Find all functions $f:Z^+\to Z^+$ satisfying $f(1)=2$, $f(2)\ne4$, and $\max\{f(m+f(n),m+n\}\mid\min\{2m+2n,f(m+n)+1\}$ for all $m,n\in Z^+$.

f(x)=x+1 is the only solution

$\max\{f(m+f(n),m+n\}\mid\min\{2m+2n,f(m+n)+1\}$ Since the max of the $L.H.S$ divides the min of the $R.H.S$, then $2m+2n\geq f(m)+f(n)$, furthermore $2m \geq f(m)$ Substituting $1, 2,$ one can easily find that $f(2)=3,$ induction assuming$ f(n)=n+1 $finishes the problem.