1982 AMC 12 第 28 题
先试着解答 1982 AMC 12 第 28 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 1982 AMC 12 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
28.
黑板上写有一组从 开始的连续正整数。擦去其中一个数后，其余各数的平均数（算术平均值）为 。擦去的是哪个数？
A set of consecutive positive integers beginning with is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is What number was erased?
无法确定
can not be determined
小提示：
若写出的最大数为 ，则剩余数的个数为
If the last written number is the remaining count is
大提示：
利用被擦去的数满足 ，缩小 的可能取值范围
Use the bounds on the erased value to narrow the possible values of
解答：
新的平均数 小于原平均数 ，但两者之差小于 ，因而 或 。又因为剩余各数之和 必须为整数，所以 是 的倍数，从而 。擦去的数为
因此，正确答案为 B。
The new average is below the original average but differs from it by less than forcing or Since the remaining sum must be integral, is divisible by so The erased number is
Therefore, the correct answer is B.
其他年份的第 28 题
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