<s>[INST] Are there any positive integers such that $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1$? 
Are there any positive integers $x,y,z$ such that $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1?$$ Prove/Disprove.

I've plugged in random positive integers for $x,y,z$ and I have not been able to get the equation to equal $1$.
 [/INST] Rearrange the equality 
$x^2z+y^2x+z^2y=xyz$
Since $x,y,z>0$, we have
$x^2z+y^2x+z^2y>x^2z$
Multiply both sides with $\frac{y}{x}$
$y^2z>xyz$
And $y>x$
On the other hand,
$x^2z+y^2x+z^2y>y^2x$
Multiply both sides with $\frac{z}{y}$ and you get
$xz^2>xyz$, and $z>y$
Lastly, 
$x^2z+y^2x+z^2y>z^2y$ Multiply both sides with $\frac{x}{z}$ and get
$x^2y>xyz$ and $x>z$
So we got, $y>x, z>y, x>z$ which is clearly not possible when $x,y,z$ are positive.
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