Find all real solutions for the system: $x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}$ given that $xyz=1$

Asked Nov 28 '17 at 11:35

Active Nov 28 '17 at 16:50

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Find all real solutions for the system: $$x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}$$ given that $xyz=1$.

This is from a math olympiad (OBM 2009).

By inspection it is easy to spot that a possible solution is $x=y=z=1$ but are there others? It looks like it is not a difficult question but I'm not finding a nice approach.

Hints and solutions are appreciated. Sorry if this is a duplicate.

Edit 1: Pointed out by helpful comments below, another answer presents a complete solution for this problem, for all possible cases. From that mentioned answer we can conclude that when $xyz=1$ there are two possible solutions:

$$(x,y,z)=(1,1,1)~~\text{and}~~(x,y,z)=(a,\frac{-1}{a+1},\frac{-(a+1)}{a}),\forall a \in \mathbb R\setminus\{0;-1\}.$$

Answers using other techniques or arguments are welcomed.

edited Nov 28 '17 at 16:50

asked Nov 28 '17 at 11:35

bluemaster

4,187

2

See https://math.stackexchange.com/questions/2078126/x-frac1y-y-frac1z-z-frac1x-then-value-of-xyz-is and the linked one : https://math.stackexchange.com/questions/2062159/if-x-frac1y-y-frac1z-z-frac1x-then-prove-x-y-z – lab bhattacharjee Nov 28 '17 at 11:37
$x=1, y=-\frac{1}{2}, z=-2$ works – JohnColtraneisJC Nov 28 '17 at 11:43
$x=x, y = -\frac1{x+1}, z = -\frac{x+1}x$ also works. – g.kov Nov 28 '17 at 16:07

Find all real solutions for the system: $x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}$ given that $xyz=1$

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