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tennis, a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}

Source: 2010 Thailand Mathematical Olympiad day 1 p5

March 12, 2021
combinatorics

Problem Statement

In a round-robin table tennis tournament between 20102010 athletes, where each match ends with a winner and a loser, let a1,...,a2010a_1,... , a_{2010} denote the number of wins of each athlete, and let b1,..,b2010b_1, .., b_{2010} denote the number of losses of each athlete. Show that a12+a22+...+a20102=b12+b22+...+b20102a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}.
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