let $a_i\ge 0,i=1,2,\cdots,2021$,and such $a_{1}+a_{2}+\cdots+a_{2021}=1$.find the maximum the value $$\sum\limits_{i=1}^{2021}\min(a_{i},a_{i+1})\cdot\min(a_{i+1},a_{i+2})$$ where $a_{2022}=a_{1},a_{2023}=a_{2}$
Sangchul Lee suspect that the maximum is $\frac{1}{8}$, and this value is achieved at a sequence of the form $(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},0,0,\cdots,0)$ so How to prove this inequality?Thanks
