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:
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E.
Solution:
The top three teams played six games against one another in which at most points were awarded, and they played games against the other three teams in which at most points were awarded. Therefore the sum of the scores of the top three teams was at most , and thus the total score of each of the top teams was at most This score is indeed obtained when between each pair of top teams, one game is won and one game is lost, and each top team wins both games against each of the other three teams.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions