This problem was asked in a test, couple of years ago. Looked interesting!
In a chess tournament, each pair of players plays exactly one game. No game is drawn. Suppose the $i^{th}$ player wins $a_{i}$ games and loses $b_{i}$ games. Show that $$\sum a_{i}^{2} = \sum b_{i}^{2}$$
I can't think of a better title. It would be nice in case someone comes up with a nice title.