SOLUTION: Let x, y, and z be nonzero real numbers, such that no two are equal, and x + {1}/{y} = y + {1}/{z} = z + {1}/{x}. Find all possible numeric values of xyz.

Question 1175883: Let x, y, and z be nonzero real numbers, such that no two are equal, and
x + {1}/{y} = y + {1}/{z} = z + {1}/{x}. Find all possible numeric values of xyz.
Found 2 solutions by CubeyThePenguin, ikleyn:
Answer by CubeyThePenguin(3113) (Show Source): You can put this solution on YOUR website!
x + 1/y = y + 1/z = z + 1/x

We have a system of three equations:

x + 1/y = y + 1/z
x + 1/y = z + 1/x
y + 1/z = z + 1/x

Solving, we get (x, y, z) = (-1, 1/2, 2) or (1, -1/2, -2). The product xyz can be or .
Answer by ikleyn(52792) (Show Source): You can put this solution on YOUR website!
.

For full analysis see this link

https://math.stackexchange.com/questions/2078126/x-frac1y-y-frac1z-z-frac1x-then-value-of-xyz-is

where you will find the detailed solution.

For given conditions, the answer is that all possible values of xyz are 1 and -1.

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