$, and $AN=AC$. The circumcircle of $CMN$ meets $\Gamma$ at distinct points $C,P$. Let lines $AB,CP$ meet at $Q$. Prove $\angle BMQ=\angle QMN$.