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Putnam 1958 February B3

Source: Putnam 1958 February

July 18, 2022
Putnamgraph

Problem Statement

In a round-robin tournament with nn players in which there are no draws, the numbers of wins scored by the players are s1,s2,…,sns_1 , s_2 , \ldots, s_n. Prove that a necessary and sufficient condition for the existence of three players A,B,CA,B,C such that AA beats BB, BB beats CC, and CC beats AA is s12+s22+…+sn2<(2n−1)(n−1)n6.s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.
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