Problem of the Week 980

Everyone's A Winner

In a round-robin tournament, a player A is a gold medal winner provided, for every other player C, either A beat C or there is a player B so that A beat B and B beat C. Is it possible for a 10-player tournament to have ten gold medal winners?

Bonus Problem: Determine the set of n for which it is possible for an n-player tournament to have n gold medalists.

Source: The recent Konhauser Problemfest had an amusing problem on champions in a tournament. Here is a variation that seems equally amusing, pointed out by Richard Guy. It is in his recent MAA book with Paul Vaderlind and Loren Larson: "The Inquisitive Problem Solver" (section 250).