CMO 2018 Problem 6

China Mathematical Olympiad 2018 Q6 Given the positive integer $n ,k$ $(n>k)$ and $a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ , $|\mathbb{I} |=k$ ,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$ Attachments: