Question

向一个整数列表中加入 $15$ 后，平均数增加了 $2$ 。再向这个扩大后的列表中加入 $1$ 后，扩大列表的平均数减少了 $1$ 。原列表中有多少个整数？

When $15$ is appended to a list of integers, the mean is increased by $2.$ When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1.$ How many integers were in the original list?

$4$

$5$

$6$

$7$

$8$

答案：A

知识点：平均数方程组

难度评级：1690

解答：

设原列表有 $n$ 个整数，平均数为 $m$ ，则总和为 $mn$ 。加入 $15$ 后， $\begin{aligned} &(m + 2)(n + 1) \\ &= mn + 15 \\ &\implies m + 2n = 13. \end{aligned}$

再加入 $1$ 后， $\begin{aligned} &(m + 1)(n + 2) \\ &= mn + 16 \\ &\implies 2m + n = 14. \end{aligned}$

解方程组 $m + 2n = 13$ 、 $2m + n = 14$ ，得 $m = 5$ 、 $n = 4$ ，所以原列表有四个整数。

所以正确答案是 A。

Let the original list have $n$ integers with mean $m,$ so its sum is $mn.$ Appending $15$ gives $\begin{aligned} &(m + 2)(n + 1) \\ &= mn + 15 \\ &\implies m + 2n = 13. \end{aligned}$

Appending $1$ to that enlarged list gives $\begin{aligned} &(m + 1)(n + 2) \\ &= mn + 16 \\ &\implies 2m + n = 14. \end{aligned}$

Solving $m + 2n = 13$ and $2m + n = 14$ yields $m = 5$ and $n = 4.$

Thus, the correct answer is A.

2002 AMC 10B 第 25 题

答案：A

难度评级：1690

解答：

其他年份的第 25 题