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