10th Mexican Mathematical Olympiad Problems 1996

10th Mexican Mathematical Olympiad Problems 1996A1.  ABCD is a quadrilateral. P and Q are points on the diagonal BD such that the points are in the order B, P, Q, D and BP = PQ = QD. The line AP meets BC at E, and the line Q meets CD at F. Show that ABCD is a parallelogram iff E and F are the midpoints of their sides. A2.  64 tokens are numbered 1, 2, ... , 64. The