Question 86
Let n$\gt$0 be an integer and let $x_1,\ldots,x_n$ be positive reals. Prove
$$\begin{aligned}&\max_{x_1\gt 0,\ldots,x_n\gt 0}\min\left(x_1,\frac1{x_1}+x_2,\ldots,\frac1{x_{n-1}}+x_n,\frac1{x_n}\right)\\&=\min_{x_1\gt 0,\ldots,x_n\gt 0}\max\left(x_1,\frac1{x_1}+x_2,\ldots,\frac1{x_{n-1}}+x_n,\frac1{x_n}\right)\\&=2\cos\left(\frac\pi{n+2}\right).\end{aligned}$$
$$\begin{aligned}&\max_{x_1\gt 0,\ldots,x_n\gt 0}\min\left(x_1,\frac1{x_1}+x_2,\ldots,\frac1{x_{n-1}}+x_n,\frac1{x_n}\right)\\&=\min_{x_1\gt 0,\ldots,x_n\gt 0}\max\left(x_1,\frac1{x_1}+x_2,\ldots,\frac1{x_{n-1}}+x_n,\frac1{x_n}\right)\\&=2\cos\left(\frac\pi{n+2}\right).\end{aligned}$$