2008 AMC 12B 第 16 题

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

小提示：

涂色矩形为

(a - 2) \times (b - 2)

，面积等于地板的一半，所以

ab = 2(a - 2)(b - 2)

。

The painted rectangle is

(a - 2) \times (b - 2)

and equals half the floor, so

ab = 2(a - 2)(b - 2)

大提示：

整理为

(a - 4)(b - 4) = 8

，再在

b \gt a \gt 0

下数因数对。

Rearrange to

(a - 4)(b - 4) = 8

and count factor pairs with

b \gt a \gt 0

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)\text{。}

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)

。

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

小提示：

大提示：

解答：

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