2012 ToT Fall Senior A p7 1 000 000$ soldiers in a line
Source:
March 22, 2020
combinatorics
Problem Statement
There are soldiers in a line. The sergeant splits the line into segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most of these numbers are different.