2023 USAMO 第 4 题

A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.

After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.

选择一个正整数$a$，并将一些正整数写在黑板上。爱丽丝和鲍勃玩以下游戏。轮到 Alice 时，她必须用 $n+a$ 替换棋盘上的某个整数 $n$，而轮到 Bob 时，他必须用 $n/2$ 替换棋盘上的某个偶数整数 $n$。爱丽丝先走，他们轮流走。如果轮到鲍勃没有有效的动作，则游戏结束。

在分析了棋盘上的整数后，鲍勃意识到，无论爱丽丝做出什么动作，他最终都能够迫使游戏结束。事实上，对于 $a$ 的值和棋盘上的这些整数，无论 Alice 或 Bob 的动作如何，游戏都一定会结束。