Question 21
Let $n$ be a positive integer and let $x_1, \dots, x_n$ be positive real numbers. Show that
$$
\begin{aligned}
\min (x_1, 1/x_1 + x_2, \dots, 1/x_{n-1} + x_n, 1/x_n) &\le 2 \cos \left(\frac{\pi}{n+2}\right) \\
&\le \max (x_1, 1/x_1 + x_2, \dots, 1/x_{n-1} + x_n, 1/x_n).
\end{aligned}
$$
$$
\begin{aligned}
\min (x_1, 1/x_1 + x_2, \dots, 1/x_{n-1} + x_n, 1/x_n) &\le 2 \cos \left(\frac{\pi}{n+2}\right) \\
&\le \max (x_1, 1/x_1 + x_2, \dots, 1/x_{n-1} + x_n, 1/x_n).
\end{aligned}
$$