2017 AIME II Problems/Problem 5
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 1
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 
. 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 
.
Solution 2
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 
.
Solution 3
Note that if are the elements of the set, then 
. Thus we can assign 
. Then 
.
See Also
| 2017 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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