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June 10th, 2012, 12:06 PM   #1
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special unique sum

Does there exist a group of n distinct integers such that the sum of them
with each integer appearing only once
is not equal to
the sum of n of them with at least one of them appearing more than once?
These numbers should be polynomial with respect to n.Thanks.
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June 11th, 2012, 06:47 AM   #2
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Re: special unique sum

Clearly not for n = 0 or n = 1.

For n = 2 you're looking for integers a < b such that a + b = 2a or a + b = 2b, impossible.

For n = 3 you need a < b < c with a + b + c = 3b, the other cases being impossible. This is possible, just take x - y, x, x + y for any integer x and any nonzero integer y. The same technique works for any odd n > 2.

A modification works for even n > 2: take x - ky, x - (k-1)y, ..., x - y, x + y, ..., x + (k-1)y, x + y and replace with k each of x - y and x + y.
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