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?

$22$

$23$

$24$

$26$

$30$

Answer: C

Concepts:optimization extremal argument

Difficulty rating: 1610

Video solution:

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Written solution:

Each top team plays $6$ games against the bottom three teams, so it can earn at most $6 \cdot 3 = 18$ points from those games.

Among the top three teams, there are $3$ pairs and hence $6$ games. Each game awards at most $3$ points total, so these games award at most $18$ points among the three teams. If their final scores are equal, each can therefore receive at most $18/3=6$ of those points.

Thus each top team has at most $18+6=24$ points. This bound is attainable: let every top team win all its games against the bottom teams, and for each pair of top teams let each team win one of their two games. Then each top team earns $18+6=24$ points.

Thus, the correct answer is C.

2019 AMC 8 Problem 19

Answer: C

Difficulty rating: 1610

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Written solution:

Problem 19 in Other Years