If $\displaystyle a,b,c \in \mathbb{Z}$ and $\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\displaystyle \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are also integers then prove that $\displaystyle |a|=|b|=|c|$.

This is one of those things that seems like it should be fairly obvious, yet I've not found a nice way of doing it after a few days. My best attempt has been to say the following, which I will only sketch as it is very messy.

Let $\displaystyle \alpha=\frac{a}{b},\beta=\frac{b}{c},\gamma=\frac{c}{a}=\frac{1}{\alpha\beta}$. Let $\displaystyle \alpha+\beta+\gamma=N\_1 \in \mathbb{Z}, \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=N\_2 \in \mathbb{Z}$.

Treating $N\_1,N\_2, \alpha$ as given we can write a quadratic for $\displaystyle \gamma=\frac{c}{a}$. $\gamma$ must be rational and hence we get the restriction (after algebra) that $(N\_1-\alpha)^2(N\_2-\alpha)^2-4(N\_1-\alpha)$ must be a perfect square, as well as $5$ more corresponding to formulating different quadratics. You can eventually get the required answer $|\alpha|=|\beta|=|\gamma|=1$ to drop out.

Any better ideas?