Question 132
Let $n \ge 2$ be a positive integer and $a_1, a_2, \dots, a_n \in (0, 1)$. Find the maximum value of the sum
$$
\sum_{i=1}^{n} \sqrt[6]{a_i(1-a_{i+1})}
$$
where $a_{n+1} = a_1$.
$$
\sum_{i=1}^{n} \sqrt[6]{a_i(1-a_{i+1})}
$$
where $a_{n+1} = a_1$.