AMC8 2012

AMC8 2012 · Q17

AMC8 2012 · Q17. It mainly tests Area & perimeter, Geometry misc.

A square with an integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?

一个边长为整数的正方形被切成10个边长均为整数的小正方形，其中至少8个面积为1。原正方形边长的最小可能值是多少？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (B)

正确答案：（B）

Solution

The first answer choice ${\textbf{(A)}\ 3}$, can be eliminated since there must be $10$ squares with integer side lengths. We then test the next smallest side length, which is $4$. The square with area $16$ can be partitioned into $8$ squares with area $1$ and two squares with area $4$, which satisfies all the conditions of the problem. Therefore, the smallest possible value of the length of the side of the original square is $\boxed{\textbf{(B)}\ 4}$.

第一个选项 ${\textbf{(A)}\ 3}$ 可以排除，因为必须切成10个整数边长的正方形。然后测试下一个最小边长 $4$。面积为 $16$ 的正方形可以分成8个面积为1的正方形和两个面积为4的正方形，这满足所有条件。因此，原正方形边长的最小可能值是 $\boxed{\textbf{(B)}\ 4}$。

Topics