2013 IMO Shortlist N5

Fix an integer $k \geq 2$. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer $n \geq k$ 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 $m^{\prime}$ with $k \leq m^{\prime}<m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \geq k$ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n, n^{\prime} \geq k$ with the property that each prime number $p \leq k$ divides $n$ if and only if it divides $n^{\prime}$. Prove that either both $n$ and $n^{\prime}$ are good or both are bad. (Italy)

修复整数 $k \geq 2$。两个玩家，Ana 和 Banana，玩以下数字游戏： 最初，一些整数 $n \geq k$ 被写在黑板上。然后他们轮流采取行动，安娜开始。进行移动的玩家擦除刚刚写在黑板上的数字 $m$，并将其替换为与 $m$ 互质的某个数字 $m^{\prime}$，其中 $k \leq m^{\prime}<m$。第一个不能再移动的玩家就输了。如果 Banana 在初始数字为 $n$ 时有获胜策略，则整数 $n \geq k$ 称为良好，否则称为不良。考虑两个整数 $n, n^{\prime} \geq k$，每个素数 $p \leq k$ 整除 $n$ 当且仅当它整除 $n^{\prime}$。证明 $n$ 和 $n^{\prime}$ 要么都是好，要么都是坏。 (意大利)