<s>[INST] integer solution of $\frac{x^3+y^3+z^3-xy(x+y)-yz(y+z)-xz(x+z)-2xyz}{(x+y+z)(x+y-z)(x-y+z)(x-y-z)}=\frac{1}{2016}$ Let $x,y,z$ be positive integers such that $\frac{x^3+y^3+z^3-xy(x+y)-yz(y+z)-xz(x+z)-2xyz}{(x+y+z)(x+y-z)(x-y+z)(x-y-z)}=\frac{1}{2016}$.
How to find all solutions ?
I have no any idea.
Thanks in advance.
 [/INST] Hint:
Since$$\frac{x^{3}+y^{3}+z^{3}-xy(x+y)-yz(y+z)-zx(z+x)-2xyz}{(x+y+z)(x+y-z)(x-y+z)(x-y-z)}=\frac{1}{2}\left(\frac{1}{-x+y+z}+\frac{1}{x-y+z}+\frac{1}{x+y-z}+\frac{1}{x+y+z}\right),$$
 we are trying to solve $$\frac{1}{-x+y+z}+\frac{1}{x-y+z}+\frac{1}{x+y-z}+\frac{1}{x+y+z}=\frac{1}{1008}.$$
 This is equivalent to solving the equation $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c}=\frac{1}{1008}.$$
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