Question 96
Let $a_1,a_2,\ldots,a_n$ be real numbers with $a_1+a_2+\cdots+a_n=0$. Prove that
$$\begin{aligned}\frac n{n-1}\min\{a_1,a_2,\ldots,a_n\}&\ge\min\{a_1,a_2\}+\min\{a_2,a_3\}\\&\quad+\cdots+\min\{a_n,a_1\}.\end{aligned}$$
$$\begin{aligned}\frac n{n-1}\min\{a_1,a_2,\ldots,a_n\}&\ge\min\{a_1,a_2\}+\min\{a_2,a_3\}\\&\quad+\cdots+\min\{a_n,a_1\}.\end{aligned}$$