Problem

Source: Mar , 2019

Tags: inequalities, China, China TST



Let $n$ be a given even number, $a_1,a_2,\cdots,a_n$ be non-negative real numbers such that $a_1+a_2+\cdots+a_n=1.$ Find the maximum possible value of $\sum_{1\le i<j\le n}\min\{(i-j)^2,(n+i-j)^2\}a_ia_j .$