Hello everyone.

I have been working on a math problem for some time now, but I can not finish the proof. In other words, I'm stuck. I hope someone here can solve it for me or perhaps we could make it into a joint effort. The problem is taken from a monthly math magazine, which features 3 math problems each month.

I am not sure whether this goes under number theory or set theory. Admins, feel free to move it around if you think it's necessary.

Here's the problem:

--------------------------------------------------------------------------------

Consider a set $\displaystyle S\subset\mathbb{Z}$ with $\displaystyle |S| = 15$ and the following property:

$\displaystyle \forall s\in S$, $\displaystyle \exists a,b\in S$ such that $\displaystyle s = a + b$.

Proof that for every such $\displaystyle S$, there exists a non-empty subset $\displaystyle T\subset S$ with $\displaystyle |T|\leq 7$ of which the elements sum up to zero.

--------------------------------------------------------------------------------

I have come up with a few observations that might be useful, but I think it's best to wait and see how things go. If noone has an idea how to approach the problem or if someone asks me for the info, then I will give it.

Until then, good luck!