Putnam 1965 B2

B-2. In a round-robin tournament with $n$ players $P_{1}, P_{2}, \cdots, P_{n}$ (where $n>1$ ), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $\omega_{r}$ and $l_{r}$ be the number of games won and lost, respectively, by $P_{r}$. Show that $$\sum_{r=1}^{n} w_{r}^{2}=\sum_{r=1}^{n} l_{r}^{2}$$