Question

在一场锦标赛中，每名选手与其他每名选手恰好比赛一场。每场比赛中，胜者得 $1$ 分，负者得 $0$ 分；若比赛打平，两名选手各得 $\frac12$ 分。锦标赛结束后发现，每名选手所得分数恰有一半来自与得分最低的十名选手之间的比赛。（特别地，这十名最低分选手中的每一名，都有一半的得分来自与其余九名最低分选手之间的比赛。）锦标赛共有多少名选手？

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac12$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest-scoring players earned half of her or his points against the other nine of the ten.) What was the total number of players in the tournament?

答案

答案：25

知识点：双重计数图论二次方程

难度评级：2720

小提示：

汇总得分最低的十名选手的分数，并计算他们相互之间的比赛数

Sum the scores of the ten lowest-scoring players and count their internal games

大提示：

令 $m$ 为其余选手的人数，并对两组之间比赛产生的分数进行双重计数

Let $m$ be the number of other players and double-count points from games across the two groups

解答：

得分最低的十名选手相互之间的比赛共贡献 $\binom{10}{2}=45$ 分。这是这十名选手总得分的一半，因此他们的总得分为 $90$ 。所以，他们在与其余 $m$ 名选手的比赛中共得到 $45$ 分。

因此，其余选手在对阵最低分十人时共得到 $10m-45$ 分。根据条件，这是其余选手总得分的一半，而所有选手的总分扣除最低分十人的总分后为 $\binom{m+10}{2}-90$ 。因而 $2(10m-45)=\binom{m+10}{2}-90\text{，}$ 得到 $(m-6)(m-15)=0$ 。若 $m=6$ ，最低分十人的平均得分为 $9$ 分，而其余六人的平均得分只有 $5$ ，后一组不可能排名在前一组之上。因此 $m=15$ ，选手总数为 $10+15=25$ 。

The games among the ten lowest players contribute $\binom{10}{2}=45$ total points. These are half of those ten players’ combined score, so their combined score is $90.$ Hence they earned $45$ points in games against the other $m$ players.

The other players therefore earned $10m-45$ points against the lowest ten. By the condition, this is half their combined score, which is $\binom{m+10}{2}-90.$ Thus $2(10m-45)=\binom{m+10}{2}-90,$ giving $(m-6)(m-15)=0.$ If $m=6,$ the lowest ten average $9$ points while the other six average only $5,$ impossible for the latter group to rank above them. Hence $m=15,$ and the total number of players is $10+15=25.$

1985 AIME 第 14 题

答案：25

难度评级：2720

小提示：

大提示：

解答：

其他年份的第 14 题