US-TST 2001 Problem 3

For a set $S$ , let $|S|$ denote the number of elements in $S$ . Let $A$ be a set of positive integers with $|A| = 2001$ . Prove that there exists a set $B$ such that

(i) $B \subseteq A$ ; (ii) $|B| \ge 668$ ; (iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$ .