Question 11
Find the greatest integer $n$ such that the cubic polynomial $$
x^3 - \frac{n}{6}x^2 + (n - 11)x - 400
$$ has roots $\alpha^2,$ $\beta^2,$ and $\gamma^2,$ where $\alpha,$ $\beta,$ and $\gamma$ are complex numbers, and there are exactly seven different possible values for $\alpha + \beta + \gamma.$
x^3 - \frac{n}{6}x^2 + (n - 11)x - 400
$$ has roots $\alpha^2,$ $\beta^2,$ and $\gamma^2,$ where $\alpha,$ $\beta,$ and $\gamma$ are complex numbers, and there are exactly seven different possible values for $\alpha + \beta + \gamma.$