It is Just a Numbers Game
Source: Indonesian National Mathematical Olympiad 2024, Problem 4 (Day 1)
August 28, 2024
Game TheoryIntegerscombinatoricsnumber theoryInamo
Problem Statement
Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from and to and , or from and to and . The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.At the beginning of the game, the two numbers on the board are and . If it is known that Kobar does not lose on his first move, determine the largest possible value of so that Borah can win this game.