Question 9
Let $E$ and $F$ be points on the sides $CA$ and $AB$, respectively, of a triangle $ABC$ with circumcircle $\Gamma$. The circumcircle of $AEF$ meets $\Gamma$ again at $X\ne A$. The circumcircles of $ABE$ and $ACF$ meet again at $K\ne A$. Let $AK$ meet $\Gamma$ again at $M\ne A$, and let $N$ be the reflection of $M$ across $BC$. Let $XN$ meet $\Gamma$ again at $S\ne X$. Prove that $SM\parallel BC$.
Source clarification and construction caveat: the English source names $K$ as $Y$ in one sentence; the Chinese source uses $K$ consistently. No nondegeneracy hypothesis is added to the original data. Some allowed choices fail to define a named circle or line, and even otherwise valid intersections may give $S=M$, so $SM$ is undefined. The solution explicitly documents this omission and proves the claim for every configuration in which all the named objects, including $SM$, exist.
Source clarification and construction caveat: the English source names $K$ as $Y$ in one sentence; the Chinese source uses $K$ consistently. No nondegeneracy hypothesis is added to the original data. Some allowed choices fail to define a named circle or line, and even otherwise valid intersections may give $S=M$, so $SM$ is undefined. The solution explicitly documents this omission and proves the claim for every configuration in which all the named objects, including $SM$, exist.