CWMO 2016 Problem 8

For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$ . Determine the smallest positive integer $k$ satisfying the following condition: for any $m$ -element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$ , then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that $$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$