IMO 2024 Shortlist C5
Let N be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers 1, 2, ..., N on a boa...
Category: Combinatorics
Problem
Let N be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers 1, 2, ..., N on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers pk,nq, where k ě 0 and n is one of the integers on the board, and then erasing every integer s on the board such that 2k | n ´ s. The game continues until the board is empty. The player who erases the last integer on the board loses. Determine all values of N for which Geoff can ensure that he wins, no matter how Ceri plays. (Indonesia) 6 Bath, United Kingdom, 10th –22nd July 2024