RS-SMO 2013

Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$ , $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$ . Prove that for every $i,j\in \mathbb{Z}$ it stands: $$|f(i)-f(j)|= |i-j|.$$

Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$ , such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$ .