Sean $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Demuestra que $$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min\{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$