I had seen this problem a long time back and wasn't able to solve it. For some reason I was reminded of it and thought it might be interesting to the visitors here.
Apparently, this problem is from a mathematics magazine of some university in the United States (sorry, no idea about either).
So the problem is:
Suppose $S \subset \mathbb{Z}$ (set of integers) such that
1) $|S| = 15$
2) $\forall ~s \in S, \exists ~a,b \in S$ such that $s = a+b$
Show that for every such $S$, there is a non-empty subset $T$ of $S$ such that the sum of elements of $T$ is zero and $|T| \leq 7$.
Update (Sep 13)
Here is an approach which seems promising and others might be able to take it ahead perhaps.
If you look at the set as a vector $s$, then there is a matrix $A$ with the main diagonal being all $1$, each row containing exactly one $1$ and one $-1$ (or a single $2$) in the non-diagonal position such that $As = 0$.
The problem becomes equivalent to proving that for any such matrix $A$ the row space of $A$ contains a vector with all zeroes except for a $1$ and $-1$ or a vector with all zeroes except $\leq 7$ ones.
This implies that the numbers in the set $S$ themselves don't matter and we can perhaps replace them with elements from a different field (like say reals, or complex numbers).