USAJMO 2014 Problem 6

Let $ABC$ be a triangle with incenter $I$ , incircle $\gamma$ and circumcircle $\Gamma$ . Let $M,N,P$ be the midpoints of sides $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$ , respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$ , respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$ .

(a) Prove that $I$ lies on ray $CV$ .

(b) Prove that line $XI$ bisects $\overline{UV}$ .