RU-ALLRUSSIAN 1987 Problem 1

Ten players play in a tournament. Each pair plays one match, which results in a win or loss. If the ith player wins $\mathbf{a}_{\mathrm{i}}$ matches and loses $\mathbf{b}_{\mathrm{i}}$ matches, show that $\sum \mathbf{a}_{\mathrm{i}}^{2} = \sum \mathbf{b}_{\mathrm{i}}^{2}$ .