From the Putnam Competition (I believe):
Suppose # is a binary operation on a set S such that the following properties hold:
- For all a, b, c in S, (a#b)#c = a#(b#c);
- For all a, b in S, if a#b = b#a then a=b.
Prove that, for all x, y, z in S, we have x#y#z = x#z.
From the 2011 South African Olympiad:
In chess tournaments, players get 1 point for a win, 0.5 points for a draw and 0 points for a loss. In a recent round-robin tournament each pair of players met exactly once, and the top four scores were 4.5, 3.5, 3 and 1.5. What was the lowest score at the tournament?