CMO 2010

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that $$|T| \leq 1+ \frac{a_n-a_1}{2n+1}$$ and for every $i \in \{1,2,\ldots , m\}$ , there exists $t \in T$ and $s \in [-n,n]$ , such that $a_i=t+s$ .

Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that $$(n + 1)a_1^n + na_2^n + (n - 1)a_3^n|(n + 1)b_1^n + nb_2^n + (n - 1)b_3^n$$ holds for all positive integers $n$ . Prove that there exists $k\in N$ such that $b_i = ka_i$ for $i = 1,2,3$ .