Hello,

Given is a positive integer $\displaystyle k$. Prove that out of every set of integers which has more than $\displaystyle 3^k$ elements one can pick out a sub-set $\displaystyle S$ with $\displaystyle k+1$ elements and the following quality:

for any two subsets $\displaystyle A,B\subseteq S$ the sum of all elements in A is different from the sum of all elements in B.

I've been trying to play with base-system representation for a while but it leads me nowhere. Your help will be appreciated.