ELMO 2024 Problem 1

In convex quadrilateral $ABCD$ , let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$ . Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$ , respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$ , and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$ . Prove that $A$ , $B$ , $R$ , and $S$ are concyclic.

Tiger Zhang