Problem:
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are , and . Find the greatest possible value of .
Solution:
Let the elements of the set be , and . Because , and have the same value, it must be possible to group the six values , and into pairs such that each of the three pairs has the same sum. There is no way to group , and into such pairs, so and are not paired with each other, but each is instead paired with one of the known values, and the two remaining known values are paired with each other. If the sum of each pair is denoted by , the total of all six values is . The total of all six values can also be seen to be . Thus , and will be maximized by maximizing . Because must be equal to the sum of two out of the four values and , the value of cannot exceed , and the maximum value of is . Actual values for can be found by either requiring , and , which results in , or by requiring 287 , and , which results in .
The problems on this page are the property of the MAA's American Mathematics Competitions