2017 AIME II Problems/Problem 5
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 .
Let these four numbers be , , , and , where . needs to be maximized, so let and because these are the two largest pairwise sums. Now needs to be maximized. Notice that . No matter how the numbers , , , and are assigned to the values , , , and , the sum will always be . Therefore we need to maximize . The maximum value of is achieved when we let and be and because these are the two largest pairwise sums besides and . Therefore, the maximum possible value of .
Let the four numbers be , , , and , in no particular order. Adding the pairwise sums, we have , so . Since we want to maximize , we must maximize .
Of the four sums whose values we know, there must be two sums that add to . To maximize this value, we choose the highest pairwise sums, and . Therefore, .
We can substitute this value into the earlier equation to find that .
Note that if are the elements of the set, then . Thus we can assign . Then .
Solution 4 ( Short Casework )
There are two cases we can consider. Let the elements of our set be denoted , and say that the largest sums and will be consisted of and . Thus, we want to maximize , which means has to be as large as possible, and has to be as small as possible to maximize and . So, the two cases we look at are:
Note we have determined these cases by maximizing the value of determined by our previous conditions. So, the answers for each ( after some simple substitution ) will be:
See the first case has our largest , so our answer will be
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