Question 25
Let $F$ be a finite nonempty set of integers and let $n$ be a positive integer. Suppose that
* Any $x \in F$ may be written as $x = y + z$ for some $y, z \in F$;
* If $1 \le k \le n$ and $x_1, \dots, x_k \in F$, then $x_1 + \dots + x_k \ne 0$.
Show that $F$ has at least $2n + 2$ distinct elements.
* Any $x \in F$ may be written as $x = y + z$ for some $y, z \in F$;
* If $1 \le k \le n$ and $x_1, \dots, x_k \in F$, then $x_1 + \dots + x_k \ne 0$.
Show that $F$ has at least $2n + 2$ distinct elements.