Question

将整数 $1$ 到 $25$ 任意分成五组，每组 $5$ 个数。找出每组的中位数，再在这五个中位数中取中位数，记为 $M$ 。 $M$ 的最小可能值是多少？

The integers from $1$ through $25$ are arbitrarily separated into five groups of $5$ numbers each. The median of each group is identified. Let $M$ equal the median of the five medians. What is the least possible value of $M?$

$9$

$10$

$12$

$13$

$14$

答案：A

知识点：中位数（数据）抽屉原理极端原理

难度评级：1710

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文字解答：

如果某组的中位数至多为 $8$ ，那么这一组至少有 $3$ 个数不超过 $8$ 。若五个中位数的中位数至多为 $8$ ，则至少有 $3$ 组满足这一点，需要至少 $9$ 个不超过 $8$ 的数，矛盾。因此 $M\ge9$ 。这个下界可以达到，例如让五组的中位数为 $5,8,9,12,17$ ： $\{3,4,5,20,21\}$ 、 $\{6,7,8,22,23\}$ 、 $\{1,2,9,24,25\}$ 、 $\{10,11,12,13,14\}$ 和 $\{15,16,17,18,19\}$ 。所以最小值是 $9$ 。

If a group has median at most $8$ , then at least $3$ numbers in that group are at most $8$ . For the median of the five medians to be at most $8$ , at least $3$ groups would need such medians, requiring at least $9$ numbers at most $8$ , impossible. Thus $M\ge9$ . This is attainable, for example with group medians $5,8,9,12,17$ : use groups $\{3,4,5,20,21\}$ , $\{6,7,8,22,23\}$ , $\{1,2,9,24,25\}$ , $\{10,11,12,13,14\}$ , and $\{15,16,17,18,19\}$ . Hence the least possible value is $9$ .

2026 AMC 8 第 22 题

答案：A

难度评级：1710

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其他年份的第 22 题