Math Gold Medalist

Lor

2023 USAMO

Problem 4

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.


Small Example

Invariant

vp(n)

Suppose the numbers are x1, x2, …, xn

Prove that

Lemma1. a is even.

Lemma2. xi is not a.

Lemma3. xi is not a.2^k .

Lemma 4. If v2(xi)<v2(a) Bob can change this number to an odd number eventually and after that step the number is always odd.

2020 IMO Problem 4

2023 BMO Round 2 Problem 2 (British Mathematical Olympiad)

   Solution