\documentclass[11pt]{scrartcl}
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\begin{document}
\title{EGMO 2012/4}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 12}
\maketitle
\section*{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?
\newpage
\section*{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.
\end{document}