AMC 8 2020 Problem 25
AMC8 2020 年第 25 题
四边形与多边形★★★☆☆
Rectangles R_1 and R_2, and squares S_1, S_2, and S_3, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of S_2 in units?
如图所示，矩形 、 和正方形 、、 拼成了一个宽 3322、高 2020 的大矩形。求 的边长是多少？
- A.651
- B.655
- C.656
- D.662
- E.666
正确答案 · Correct Answer
A
解析 · Solution
解法1
常规解法 · 横竖各列一条等式
看图：大长方形的顶边由 、、 三个正方形并排拼成，所以宽
沿左侧竖直向下，先经过 ，中间退掉一个 ，再接上 ，所以高
两式相减， 与 同时消去，只剩 ：
所以 。
巧解 · 直接取宽高之差
根本不必把 、 单独解出来。宽和高都是「 加 」再对 一加一减得到的——宽多一个 ，高少一个 。所以宽与高的差，恰好是两条 的边长：
这也说明：无论 、 各自多大，答案都被唯一确定。
Solution 1
Let the side length of each square be . Then, from the diagram, we can line up the top horizontal lengths of , , and to cover the top side of the large rectangle, so . Similarly, the short side of will be , and lining this up with the left side of to cover the vertical side of the large rectangle gives . We subtract the second equation from the first to obtain , and thus .
Assuming that the problem is well-posed, it should be true in the particular case where and . Let the sum of the side lengths of and be , and let the length of square be . We then have the system
$\begin{dcases}x+y =3322 \\x-y=2020\end{dcases}
$ which we solve to determine $y=651$. Since, for each pair of rectangles, the side lengths have a sum of 3322 or 2020 and a difference of $$, the answer must be $$.继续练习