AMC 8 2015 Problem 25
AMC8 2015 年第 25 题
面积与周长（综合）★★★★☆
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can fit into the remaining space?
从边长为 5 英寸的大正方形的四个角落裁去 4 个边长为 1 英寸的小正方形，则能够放入剩余图形中的最大正方形的面积是多少平方英寸？
- A.9
- B.
- C.15
- D.
- E.17
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 设距离 + 边界约束
取斜置内接方，四顶点落在大方四边、距大方角各 ，顶点为 (a,0),(5,a),(5-a,5),(0,5-a)，其面积 。约束：边 不能切进右下缺口 ——临界是这条边恰过缺口内角 (4,1)，代入得 ，于是 a(5-a)=5。故
巧解 · 边过缺口内角，凑配方
要方尽量大就把顶点尽量推向角，极限是每条边都擦过一个缺口的内角（如 (4,1)）。此时距离满足 a(5-a)=5（边过该点的条件）。直接套 ，不必解出 。
Solution 1
Elementary · set the distance + boundary constraint
Take the tilted inscribed square with a vertex a distance from each corner of the big square: (a,0),(5,a),(5-a,5),(0,5-a), with area . The edge must not cut into the bottom-right notch ; the critical case is when it passes exactly through the inner corner (4,1), giving , so a(5-a)=5. Then
Slick · edge through the notch corner, complete the square
To make the square as large as possible, push the vertices toward the corners; the limit is each edge grazing a notch's inner corner such as (4,1), where a(5-a)=5. Then , without solving for .
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