TITLE: Find answer of $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$
QUESTION [18 upvotes]: If I had a very shallow question, then I am sorry.
$x,y,z\in\mathbb{N}^{+}$ and$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$$find $x,y,z$.
I try with AM-GM, just get$$ \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq\frac{3}{2}$$
This means that the equation must have a real solution, but can not be sure there is an integer solution.
Let: $x=ay=abz$, then the equation becomes:$$\frac{ab}{a+b}+\frac{b}{ab+1}+\frac{1}{a^2+ab}=4$$
Which makes the problem become non-homogeneous, and seems to become more difficult. I have no more ideas. Could anyone help me? Thanks a lot.

REPLY [13 votes]: There are indeed positive integer solutions. The smallest solution with $x, y, z$ in $\mathbb{N}_+$ is
$\small x = 154476802108746166441951315019919837485664325669565431700026634898253202035277999\\ 
\small y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579\\
\small z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036$
(Source: https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4 )
This solution was found using elliptic curves. Quite cool, eh?