CN-TST 2012 Problem 14

Given an integer $k\ge 2$ . Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$ , we have $$k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.$$