Problem

Source: European Mathematical Cup 2012, Junior Division, Problem 4

Tags: induction, strong induction, combinatorics proposed, combinatorics, Tournament graphs, Hamiltonian path



Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. Proposed by Matija Bucić.