Problem:
There are exactly distinct rational numbers such that and
has at least one integer solution for . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Solve the equation for to obtain . For each integer value of except , there is a corresponding rational value for . As a function of is increasing for . Thus by inspection, the integer values of that ensure satisfy the inequality . There are 78 such values. Assume that and are two different integer values of that produce the same . Then , which simplifies to . Because , it follows that , but there are no integers satisfying this equation. Thus the values of corresponding to the 78 values of are all distinct, and the answer is therefore .
The problems on this page are the property of the MAA's American Mathematics Competitions