Problem: A set $A$ of integers is called \emph{sum-full} if $A \subseteq A + A$.
A set $A$ of integers is said to be \emph{zero-sum-free} if $0$
is the only integer that cannot be expressed as the
sum of the elements of a finite nonempty subset of $A$.
Does there exist a sum-full zero-sum-free set of integers?

Solution: The answer is YES, such a set exists, and is given by
\[ A = \left\{ 1, -2, 3, -5, 8, -13, 21, -34, 55, \dots \right\}. \]

To prove every number can be expressed as a
(possibly empty) subset of $A$, we have the following stronger claim.
\begin{claim*}
  Let $N$ be an integer (possibly zero or negative).
  Let $F$ denote the smallest Fibonacci number
  which is strictly greater than $|N|$.
  Then there exists a representation of $N$ as the sum of
  a (possibly empty) subset of $A$,
  where no element used has absolute value greater than $N$.
\end{claim*}
\begin{proof}
  By induction on $|N|$.
  Omitted.
\end{proof}

On the other hand, the \emph{Zeckendorf representation theorem}
implies that it's not possible to express $0$
as a nonempty subset;
such a representation would amount to an identity of the form
\[ F_{i_1} + F_{i_2} + \dots = F_{j_1} + F_{j_2} + \dots \]
where no terms on either side are consecutive,
and both sides have no common terms; and this is impossible.