Question

图形 $0, 1, 2$ 和 $3$ 分别由 $1, 5, 13$ 和 $25$ 个互不重叠的单位正方形组成。如果继续这个规律，图形 $100$ 中会有多少个互不重叠的单位正方形？

Figures $0, 1, 2,$ and $3$ consist of $1, 5, 13,$ and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure $100?$

$10401$

$19801$

$20201$

$39801$

$40801$

答案：C

知识点：找规律完全平方数

难度评级：1370

解答：

图形 $n$ 是一个菱形，各行长度按奇数增大后再减小，所以共有 $n^2 + (n + 1)^2$ 个单位正方形。这与 $n = 0, 1, 2, 3$ 时的 $1, 5, 13, 25$ 相符。

因此图形 $100$ 中单位正方形的个数为个。 $\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201 \end{aligned}$

因此，正确答案是 C。

Figure $n$ is a diamond whose row lengths increase through the odd numbers and back down, giving a total of $n^2 + (n + 1)^2$ unit squares. This matches $1, 5, 13, 25$ for $n = 0, 1, 2, 3.$

Therefore figure $100$ has $\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201 \end{aligned}$ unit squares.

Thus, the correct answer is C.

2000 AMC 12 第 8 题

答案：C

难度评级：1370

解答：

其他年份的第 8 题