IT-CESENATICO 2018 Problem 6

Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$ . Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$ , and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$ , respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively.

(i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$ (ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$