ELMO 2024 Problem 5

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$ , let $M$ be the midpoint of $\overline{BC}$ . Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$ . Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$ . Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$ , and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$ . Prove that $A$ , $Y$ , and $Z$ are collinear.

Tiger Zhang