KMO 2007 Problem 6

Let f:N→N be a function satisfying $kf(n)\le f(kn)\le kf(n) + k - 1$ for all $k, n\in N$ .

(a)Prove that $f(a) + f(b)\le f(a + b)\le f(a) + f(b) + 1$ for all $a, b\in N$ .

(b)If $f$ satisfies $f(2007n)\le 2007f(n) + 200$ for every $n\in N$ , show that there exists $c\in N$ such that $f(2007c) = 2007f(c)$ .