SILK-ROAD 2021 Problem 3

In a triangle $ABC$ , $M$ is the midpoint of the $AB$ . A point $B_1$ is marked on $AC$ such that $CB=CB_1$ . Circle $\omega$ and $\omega_1$ , the circumcircles of triangles $ABC$ and $BMB_1$ , respectively, intersect again at $K$ . Let $Q$ be the midpoint of the arc $ACB$ on $\omega$ . Let $B_1Q$ and $BC$ intersect at $E$ . Prove that $KC$ bisects $B_1E$ .

M. Kungozhin