2012 AMC 8 第 17 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
17.
一个边长为整数的正方形被切成 个正方形，所有小正方形边长都是整数，并且至少 个小正方形的面积为 。原正方形边长的最小可能值是多少？
A square with an integer side length is cut into squares, all of which have integer side length and at least of which have area What is the smallest possible value of the length of the side of the original square?
小提示：
边长为 时，总面积太小，无法切成十个整数边长正方形。
A side length of gives area too small for ten integer squares.
大提示：
一个 乘 正方形可以按图示方式切分。
A by square can be cut as shown.
视频讲解：
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文字解答：
因为这 个小正方形的边长都是正整数，所以每个小正方形的面积至少为 ，总面积至少为 。边长不超过 的整数边长正方形面积至多为 ，所以大正方形的边长不能小于 。
下图把一个 正方形切成十个整数边长的正方形，其中八个是单位正方形。因此边长 可以达到，而且是最小值。
所以正确答案是 B。
Since all squares have positive integer side lengths, each has area at least . Their total area is therefore at least . A square with integer side length at most has area at most , so its side length cannot be less than .
The following configuration cuts a square into ten integer-sided squares, eight of which are unit squares. Therefore side length is attainable and is the minimum.
Thus, the answer is B .
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