Question

一个礼堂有 $20$ 排座位，第一排有 $10$ 个座位。之后每一排都比前一排多一个座位。如果参加考试的学生可以坐在任何一排，但同一排中不能与另一个学生相邻，那么最多能安排多少名学生考试？

An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is

$150$

$180$

$200$

$400$

$460$

答案：C

知识点：有限制的排列等差数列取整函数

难度评级：1140

解答：

第 $k$ 排有 $9+k$ 个座位，所以最多坐 $\left\lceil (9+k)/2 \right\rceil$ 人。第 $1$ 到第 $20$ 排的座位数从 $10$ 到 $29$ ，对应最大人数为 $5, 6, 6, 7, 7, 8, 8, \ldots, 14, 14, 15$ 。

这些数相加为 $200$ 。

所以正确答案是 C。

Row $k$ has $9+k$ seats, so it holds $\left\lceil (9+k)/2 \right\rceil$ students. For rows $1$ through $20$ (seats $10$ through $29$ ) the maxima are $5, 6, 6, 7, 7, 8, 8, \ldots, 14, 14, 15.$

These sum to $200.$

Thus, the correct answer is C .

1991 AMC 8 第 17 题

答案：C

难度评级：1140

解答：

其他年份的第 17 题