IMO 2017 Shortlist C3
Sir Alex plays the following game on a row of 9 ells. Initially, all ells are empty. In ea h move, Sir Alex is allowed t...
Category: Combinatorics
Problem
Sir Alex plays the following game on a row of 9 ells. Initially, all ells are empty. In ea h move, Sir Alex is allowed to perform exa tly one of the following two operations: (1) Choose any number of the form 2j , where j is a non-negative integer, and put it into an empty ell. (2) Choose two (not ne essarily adja ent) ells with the same number in them; denote that number by 2j . Repla e the number in one of the ells with 2j`1 and erase the number in the other ell. At the end of the game, one ell ontains the number 2n , where n is a given positive integer, while the other ells are empty. Determine the maximum number of moves that Sir Alex ould have made, in terms of n. (Thailand)