1991 AMC 12 第 15 题

先试着解答 1991 AMC 12 第 15 题,然后核对你的答案与精心整理的解答,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 1991 AMC 12 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。

15.

一张圆桌周围恰有 6060 把椅子。桌边已经坐了 NN 个人,使得下一位入座者无论坐哪一把空椅子,都必须挨着某个人。NN 的最小可能值为
A circular table has exactly 6060 chairs around it. There are NN people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of NN is
1515
2020
3030
4040
5858
答案:B
知识点:circular arrangementcoveringextremal counting
小提示:
任何空椅子的两把相邻椅子不能同时为空
No empty chair can have both neighboring chairs empty
大提示:
相邻的两把已占椅子之间至多有两把空椅子
Between consecutive occupied chairs there can be at most two empty chairs
解答:
条件说明每把空椅子都与一把已占椅子相邻,所以不可能有三把连续的椅子都空着。因此,每把已占椅子至多可以对应它自己以及其后的两把空椅子,从而 60≤3N60\le3N,即 N≥20N\ge20。沿圆桌重复“已占、空、空”的排列即可达到 N=20N=20。
因此正确答案为 B。
The condition says that every empty chair has an occupied neighbor, so no three consecutive chairs may all be empty. Therefore each occupied chair can account for at most itself and the two empty chairs following it, giving 60≤3N60\le3N and N≥20.N\ge20. Repeating the pattern occupied-empty-empty around the table attains N=20.N=20.
Thus the correct answer is B.
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