Problem:
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost games; there were no ties. How many sets of three teams were there in which beat beat , and beat ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There must have been teams, and therefore there were subsets of three teams. If such a subset does not satisfy the stated condition, then it consists of a team that beat both of the others. To count such subsets, note that there are choices for the winning team and choices for the other two teams in the subset. This gives such subsets. The required answer is . To see that such a scenario is possible, arrange the teams in a circle, and let each team beat the teams that follow it in clockwise order around the circle.
The problems on this page are the property of the MAA's American Mathematics Competitions