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Saudi Arabian Mathematical Competitions

Algebra Difficulty 4.8 AIME Prove it Saudi Arabia

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 $\frac{1}{x}$ .

Solution

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 $|S| \leq 5$ , 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
$S=\left\{-1, \frac{1}{2}, 2\right\}$

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