Distinct Pairwise Sums

Algebra Level 4

4 numbers are given such that each of the 6 pairwise sums are distinct. If the 4 smallest sums are \(7, 9, 10\) and \(14\), what is the largest possible product of these 4 numbers?

Details and assumptions

The numbers are not necessarily integers.

The the numbers were \(1, 2, 3, 4\), then the 6 pairwise sums are \[ 3 = 1+2, 4 = 1+3, 5 = 1+4, 5 = 2+3, 6 = 2+4, 7= 3+4. \]

2 numbers are distinct if they are not the same.

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