15th All Soviet Union Mathematical Olympiad Problems 1981

1.  A chess board is placed on top of an identical board and rotated through 45 degrees about its center. What is the area which is black in both boards? 2.  AB is a diameter of the circle C. M and N are any two points on the circle. The chord MA' is perpendicular to the line NA and the chord MB' is perpendicular to the line NB. Show that AA' and BB' are