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

(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.