2017 AIME II Problems/Problem 5
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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
. 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
.