Question 149
Two positive integers are written on the board. Two players take turns in a game changing the numbers on the board. If the numbers on the board are $A$ and $B$ ($A \ge B$), the player who has the turn chooses a positive integer $k$ such that $A - kB \ge 0$, erases the number $A$ and writes $A - kB$ on the board. The winner is the player who writes number $0$.
Determine all ratios of the starting two numbers for which the first player can win independently of the choices of the second player. (Brazil 1987)
Determine all ratios of the starting two numbers for which the first player can win independently of the choices of the second player. (Brazil 1987)