RIOPLATENSE 2017 Problem 3

Show that there are infinitely many pairs of positive integers $(m,n)$ , with $m<n$ , such that $m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$ .