Question

下列 $4040$ 个数的列表的中位数是多少？ $\begin{aligned} &1,2,3,\ldots,2020,\\ &1^2,2^2,3^2,\ldots,2020^2 \end{aligned}$

What is the median of the following list of $4040$ numbers? $\begin{aligned} &1,2,3,\ldots,2020,\\ &1^2,2^2,3^2,\ldots,2020^2 \end{aligned}$

$1974.5$

$1975.5$

$1976.5$

$1977.5$

$1978.5$

答案：C

知识点：中位数（数据）完全平方数区间内整数计数

难度评级：1480

视频讲解：

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文字解答：

因为 $44^2=1936$ 且 $45^2=2025$ ，所以共有 $44$ 个平方数不超过从 $1936$ 到 $2020$ 之间的任一数。

在 $1975$ 处，不超过它的项有 $1975+44=2019$ 个，其中普通整数截至 $1975$ 。在 $1976$ 处，不超过该数的项有 $2020$ 个，其中普通整数截至 $1976$ ，所以第 $2020$ 项为 $1976$ ，下一项为 $1977$ 。中位数为 $1976.5$ 。正确答案是 C。

For a number near the median, the sorted list includes all ordinary integers up to that number and all squares up to that number. Since $44^2=1936$ and $45^2=2025$ , there are $44$ squares not exceeding any number from $1936$ through $2020$ .

At $1975$ , there are $1975+44=2019$ list entries at most $1975$ . At $1976$ , there are $2020$ entries at most $1976$ , so the $2020$ th entry is $1976$ , and the next is $1977$ . The median is $1976.5$ . Thus, C is the correct answer.

2020 AMC 10A 第 11 题

答案：C

难度评级：1480

视频讲解：

文字解答：

其他年份的第 11 题