Find all non-empty sets $S$ of nonzero real numbers such that a) $S$ has at most $5$ elements b) If $x$ is in $S$, then so are $1- x$ and $\frac{1}{x}$.
Source: 2010 Saudi Arabia BMO TST 3.1 - Balkan Math Olympiad
Tags: algebra
Find all non-empty sets $S$ of nonzero real numbers such that a) $S$ has at most $5$ elements b) If $x$ is in $S$, then so are $1- x$ and $\frac{1}{x}$.