IMO 2013 Shortlist N5
Fix an integer k ě 2. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer n ...
Category: Number Theory
Problem
Fix an integer k ě 2. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer n ě k gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number m just written on the blackboard and replaces it by some number m1 with k ď m1 ă m that is coprime to m. The first player who cannot move anymore loses. An integer n ě k is called good if Banana has a winning strategy when the initial number is n, and bad otherwise. Consider two integers n,n1 ě k with the property that each prime number p ď k divides n if and only if it divides n1 . Prove that either both n and n1 are good or both are bad. (Italy)