AMC12 2008 B

AMC12 2008 B · Q16

AMC12 2008 B · Q16. It mainly tests Linear equations, Area & perimeter.

A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?

一个矩形地板的尺寸为 $a$ 英尺乘 $b$ 英尺，其中 $a$ 和 $b$ 为正整数且 $b > a$。一位艺术家在地板上涂画了一个矩形，且该矩形的边与地板的边平行。未涂画的部分在涂画矩形周围形成宽度为 $1$ 英尺的边框，并且其面积占整个地板面积的一半。有多少种有序对 $(a,b)$ 的可能？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (B)

正确答案：（B）

Solution

$A_{outer}=ab$ $A_{inner}=(a-2)(b-2)$ $A_{outer}=2A_{inner}$ $ab=2(a-2)(b-2)=2ab-4a-4b+8$ $0=ab-4a-4b+8$ By Simon's Favorite Factoring Trick: $8=ab-4a-4b+16=(a-4)(b-4)$ Since $8=1\times8$ and $8=2\times4$ are the only positive factorings of $8$. $(a,b)=(5,12)$ or $(a,b)=(6,8)$ yielding $\Rightarrow\textbf{(B)}$ $2$ solutions. Notice that because $b>a$, the reversed pairs are invalid.

$A_{outer}=ab$ $A_{inner}=(a-2)(b-2)$ $A_{outer}=2A_{inner}$ $ab=2(a-2)(b-2)=2ab-4a-4b+8$ $0=ab-4a-4b+8$ 用 Simon's Favorite Factoring Trick： $8=ab-4a-4b+16=(a-4)(b-4)$ 由于 $8=1\times8$ 和 $8=2\times4$ 是 $8$ 的仅有的正因数分解。 $(a,b)=(5,12)$ 或 $(a,b)=(6,8)$，因此 $\Rightarrow\textbf{(B)}$ 有 $2$ 个解。注意因为 $b>a$，交换后的有序对不合法。

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