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 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.

2008 AMC 10B Problem 23

Solution:

Problem 23 in Other Years