2004 IMO Shortlist S06

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

是否存在函数 $s: \mathbb{Q} \rightarrow \{-1,1\}$ 使得如果 $x$ 和 $y$ 是满足 ${xy=1}$ 或 ${x+y\in \{0,1\}}$ 的不同有理数，则 ${s(x)s(y)=-1}$ ？证明你的答案合理。

*由加拿大 Dan Brown 提出*