Problem:
Brian writes down four integers whose sum is . The pairwise positive differences of these numbers are , and . What is the sum of the possible values for ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The largest pairwise difference is , so . Let be either or . Because is between and ,
Therefore the positive differences and must sum to . The given pairwise differences that sum to are and . The remaining pairwise difference must be .
The second largest pairwise difference is , so either or . In the first case the set of four numbers may be expressed as . Hence , so . In the second case , and the four numbers may be expressed as . Therefore , so . The sum of the possible values for is .
Note: The possible sets of four numbers are and .
The problems on this page are the property of the MAA's American Mathematics Competitions