2002 AMC 12A 第 15 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。
15.
一组八个整数的平均数、中位数、唯一众数和极差都等于 。这组数中可能出现的最大整数是多少？
The mean, median, unique mode, and range of a collection of eight integers are all equal to The largest integer that can be an element of this collection is
小提示：
八个整数的和为 ，且众数 必须重复出现。
The eight integers sum to and the mode must repeat
大提示：
测试最大值 ；此时最小值应为 ，再检查剩余和。
Test the largest value the smallest would then be and check the remaining sum
解答：
集合 的平均数、中位数、唯一众数和极差都等于 ，所以 可以达到。
假设最大值为 。极差 迫使最小值为 。因为 是众数，它在排好序的八个数中重复出现；再结合中位数 ，中间两个数只能是 。于是 ，剩下四个数的和为 ，平均为 。这样至少有一个数会小于 ，与最小值矛盾。因此 不可能。
因此，正确答案是 D。
The collection has mean, median, unique mode, and range all equal to so is attainable.
Suppose the largest were The range forces the smallest to be Because is the mode, it occurs in the sorted list; together with median this forces the two middle values to be Then so the remaining four values sum to averaging At least one would be below contradicting the minimum. So is impossible.
Thus, the correct answer is D.
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