KMO 1999

Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$ , $n$ . Show that for all positive integers $k$ : $$\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}$$.

A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations.