ISL 2004 A3

Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ 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.

Proposed by Dan Brown, Canada