Problem:
In a round-robin tournament with teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
A total of games are played, so all teams could not be tied for the most wins as this would require wins per team. However, it is possible for teams to be tied, each with wins and losses. One such outcome can be constructed by labeling of the teams , and , and placing these labels at distinct points on a circle. If each of these teams beat the labeled teams clockwise from its respective labeled point, as well as the remaining unlabeled team, all would tie with wins and losses.
The problems on this page are the property of the MAA's American Mathematics Competitions