<s>[INST] Evaluate an expression with assumption Let $a,b,c$ be non-zero real numbers and satisfy
\begin{equation}
\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-\frac{a^3+b^3+c^3}{abc}=2.
\end{equation}
Evaluate
\begin{equation}
M=[(a+b)^{2018}-c^{2018}][(b+c)^{2018}-a^{2018}][(a+c)^{2018}-b^{2018}].
\end{equation}
My impression is to divide $M$ by $abc$, but I'm stuck.  I have no idea what to do.  I tried Mathematica, but it didn't give me an answer.  Could someone please help me?
Thank you.
EDIT:
I think I got it. The assumption is equivalent to
\begin{equation}
(a+b-c)(b+c-a)(a+c-b)=0.
\end{equation}
From here, it is easy to manipulate $M$.
 [/INST] For a way to factor the original condition without using hints from the second part of the question, eliminate denominators and collect the powers of $\,a\,$ to get:
$$
\begin{align}
0 &= 2abc + (a^3+b^3+c^3) - ab(a+b) - bc(b+c) - ca(c+a) \\
 &= a^3 - a^2 (b + c) - a (b^2 - 2 b c + c^2) + b^3 - b c (b + c) + c^3 \\
 &= \underbrace{a^3 - a^2(b+c)}_{\large{a^2\left(a-\left(b+c\right)\right)}} \;\underbrace{- a(b-c)^2+(b+c)(b-c)^2}_{\large{-(b-c)^2\left(a-\left(b+c\right)\right)}} = \ldots
\end{align}
$$
</s>