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 Solution by John Dounis)

I now follow.

The distinct sets are (in the case e.g. of numbers a to g):

a b c d e f g, then:
g+a                    g+b                g+c         g+d       g+e     g+f 
g+f+a                 g+f+b             g+f+c      g+f+d    g+f+e  
g+f+e+a             g+f+e+b         g+f+e+c  g+f+e+d   
g+f+e+d+a         g+f+e+d+b     g+f+e+d+c    
g+f+e+d+c+a     g+f+e+d+c+b     
g+f+e+d+c+b+a

Very  nice!

Edited on December 8, 2011, 7:02 am

Posted by broll on 2011-12-08 02:13:23