Problem:
Let be the set of all positive rational numbers such that when the two numbers and are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of can be expressed in the form , where and are relatively prime positive integers. Find .
Solution:
Answer (719):
Suppose that the rational number satisfies the condition in the problem statement and write , where and are relatively prime positive integers. Let . Then , so is either , or 55 . Observe that
and the latter representation is in lowest terms. Hence
which simplifies to
The values and do not yield positive rational numbers , but and yield and , respectively. Both of these fractions satisfy the condition in the problem statement. Their sum is . The requested sum is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions