Mathematica competitions in Croatia

Without loss of generality we can assume that $m\ge n$. Obviously, $m^2<m^2+n$. On the other hand, $(m+1{)}^2=m^2+2m+1>m^2+n$ because $2m>n$. Hence we have $$m^2<m^2+n<(m+1{)}^2,$$ which means that $m^2+n$ is between two consecutive squares, so it can not be a square of an integer. Such positive integers $m$ and $n$ do not exist.