TITLE: Estimating the size of solutions of a diophantine equation
QUESTION [155 upvotes]: A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers). 
B. Suppose that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an even natural number ($a,b,c $ are still natural numbers) then is there any way to estimate the minimum of $a,b $ and $c$ ?
The smallest solution that I know for  $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4 $ is:

REPLY [73 votes]: This exact problem is the subject of the paper "An Unusual Cubic Representation Problem" by Andrew Bremner (ASU) and myself. It was published in Volume 43 (2014) of Annales Mathematicae et Informaticae, pages 29-41.
It is proven that strictly positive solutions never exist for $n$ odd. They sometimes do not exist for $n$ even, and, even if they do, they can be of truly enormous size - much larger than the example given.