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Let $n>1$ be an integer and $S \subseteq \mathbb Z$ be such that $|S|=n$ and $S \subseteq S+S:=\{a+b : a,b \in S\}$ ; then does there exist $T \subseteq S$ with $1 \le |T| \le n/2$ such that $\sum_{a \in T}=0$ ?

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    What is the source of this problem? – Seva Oct 28 '17 at 16:55
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    Could you prove it with the requirement $|T|\le n/2$ dropped? – Seva Oct 28 '17 at 17:12
  • @Seva : It was posed to me by a friend who likes combinatorial problems ... I didn't ask for a reference ... and no I couldn't prove it even without the size bound ... note that it is enough to consider $S$ with $0 \ne S$ ... I think I can show that sum of some elements of $S$ , with repetition , is $0$ ... –  Oct 28 '17 at 17:42
  • I vaguely recall that a very similar (if not very same) problem has been once discussed here, on MO - with references, computations, partial results etc. Unfortunately, I was unable to locate that discussion now. – Seva Oct 28 '17 at 18:06
  • It should be noted that it is a great question and quite possibly an open one. The six “answers” provide valuable reformulations but not much more. – Aaron Meyerowitz Oct 29 '17 at 20:43

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