IMO 2004 — Complete Shortlist Question 3

Does there exist a function $ s\colon \mathbb{Q} \to \{-1,1\} $ such that if $ x $ and $ y $ are distinct rational numbers satisfying $ xy = 1 $ or $ x + y \in \{0,1\} $, then $ s(x)s(y) = -1 $? Justify your answer.