Question 6
Let $S$ be a set with $2006$ elements. A subset $T$ of $S$ is called “stubborn” if for any $u,v\in T$, allowing $u=v$, one has $u+v\notin T$. Prove that:
(1) If $S=\{1,2,\ldots,2006\}$, every stubborn subset has at most $1003$ elements.
(2) If $S$ consists of any $2006$ positive integers, it has a stubborn subset with $669$ elements.
(1) If $S=\{1,2,\ldots,2006\}$, every stubborn subset has at most $1003$ elements.
(2) If $S$ consists of any $2006$ positive integers, it has a stubborn subset with $669$ elements.