Combinatorics · Tournaments · Double counting · Algebraic identities
Ten teams took part in a volleyball tournament, and each team played exactly one match against each of the other nine. When the tournament ended, the first team had \(x_{1}\) wins and \(y_{1}\) losses, the second team \(x_{2}\) wins and \(y_{2}\) losses, and so on. Prove that
\[ x_{1}^{2} + x_{2}^{2} + \dots + x_{10}^{2} = y_{1}^{2} + y_{2}^{2} + \dots + y_{10}^{2}. \]
(A volleyball match cannot end in a draw.)
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Serbian Municipal Competition 2014, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source