Problem:
Call a fraction ba​, not necessarily in the simplest form special if a and b are positive integers whose sum is 15. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
Answer Choices:
A. 9
B. 10
C. 11
D. 12
E. 13
Solution:
The special fractions are
141​,132​,123​,114​,105​,96​,87​,78​,69​,510​,411​,312​,213​,114​​
141​,132​,41​,114​,21​,32​,87​,171​,121​,2,243​,4,621​,14​
Note that two unlike fractions in the simplest form cannot sum to an integer. So, we only consider the fractions whose denominators appear more than once:
41​,21​,121​,2,243​,4,621​,14​
For the set {2,4,14}, two elements (not necessarily different) can sum to 4,6,8,16,18,28.
For the set {21​,121​,621​}, two elements (not necessarily different) can sum to 1,2,3,7,8,13.
For the set {41​,243​}, two elements (not necessarily different) can sum to 3
Together, there are (C) 11​ distinct integers that can be written as the sum of two, not necessarily different,
special fractions:
1,2,3,4,6,7,8,13,16,18,28.
The problems on this page are the property of the MAA's American Mathematics Competitions