KMO 2008 Problem 4

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$ .

(1) For every $m\ge2$ , prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$ .

(2) For a certain $m\ge2$ , let $a, b$ be positive integers that satisfy $2^a\in A_m$ , $2^b+1\in A_m$ . Let $a_0, b_0$ be the least such pair $a, b$ . Find the relation between $a_0$ and $b_0$ .