2014 AMC 8 第 24 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。
24.
某天 Beverage Barn 卖出了 罐汽水，购买者共有 位，并且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 cans of soda to 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?
小提示：
中位数是按顺序排列后第 项和第 项的平均数
The median is the average of the th and st ordered purchases.
大提示：
让前 位顾客购买数尽可能小
Make the first purchases as small as possible.
解答：
将 位顾客的购买数从小到大排列。中位数是第 项和第 项的平均数。为了最大化它，让前 项尽可能小，都等于 。
如果第 项至少为 ，那么总数至少为 ，超过二百五十二。所以第 项至多为 。
这个最大值可以达到：取 项等于 ，第 项等于 ，最后 项都等于 。总数为 ，中位数为 。
所以正确答案是 C。
Order the purchases from least to greatest. The median is the average of the th and st entries. To maximize it, make the first entries as small as possible, so set them all equal to .
If the th entry were at least , then the total would be at least , too many cans. So the th entry is at most .
This maximum is attainable: use entries equal to , the th entry equal to , and the final entries equal to . The total is , and the median is .
Thus, C is the correct answer.
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