I've advanced a little with this question but I'm not sure that I'm in the right direction.
For any set $X$ with $n$ positive numbers, $n>5$, prove the existiance of subset $Y \subset X$ so that we can sum the elements of $Y$ with different signs (i.e. for $y \in Y$ we can add to our sum $y$ or $-y$) and the sum of the elements from Y will be multiple of $n^2$ ($n$ is the size of the original set - $X$).
I've noticed that the number of 'sums' that we can create is $\displaystyle\sum_{k=1}^n{n \choose k}2^k$.
Using the binomial formula I've get that $\displaystyle\sum_{k=1}^n{n \choose k}2^k = 3^n$, but I didn't advanced further.