AMC8 2026

AMC8 2026 · Q22

AMC8 2026 · Q22. It mainly tests Casework.

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 $M$?

将从 1 到 25 的整数任意分成五组，每组 5 个数。找出每组的中位数。设 $M$ 为这五个中位数的中位数。问 $M$ 的最小可能值是多少？

(A) 9 9

(B) 10 10

(D) 13 13

(E) 14 14

Answer

Correct choice: (A)

正确答案：（A）

Solution

Consider using a five by five square table to solve this problem. The rows represent the five groups, the center column is the median of each group and the center square cell is the least possible value of $M$. The only numbers that really matter is the ones in the shaded area, the rest can be ignored. In order to find the least possible value of $M$ the smallest numbers, $1\ldots 9$, need to be in these nine square cells. $9$ is the smallest number that is able to satisfy both median of a group and median of five medians. Final answer, the least possible value of $M$ is (A) $9$.

考虑用一个 $5\times 5$ 的方格表来解决这个问题。行表示五个组，中间一列是每组的中位数，而中心那个格子就是 $M$ 的最小可能值。真正重要的只有阴影区域里的数字，其余部分可以忽略。为了使 $M$ 尽可能小，最小的数字 $1\ldots 9$ 必须放在这九个格子里。$9$ 是能同时满足“某组的中位数”和“五个中位数的中位数”这两个条件的最小数字。最终答案：$M$ 的最小可能值是 (A) $9$。~TutorJack

Topics

CP.010 Casework