Question 4
Let $F$ be a finite nonempty set of integers and let $n$ be a positive integer. Suppose that:
- Every $x\in F$ can be written as $x=y+z$ for some $y,z\in F$.
- If $1\le k\le n$ and $x_1,\ldots,x_k\in F$, then $x_1+\cdots+x_k\ne0$.
Show that $F$ has at least $2n+2$ distinct elements.
- Every $x\in F$ can be written as $x=y+z$ for some $y,z\in F$.
- If $1\le k\le n$ and $x_1,\ldots,x_k\in F$, then $x_1+\cdots+x_k\ne0$.
Show that $F$ has at least $2n+2$ distinct elements.