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 all of the following conditions are fulfilled: a) $B \subseteq A$; b) $|B| \ge 668$; c) for any $x, y \in B$ we have $x + y \notin B$.