WLOG $m \ge n$ $m=kn$ with $k\ge 1$ and $k \in \mathbb{R}$ a) $n>1$ then: Notice that $n < k^2 (n^2-1)$ $k^2+n/k < kn^2$ $k^2+n/k < kn^2+|k+1/k|$ On the other hand: $|k^2| \le k^2$ and $|n/k|\le n/k$ then $|k^2|+|n/k| \le k^2+n/k$ $|k^2|+|n/k|<kn^2+|k+1/k|$ Contradiction b) $n=1$ then $m^2+|1/m|=|m+1/m|+m$ then no solution is it ok?

see here http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=382453