Question 66
The managers of two companies are 1000 km apart. The segment that connects them contains $n$ points, in each of which sits a mathematician. Every second each mathematician moves to a point which is a middle of a segment that connects him/her and the closest mathematician or manager. If there is more than one such point, mathematician chooses any of them. If two mathematicians arrive to the same point according to the rules above, younger of them leaves the game. Show that in finitely many seconds every mathematician which is still in the game can shake hands with one of the managers. A mathematician can shake hands when the distance from the manager does not exceed 1 m.