2008 AMC 10B Problem 23

Problem 23 of 25HarderAlgebra Geometry Number Theory

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)?

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Solution

The painted rectangle is

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

and it is half 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 factorizations

8=1\cdot 8=2\cdot 4

give

(a,b)=(5,12)

and

(6,8).

So there are

2

possibilities. Thus, the correct answer is B.

Answer choices

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