Problem:
In a tournament there are six teams that play each other twice. A team earns points for a win, point for a draw, and points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
We can assume that the top teams won every game against every team not amongst themselves. They play games in total, getting a total of points. Now, amongst the top , each team plays each other team twice. To even out the scores, we can let one team win one game and let the other team win the other game. This means that for every pair of the top teams, each team in the pair gets points. There are such pairs, with each team appearing in pairs. This means that each team will get an extra points. Therefore, their maximum score is .
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions