IT-CESENATICO 1988 Problem 2

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ( $i = 1,2,...,n$ ), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$