The question is worded like this: suppose that we have non-zero real numbers $x, y, z$ and integers $a, b, c$ such that $$a = \frac{xy}{x + y}, b = \frac{xz}{x + z}, c = \frac{yz}{y + z}$$
Then the problem is to show that $x, y, z$ must necessarily be rational.
I was wondering if there was a slick way to do this problem (say, via a clever application of the RRT).
Also, I was wondering: if you try to do this naively (say, by substituting variables and solving for $x$ in terms of $a, b, c$) you get an expression of the form $$x = \frac{\text{stuff}}{ac + ab - bc},$$ or an expression which looks similar to this. No one I have talked to so far has been able to successfully answer why the denominator expression must be a non-zero number (so that the value we get for $x$ is well-defined).
If $x$ and $y$ are non-zero then $a\neq 0$ and $$\frac{1}{x} + \frac{1}{y} = \frac{1}{a}.$$ Write the other two relations and solve for $\frac{1}{x}$, $\frac{1}{y}$, and $\frac{1}{z}$.