1990 AMC 8 第 19 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
19.
一排有 个座位。至少要占用多少个座位，才能保证下一个入座的人必须坐在某个人旁边？
There are seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?
小提示：
要迫使下一个人坐在别人旁边，就不能有一个空座位的相邻座位也都为空
To force the next person to sit adjacent to someone, no empty seat may have both neighbors empty
大提示：
让已坐座位之间尽量隔开：每三个座位中坐一个人
Seat people so there is exactly one empty seat between occupied ones: occupy every third seat
解答：
要迫使下一个人坐在别人旁边，每个空座位都必须与至少一个已坐座位相邻。每个已坐座位最多覆盖它自己及左右两个相邻座位，所以至少要坐满 个座位。
这个下界可以达到：重复采用“空、已坐、空”的排列，每三个座位只坐中间一个。这样恰好有 个已坐座位。
所以正确答案是 B。
To force the next person next to someone, every empty seat must be adjacent to an occupied one. Each occupied seat can account for at most itself and its two neighboring seats, so at least seats must be occupied.
This bound is attainable: use a repeating pattern of empty-occupied-empty, filling the middle seat of every group of three. This uses exactly occupied seats.
Thus, the correct answer is B .
其他年份的第 19 题
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- 1989 AMC 8
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