2001 AMC 8 第 21 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。
21.
一组五个互不相同的正整数的平均数是 ，中位数是 。这五个整数中最大数的最大可能值是
The mean of a set of five different positive integers is The median is The maximum possible value of the largest of these five integers is
小提示：
五个数的和是
The five numbers sum to
大提示：
让另外四个数尽可能小
Make the other four numbers as small as possible
解答：
中位数是第三大的数，即 。有两个数小于 ，两个数大于它。
平均数为 ，所以五个数的和为 。要在这个总和下使最大数尽量大，其余数必须尽量小。
小于 的两个数必须是不同的正整数，所以最小为 和 。
紧接在 后的数也应尽量小，所以是 。
因此剩下的数，也就是最大可能值，为
所以正确答案是 D。
The median of the set of numbers is the third largest number, which is There are two numbers less than and two numbers greater than it.
The mean of the set is so the sum of all the numbers is In order to maximize the largest number with this sum, the other numbers must be as small as possible.
The two numbers less than must be positive and distinct, so they must be and
The number immediately after must also be as small as possible, so it must be
Therefore, the remaining number, the maximum possible value in the set, is
Thus, D is the correct answer.
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