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Not all are equal (Posted on 2011-12-03) Difficulty: 3 of 5
Given n distinct positive numbers a1,a2,...,an.
We construct all the possible sums (from 1 to n terms).

Prove that among those 2^n-1 sums there are at least n(n+1)/2 different ones.

Source: a problem from Soviet Union 1963 contest

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: possible solution | Comment 2 of 8 |
(In reply to possible solution by broll)

I think you misread the problem, broll.  We are starting with n distinct integers, but they are not necessarily the ones from 1 to n.

If we start with n consecutive integers, then there are exactly n(n+1)/2 different sums.

If we start with n different powers of k (where k > 2), then there are exactly 2^n-1 different sums.  

The problem is to prove that there are at least n(n+1)/2 different sums, no matter what numbers we start with. 

  Posted by Steve Herman on 2011-12-04 09:34:30
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