CMO 2012 Problem 3

Prove for any $M>2$ , there exists an increasing sequence of positive integers $a_1<a_2<\ldots$ satisfying:

1. $a_i>M^i$ for any $i$ ;
2. There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$ , satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$ .