专家免费评估留学案例雅思保分免费备考资料留学热线 400-880-6200

AMC10每日一题（2001年真题#10）

2018-05-25
853 人浏览
分享
收藏

2001 AMC 10 竞赛试题/第10题

Problem

Consider the dark square in an array of unit squares, part of which is shown. The ﬁrst ring of squares around this center square contains $8$ unit squares. The second ring contains $16$ unit squares. If we continue this process, the number of unit squares in the $100^\text{th}$ ring is

$\textbf{(A)}\ 396 \qquad \textbf{(B)}\ 404 \qquad \textbf{(C)}\ 800 \qquad \textbf{(D)}\ 10,\!000 \qquad \textbf{(E)}\ 10,\!404$

Solutions

Solution 1

We can partition the $n^\text{th}$ ring into $4$ rectangles: two containing $2n+1$ unit squares and two containing $2n-1$ unit squares.

There are $2(2n+1)+2(2n-1)=4n+2+4n-2=8n$ unit squares in the $n^\text{th}$ ring.

Thus, the $100^\text{th}$ ring has $8 \times 100 = \boxed{\textbf{(C)} 800}$ unit squares.

Solution 2 (Alternate Solution)

We can make the $n^\text{th}$ ring by removing a square of side length $2n-1$ from a square of side length $2n+1$ .

This ring contains $(2n+1)^2-(2n-1)^2=(4n^2+4n+1)-(4n^2-4n+1)=8n$ unit squares.

Thus, the $100^\text{th}$ ring has $8 \times 100 = \boxed{\textbf{(C)}\ 800}$ unit squares.

上一篇： AMC10每日一题（2000年真题#13）
下一篇： AMC10每日一题（2001年真题#11）

课窝考试网（http://www.ikewo.cn）声明

本站凡注明原创和署名的文章，未经课窝考试网许可，不得转载。课窝考试网的部分文章素材来自于网络，版权归原作者所有，仅供学习与研究，如果侵权，请提供版权证明，以便尽快删除。

专家答疑

确认提交

更多相关

在线咨询

扫一扫获取最新考试资讯

400-880-6200

立即咨询