Question 1
The three integers $2000$, $19$ and $n$ are written on the board. Ana and Beto play the following game:
Ana starts and then they play in turns. Each move consists of erasing one of the numbers on the board and replacing it with the difference of the other two (the larger minus the smaller). Only moves in which one of the written numbers changes are allowed. The player who cannot move on their turn loses.
Prove that for every value of $n$ the game has a winner, and determine who wins if the numbers on the board are $2000$, $19$ and $2019$.
Ana starts and then they play in turns. Each move consists of erasing one of the numbers on the board and replacing it with the difference of the other two (the larger minus the smaller). Only moves in which one of the written numbers changes are allowed. The player who cannot move on their turn loses.
Prove that for every value of $n$ the game has a winner, and determine who wins if the numbers on the board are $2000$, $19$ and $2019$.