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Let us play volleyball, shall we?

Source: Tuymaada 2001, day 1, problem 1.

April 30, 2007
inequalitiescombinatorics proposedcombinatorics

Problem Statement

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored nn-th has xnx_{n} points (n=1,…,10n=1, \dots, 10), prove that x1+2x2+⋯+10x10≥165x_{1}+2x_{2}+\dots+10x_{10}\geq 165.
Proposed by D. Teryoshin
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