2023 USAJMO

Problem 5

A positive integer  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  on the board with , and on Bob’s turn he must replace some even integer  on the board with . 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  and these integers on the board, the game is guaranteed to end regardless of Alice’s or Bob’s moves.