VMO 2006 Problem 6

Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ naughty if for any two arbitrary numbers $u$ , $v$ (not neccesary distinct) in $T$ , $u+v$ is not in $T$ . Prove that

1. If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements;

2. If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.