Question

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 total points for each of the top three teams?

$22$

$23$

$24$

$26$

$30$

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Written solution:

We can assume that the top $3$ teams won every game against every team not amongst themselves.

They play $3 \cdot 2 = 6$ games in total, getting a total of $6 \cdot 3 = 18$ points.

Now, among the top $3,$ each pair of teams plays twice. To even out the scores, we can let one team win one game and let the other team win the other game.

To make the three scores equal while keeping all possible points, split the two games in each pair so each team gets $3$ points from that pair. There are $3$ such pairs, with each team appearing in $2$ pairs.

This means that each team will get an extra $3 \cdot 2 = 6$ points. Therefore, their maximum score is $18 + 6 = 24.$

Thus, the correct answer is C.

2019 AMC 8 Problem 19

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Problem 19 in Other Years