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Let $x,$ $y,$ and $z$ be nonzero real numbers, such that no two are equal, and

\[x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}.\]

Find all possible numeric values of $xyz.$

 Feb 26, 2021
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Try letting x = a and y = 1/b.  Then you should get the result that xyz can be -2, -1, 1 or 2.

 Feb 26, 2021

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