June 10th, 2012, 12:06 PM  #1 
Newbie Joined: Jul 2010 Posts: 3 Thanks: 0  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. 
June 11th, 2012, 06:47 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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  (k1)y, ..., x  y, x + y, ..., x + (k1)y, x + y and replace with k each of x  y and x + y. 

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