How many positive integers $p$, $q$, $r$ exists?

Question

How many positive integers $p$, $q$, $r$ exists satisfying the relation below?

$$p/q + q/r + r/p = 2$$

Thank in advance. I tried so much but could not think how to proceed.

I just need a hint from where to begin.

Answer 1

HINT : Note that from AM-GM inequality, we have that $a+b+c \geq 3 \sqrt[3]{abc}$. If you wish to view the answer, hover your cursor over the grey region directly below.

Answer: Since $p$, $q$ and $r$ are positive, so are $\frac{p}{q}$, $\frac{q}{r}$, $\frac{r}{p}$. From AM-GM inequality, we have that $$\displaystyle \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \geq 3 \sqrt[3]{\frac{p}{q} \times \frac{q}{r} \times \frac{r}{p}} = 3.$$ Hence, there doesn't exists any $p$, $q$ and $r$ satisfying $\displaystyle \frac{p}{q} + \frac{q}{r} + \frac{r}{p} = 2$.