Combinatorics · Round robin tournament · Double counting · Sums of squares
Ten teams took part in a volleyball tournament, and every team played exactly one match against each of the others. When the tournament ended, the first team had \(x_1\) wins and \(y_1\) losses, the second \(x_2\) wins and \(y_2\) losses, and so on, the tenth having \(x_{10}\) wins and \(y_{10}\) losses. Prove that
(Volleyball has no drawn matches: every match ends with a win for one of the two teams.)
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Serbian Regional Competition (Okruzno takmicenje) 2002, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source