UA-UMO 2023 Problem 116

Let $K$ be the midpoint of the median $AM$ of a triangle $ABC$ . Points $X, Y$ lie on $AB, AC$ , respectively, such that $\angle KXM =\angle ACB$ , $AX>BX$ and similarly $\angle KYM =\angle ABC$ , $AY>CY$ . Prove that $B, C, X, Y$ are concyclic.

Proposed by Mykhailo Shtandenko