BY-MO 2019 Problem 96

The point $M$ is the midpoint of the side $BC$ of triangle $ABC$ . A circle is passing through $B$ , is tangent to the line $AM$ at $M$ , and intersects the segment $AB$ secondary at the point $P$ . Prove that the circle, passing through $A$ , $P$ , and the midpoint of the segment $AM$ , is tangent to the line $AC$ .

(A. Voidelevich)