Question

一排有 $120$ 个座位。至少要坐下多少人，才能保证下一个入座的人必须坐在某个人旁边？

There are $120$ 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?

$30$

$40$

$41$

$60$

$119$

答案：B

知识点：最优化抽屉原理

难度评级：1090

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 $120/3 = 40$ 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 $40$ 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 $120/3 = 40$ 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 $40$ occupied seats.

Thus, the correct answer is B .

1990 AMC 8 第 19 题

答案：B

难度评级：1090

解答：

其他年份的第 19 题