Let $(a,b,c)$ be the [real](https://artofproblemsolving.com/wiki/index.php/Real_number) solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. 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](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$.
Solution 1
Add the three equations to get $a^3 + b^3 + c^3 = 28 + 3abc$.  Now, let $abc = p$.  $a = \sqrt [3]{p + 2}$, $b = \sqrt [3]{p + 6}$ and $c = \sqrt [3]{p + 20}$, so $p = abc = (\sqrt [3]{p + 2})(\sqrt [3]{p + 6})(\sqrt [3]{p + 20})$.  Now [cube](https://artofproblemsolving.com/wiki/index.php/Cube) both sides; the $p^3$ terms cancel out.  Solve the remaining [quadratic](https://artofproblemsolving.com/wiki/index.php/Quadratic) to get $p = - 4, - \frac {15}{7}$.  To maximize $a^3 + b^3 + c^3$ choose $p = - \frac {15}{7}$ and so the sum is $28 - \frac {45}{7} = \frac {196 - 45}{7}$ giving $151 + 7 = \fbox{158}$.

Solution 2
This is almost the same as solution 1. Note $a^3 + b^3 + c^3 = 28 + 3abc$. Next, let $k = a^3$. Note that $b = \sqrt [3]{k + 4}$ and  $c = \sqrt [3]{k + 18}$, so we have $28 + 3\sqrt [3]{k(k+4)(k+18)} = 28+3abc=a^3+b^3+c^3=3k+22$. Move 28 over, divide both sides by 3, then cube to get $k^3-6k^2+12k-8 = k^3+22k^2+18k$. The $k^3$ terms cancel out, so solve the quadratic to get $k = -2, -\frac{1}{7}$. We maximize $abc$ by choosing $k = -\frac{1}{7}$, which gives us $a^3+b^3+c^3 = 3k + 22 = \frac{151}{7}$. Thus, our answer is $151+7=\boxed{158}$.

Solution 3
We have that $x^3 = 2 + xyz$, $y^3 = 6 + xyz$, and $z^3 = 20 + xyz$. Multiplying the three equations, and letting $m = xyz$, we have that $m^3 = (2+m)(6+m)(20+m)$, and reducing, that $7m^2 + 43m + 60 = 0$, which has solutions $m = -\frac{15}{7}, -4$. Adding the three equations and testing both solutions, we find the answer of $\frac{151}{7}$, so the desired quantity is $151 + 7 = \fbox{158}$.