CN-SMO 2012 Problem 8

Let positive integers $m,n$ satisfy $n=2^m-1$ . $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions: (a) $x, y$ are two arbitrary points in $P_n$ ; (b) starting at point $x$ , the grasshopper leaps $2012$ times and finishes at point $y$ ; (the grasshopper is allowed to travel $x$ and $y$ more than once) (c) there are even number ways for the grasshopper to do (b).