Question

某天 Beverage Barn 向 $100$ 位顾客卖出了 $252$ 罐汽水，每位顾客至少买了一罐。当天每位顾客购买罐数的中位数最大可能是多少？

One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

$2.5$

$3.0$

$3.5$

$4.0$

$4.5$

答案：C

知识点：中位数（数据）最优化

难度评级：1560

解答：

将 $100$ 位顾客的购买数从小到大排列。中位数是第 $50$ 项和第 $51$ 项的平均数。为了最大化它，让前 $49$ 项尽可能小，都等于 $1$ 。

如果第 $50$ 项至少为 $4$ ，那么总数至少为 $49+51\cdot4=253$ ，超过二百五十二。所以第 $50$ 项至多为 $3$ 。

这个最大值可以达到：取 $49$ 项等于 $1$ ，第 $50$ 项等于 $3$ ，最后 $50$ 项都等于 $4$ 。总数为 $49+3+50\cdot4=252$ ，中位数为 $(3+4)/2=3.5$ 。

所以正确答案是 C。

Order the $100$ purchases from least to greatest. The median is the average of the $50$ th and $51$ st entries. To maximize it, make the first $49$ entries as small as possible, so set them all equal to $1$ .

If the $50$ th entry were at least $4$ , then the total would be at least $49+51\cdot4=253$ , too many cans. So the $50$ th entry is at most $3$ .

This maximum is attainable: use $49$ entries equal to $1$ , the $50$ th entry equal to $3$ , and the final $50$ entries equal to $4$ . The total is $49+3+50\cdot4=252$ , and the median is $(3+4)/2=3.5$ .

Thus, C is the correct answer.

2014 AMC 8 第 24 题

答案：C

难度评级：1560

解答：

其他年份的第 24 题