China Western Mathematical Olympiad

By the AM-GM Inequality, we deduce that $$\begin{array}{rl}& \sqrt[6]{a_i(1-a_{i+1})}\\ & =2^{\frac{4}{6}}\sqrt[6]{a_i(1-a_{i+1})\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}}\\ & \le 2^{\frac{2}{3}}\cdot \frac{1}{6}\cdot (a_i+1-a_{i+1}+2)\\ & =2^{\frac{2}{3}}\cdot \frac{1}{6}\cdot (a_i-a_{i+1}+3).\end{array}$$ So $$\begin{array}{rl}& \sum _{i=1}^n\sqrt[6]{a_i(1-a_{i+1})}\\ & \le 2^{\frac{2}{3}}\cdot \frac{1}{6}\sum _{i=1}^n(a_i-a_{i+1}+3)\\ & =2^{\frac{2}{3}}\cdot \frac{1}{6}\cdot 3n\\ & =\frac{n}{\sqrt[3]{2}}.\end{array}$$ The equality holds if and only if $a_1=a_2=\cdots =a_n=\frac{1}{2}$. So $$y_{\max }=\frac{n}{\sqrt[3]{2}}.$$