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Algebra Difficulty 4.8 AIME Prove it Saudi Arabia

Find all non-empty sets SS of nonzero real numbers such that

a) SS has at most 5 elements;

b) If xx is in SS, then so are 1−x1-x and 1x\frac{1}{x}.

Solution

Let x∈Sx \in S. Then
1−x,1x,11−x,1−1x,1−11−x∈S. 1-x, \frac{1}{x}, \frac{1}{1-x}, 1-\frac{1}{x}, 1-\frac{1}{1-x} \in S.
(1) Since ∣S∣≤5|S| \leq 5, it follows that at least two of these numbers are equal. Considering all 15 possible cases it follows x=−1x=-1 or x=12x=\frac{1}{2}. We get
S={−1,12,2} S=\left\{-1, \frac{1}{2}, 2\right\}

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