Question 9
Let $(a, b, c)$ be a real solution of the system of equations $$
\begin{aligned} x^3 - xyz &= 2, \\ y^3 - xyz &= 6, \\ z^3 - xyz &= 20. \end{aligned}
$$ The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\begin{aligned} x^3 - xyz &= 2, \\ y^3 - xyz &= 6, \\ z^3 - xyz &= 20. \end{aligned}
$$ The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$