The following problem has been bothering me for a while, and I finally gave up to solve it on my own. However, I still would like to see a solution:
For an arbitrary integer $n$ consider a set of all n-vectors with coordinates $1$ and $-1$, e.g. for $n=2$:
$(1,1),\ (1,-1),\ (-1,1),\ (-1,-1)$
The sum of the vectors is obviously $0$ (zero-vector). Now, arbitrarily change some of the coordinates to
0 (it may be different coordinates in different vectors), e.g.:
$(0,1),\ (1,0),\ (-1,1),\ (-1,-1)$
Prove that there exists a non-empty subset of vectors whose sum is still $0$.
(In our case: $(0,1) + (1,0) + (-1,-1)=(0,0)$.