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2019 AMC 8 Question 19

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2019 AMC 8 Question 19
by Ellen Zong - Thursday, April 9, 2020, 1:13 AM

In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 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 the total points for each of the top three teams?

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Re: 2019 AMC 8 Question 19
by John Lensmire - Thursday, April 9, 2020, 4:55 PM

This ones a little tricky and will probably require you to write out some examples and experiment a little bit.

Some things to get started:

1) $6$ teams each play each other twice, so each team plays each of the $5$ other teams twice. Hence each team plays $10$ games in total.

2) It might be helpful to think of the six teams as divided into two groups, three top teams and three bottom teams. If the top teams need a lot of total points, that probably means the top teams always beat the bottom teams.

From here, how many points does a top team already have? Then consider what can happen when the top teams play each other.