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problem of grasshopper leaping

Source: China south east mathematical olympiad 2012 day2 problem 8

July 17, 2013
combinatorics unsolvedcombinatorics

Problem Statement

Let positive integers m,nm,n satisfy n=2m−1n=2^m-1. Pn={1,2,⋯ ,n}P_n =\{1,2,\cdots ,n\} is a set that contains nn points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of mm satisfying following conditions: (a) x,yx, y are two arbitrary points in PnP_n; (b) starting at point xx, the grasshopper leaps 20122012 times and finishes at point yy; (the grasshopper is allowed to travel xx and yy more than once) (c) there are even number ways for the grasshopper to do (b).
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