BALKAN 1999 Problem 1

Let $O$ be the circumcenter of the triangle $ABC$ . The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$ . The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$ . The point $W$ is the midpoint of $AZ$ .

a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$ ;

b) Show that $MW$ is perpendicular to $AY$ .