Question 132
For an integer $n \ge 4$, if a sequence of real numbers $x_1, x_2, \dots, x_n$ satisfies the following three equalities:
$$
\begin{aligned}
x_1 + x_2 + \dots + x_n &= n, \\
x_1^2 + x_2^2 + \dots + x_n^2 &= 2n, \\
x_1^3 + x_2^3 + \dots + x_n^3 &= 3n,
\end{aligned}
$$
then we call $(x_1, x_2, \dots, x_n)$ a *regular* $n$-tuple. We call the difference between the maximal and the minimal elements in $x_1, x_2, \dots, x_n$ the *width* of the regular $n$-tuple.
(1) Find the maximal real number $C$ such that, for any integer $n \ge 4$, the width of every regular $n$-tuple is at least $C$.
(2) For the maximum $C$ in (1), prove that there exists a real number $\lambda \gt 0$ such that, for any integer $n \ge 4$, the width of every regular $n$-tuple exceeds $C + \frac{\lambda}{n^{1.5}}$.
$$
\begin{aligned}
x_1 + x_2 + \dots + x_n &= n, \\
x_1^2 + x_2^2 + \dots + x_n^2 &= 2n, \\
x_1^3 + x_2^3 + \dots + x_n^3 &= 3n,
\end{aligned}
$$
then we call $(x_1, x_2, \dots, x_n)$ a *regular* $n$-tuple. We call the difference between the maximal and the minimal elements in $x_1, x_2, \dots, x_n$ the *width* of the regular $n$-tuple.
(1) Find the maximal real number $C$ such that, for any integer $n \ge 4$, the width of every regular $n$-tuple is at least $C$.
(2) For the maximum $C$ in (1), prove that there exists a real number $\lambda \gt 0$ such that, for any integer $n \ge 4$, the width of every regular $n$-tuple exceeds $C + \frac{\lambda}{n^{1.5}}$.