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3n+1 members in a club

Source: Kürschák 1987, problem 3

July 27, 2014
combinatorics unsolvedcombinatorics

Problem Statement

Any two members of a club with 3n+13n+1 people plays ping-pong, tennis or chess with each other. Everyone has exactly nn partners who plays ping-pong, nn who play tennis and nn who play chess.
Prove that we can choose three members of the club who play three different games amongst each other.