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 $1$.

Solution 1:

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.

Solution 2:

Assume, that a solution exists. Notice, that in each of the summands in $$ \frac xy + \frac yz + \frac zx = 1 $$ has to lie in the open intervall $(0,1)$, because $x,y,z$ are required to be positive. Therefore, we have $\frac xy < 1$, i. e. $x < y$. Similarly, we conclude $y < z$ and $z < x$. Hence $$ x < y < z < x, $$ a contradiction. Thus, no solution exists.

Solution 3:

Here is an alternative.

Multiply both sides by $xyz$

$$x^2z + xy^2 + yz^2 = xyz$$

Subtract the third term $yz^2$ from both sides

$$x^2z + xy^2 = (x - z)yz$$

The LHS is clearly positive so the RHS must be as well. Hence $x > z$.

Do the same for the other two terms gives: $z > y$ and $y > x$.

These cannot all be true.