Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:


Given a finite (nonempty) set of real numbers, $S=\{a\_1,a\_2,\dots, a\_n\}$, with the property that for each $i$ there exist $j,k$ (not necessarily distinct) so that $a\_i=a\_j+a\_k$ (i.e. every element in $S$ can be written as a sum of two elements in $S$, note that this condition is trivially satisfied if $0\in S$ as then every $x\in S$ can be written as $x+0$).


Must there exist $\{i\_1,i\_2,\dots, i\_m\}$ (distinct) so that $a\_{i\_1}+a\_{i\_2}+\cdots +a\_{i\_m}=0$?


ETA: A possible reformulation can be made in terms of graphs. We can take the vertex set $\{1,\dots ,n\}$ and for each equation $a\_i=a\_j+a\_k$ in S add an edge $[ij]$ and its "dual" $[ik]$. The idea is to find a cycle in this graph, whose dual is a matching.



Existence of a zero-sum subset
The answer is in the affirmative; indeed,



> 
> If $S$ is a finite non-empty subset of *any* abelian group such that every element of $S$ is a sum of two other (possibly, equal to each other) elements, then $S$ has a non-empty, zero-sum subset.
> 
> 
> 


For a complete proof, see this [recent preprint](http://arxiv.org/abs/2101.02586) by János Nagy, Péter Pach, and myself. The proof is a little too long to be presented here but at least, to indicate the general direction, here is our main lemma.


**Lemma.**
Suppose that $M$ is an integer square matrix of order $n$, representable as $A-I$ where $A$ has all its elements non-negative and all row sums equal to $2$, and $I$ is the identity matrix. Then there exist nonzero vectors $u,v\in\{0,1\}^n$ such that $M^Tu=v$; that is, there exists a system of rows of $M$ such that their sum is a nonzero, zero-one vector.


In fact, for our purposes it would suffice to have *any* nonzero vector $u$ satisfying $M^Tu\in\{0,1\}^n$; having $u$ itself to be zero-one is an extra (compare with Bill Thurston's answer above).


The proof of the main lemma is completely elementary, by induction on $n$.


Historically, the question seems to originate from a problem contributed by Dan Schwarz to the EGMO 2012 (European Girls’ Mathematical Olympiad):



> 
> Does there exist a set of integers such that every element of the set
> is a sum of two other elements, while the set does not contain a
> finite nonempty zero-sum subset?
> 
> 
> 


The answer to this question is positive since the set here is allowed to be infinite.


