<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] By the AM-GM inequality, one has
 $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} \ge 3\sqrt[3]{\frac{x}{y} \cdot\frac{y}{z} \cdot\frac{z}{x} }=3>1 $$
and hence it is impossible.
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