Question

Hiram 的代数笔记共有 $50$ 页，印在 $25$ 张纸上；第一张纸含第 $1$ 页和第 $2$ 页，第二张纸含第 $3$ 页和第 $4$ 页，依此类推。有一天他午饭前把笔记留在桌上，室友决定从笔记中间借走一些页。Hiram 回来时发现，室友拿走的是连续若干张纸，并且所有剩余纸张上的页码平均数恰好为 $19$ 。室友借走了多少张纸？

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2,$ the second sheet contains pages $3$ and $4,$ and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19.$ How many sheets were borrowed?

$10$

$13$

$15$

$17$

$20$

答案：B

知识点：平均数丢番图方程因式分解

难度评级：1820

视频讲解：

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文字解答：

设借走的是第 $a$ 到第 $b,$ 张纸，并设 $s=b-a+1.$ 被借走的页码从 $2a-1$ 到 $2b,$ 所以共有 $2s$ 页，页码之和为 $s(2a+2b-1).$

所有页码之和为 $50\cdot51/2=1275.$ 若剩余页码的平均数为 $19,$ 则 $\begin{aligned} 1275&-s(2a+2b-1)\\ &=19(50-2s),\\ s(2a+2b-39)&=325. \end{aligned}$

因为 $b-a+1$ 是 $325$ 的正因数，且不超过 $25,$ 所以只有 $1,5,13,25.$ 四种可能。前两种会迫使 $b>25,$ 而 $25$ 会借走所有纸张。因此唯一有效的可能是

$\begin{aligned} 2a+2b-39 &=25, \\ b-a+1 &=13. \end{aligned}$

所以 $a+b=32$ ，且 $b-a=12,$ 得到 $a=10$ 、 $b=22.$ 因此借走了 $13$ 张纸。

所以正确答案是 B。

Suppose the borrowed sheets are sheets $a$ through $b,$ and let $s=b-a+1.$ The borrowed pages run from $2a-1$ through $2b,$ so there are $2s$ borrowed pages and their sum is $s(2a+2b-1).$

The total sum of all page numbers is $50\cdot51/2=1275.$ If the remaining pages have mean $19,$ then $\begin{aligned} 1275&-s(2a+2b-1)\\ &=19(50-2s),\\ s(2a+2b-39)&=325. \end{aligned}$

Because $b-a+1$ is a positive divisor of $325$ and is at most $25,$ its only possibilities are $1,5,13,25.$ The first two would force $b>25,$ and $25$ would remove every sheet. Thus the only valid possibility is

$\begin{aligned} 2a+2b-39 &=25, \\ b-a+1 &=13. \end{aligned}$

Thus $a+b=32$ and $b-a=12,$ so $a=10$ and $b=22.$ Therefore $13$ sheets were borrowed.

Thus, B is the correct answer.

2021 AMC 10A Spring 第 22 题

答案：B

难度评级：1820

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