Question

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 $(a, b)?$

$1$

$2$

$3$

$4$

$5$

Solution:

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 Problem 16

Solution:

Problem 16 in Other Years