Question

一个矩形地板尺寸为 $a$ 英尺乘 $b$ 英尺，其中 $a$ 和 $b$ 是正整数且 $b \gt a$ 。一位艺术家在地板上画一个矩形，画出的矩形边与地板边平行。未涂色部分在画出的矩形周围形成宽 $1$ 英尺的边框，并占整个地板面积的一半。有多少个有序对 $(a, b)$ 满足条件？

A rectangular floor measures $a$ feet by $b$ feet, where $a$ and $b$ are positive integers with $b \gt a.$ 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 the area of the entire floor. How many possibilities are there for the ordered pair $(a, b)?$

$1$

$2$

$3$

$4$

$5$

答案：B

知识点：面积西蒙最爱的因式分解技巧因数

难度评级：1660

解答：

涂色矩形尺寸为 $(a - 2)$ 乘 $(b - 2)$ ，面积是地板面积的一半，所以 $ab = 2(a - 2)(b - 2).$

展开得 $0 = ab - 4a - 4b + 8$ ，再两边加 $8$ 得到 $(a - 4)(b - 4) = 8$ 。

在 $b \gt a \gt 0$ 下， $8$ 的有效因数对只有 $(a - 4, b - 4) = (1, 8)$ 和 $(2, 4)$ ，给出 $(a, b) = (5, 12)$ 和 $(6, 8)$ 。

共有 $2$ 种可能。

因此，正确答案是 B。

The painted rectangle measures $(a - 2)$ by $(b - 2)$ and has half the area of the floor, so $ab = 2(a - 2)(b - 2).$

Expanding gives $0 = ab - 4a - 4b + 8,$ and adding $8$ yields $(a - 4)(b - 4) = 8.$

With $b \gt a \gt 0,$ the only valid factor pairs of $8$ are $(a - 4, b - 4) = (1, 8)$ and $(2, 4),$ giving $(a, b) = (5, 12)$ and $(6, 8).$

There are $2$ possibilities.

Thus, the correct answer is B.

2008 AMC 12B 第 16 题

答案：B

难度评级：1660

解答：

其他年份的第 16 题