2019 AMC 8 第 19 题
先试着解答 2019 AMC 8 第 19 题，然后核对你的答案与视频与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2019 AMC 8 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
19.
一个锦标赛中有六支队伍，每两队互相比赛两次。胜一场得 分，平一场得 分，负一场得 分。所有比赛结束后，前三名队伍的总分相同。前三名每队可能得到的最大总分是多少？
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?
小提示：
前三名应赢下所有对后三名的比赛
The top three should win all games against the bottom three
大提示：
前三名之间的比赛把积分全部留在这三队内，并让每一对平分两场比赛
Among the top three, keep all points while splitting each pair evenly
视频讲解：
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文字解答：
每支前三名队伍与后三名队伍比赛 场，所以从这些比赛中最多得到 分。
前三名队伍之间有 对，因此共有 场比赛。每场比赛总共最多产生 分，所以这六场比赛给三支队伍的总分至多为 分。若三队最终同分，每队从中至多得到 分。
因此每支前三名队伍最多得到 分。这个上界可以达到：让每支前三名队伍赢下所有对后三名队伍的比赛，并让每对前三名队伍在两场交锋中各赢一场。这样每支前三名队伍都得到 分。
所以正确答案是 C。
Each top team plays games against the bottom three teams, so it can earn at most points from those games.
Among the top three teams, there are pairs and hence games. Each game awards at most points total, so these games award at most points among the three teams. If their final scores are equal, each can therefore receive at most of those points.
Thus each top team has at most 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 points.
Thus, the correct answer is C.
其他年份的第 19 题
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