Problem 4

Assume $n>3$. If $m+k\ge3n$, one of $m,k$ is at least $n+2$, while the other is at least $1$; hence $(mn-1)(kn-1)\ge(n^2+2n-1)(n-1)>n^3+1$, impossible. Since $n$ divides $m+k$, only $m+k=n$ or $m+k=2n$ remain.