USAMO 2017 Problem 3

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$ . Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$ ; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$ . Lines $MK$ and $BC$ meet at $S$ , and $N$ is the midpoint of $\overline{IS}$ . The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$ . Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$ .

Proposed by Evan Chen