TUYMAADA 2002 Problem 8

The circle with the center of $O$ touches the sides of the angle $A$ at the points of $K$ and $M$ . The tangent to the circle intersects the segments $AK$ and $AM$ at points $B$ and $C$ respectively, and the line $KM$ intersects the segments $OB$ and $OC$ at the points $D$ and $E$ . Prove that the area of the triangle $ODE$ is equal to a quarter of the area of a triangle $BOC$ if and only if the angle $A$ is $60^\circ$ .