Question 126
Given an integer $n \ge 2$, let $x_1, x_2, \dots, x_n$ be non-negative real numbers satisfying $x_1 + x_2 + \dots + x_n = n$. Find the minimum and maximum values of
$$
\sum_{k=1}^{n} \frac{1 + x_k^2 + x_k^4}{1 + x_{k+1} + x_{k+1}^2 + x_{k+1}^3 + x_{k+1}^4},
$$
where $x_{n+1} = x_1$.
$$
\sum_{k=1}^{n} \frac{1 + x_k^2 + x_k^4}{1 + x_{k+1} + x_{k+1}^2 + x_{k+1}^3 + x_{k+1}^4},
$$
where $x_{n+1} = x_1$.