BALTIC-WAY 1998 Problem 18

Determine all positive integers $n$ for which there exists a set $S$ with the following properties:

(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;

(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.