21st All Soviet Union Mathematical Olympiad Problems 1987

1.  Ten players play in a tournament. Each pair plays one match, which results in a win or loss. If the ith player wins ai matches and loses bi matches, show that ∑ ai2 = ∑ bi2. 2.  Find all sets of 6 weights such that for each of n = 1, 2, 3, ... , 63, there is a subset of weights weighing n. 3.  ABCDEFG is a regular 7-gon. Prove that 1/AB = 1/AC + 1/AD. 4.