SMO 2013 Problem 4

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$ , $z \in F$ ; $\bullet$ If $1 \leq k \leq n$ and $x_1$ , ..., $x_k \in F$ , then $x_1+\cdots+x_k \neq 0$ .

Show that $F$ has at least $2n+2$ elements.