SAMC

Let $x\in S$. Then $$1-x,\frac{1}{x},\frac{1}{1-x},1-\frac{1}{x},1-\frac{1}{1-x}\in S.$$ (1) Since $|S|\le 5$, it follows that at least two of these numbers are equal. Considering all 15 possible cases it follows $x=-1$ or $x=\frac{1}{2}$. We get $$S=\left\{-1,\frac{1}{2},2\right\}$$