Problem:
Brian writes down four integers whose sum is 44 . The pairwise positive differences of these numbers are , and 9 . What is the sum of the possible values for ?
Answer Choices:
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B.
C.
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E.
Solution:
The largest pairwise difference is 9 , so . Let be either or . Because is between and ,
Therefore the positive differences and must sum to 9 . The given pairwise differences that sum to 9 are and . The remaining pairwise difference must be .
The second largest pairwise difference is 6 , 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