Problem:
A rectangular floor measures feet by feet, where and are positive integers with . 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 foot around the painted rectangle and occupies half the area of the entire floor. How many possibilities are there for the ordered pair ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because the area of the border is half the area of the floor, the same is true of the painted rectangle. The painted rectangle measures by feet. Hence , from which . Add to each side of the equation to produce
Because the only integer factorizations of are
and because , the only possible ordered pairs satisfying this equation for are and . Hence must be one of the two ordered pairs , or .
The problems on this page are the property of the MAA's American Mathematics Competitions