Question 3
Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = \min(x_n, y_n)$ and $y_{n+1} = \max(x_n, y_n) - k \cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $\frac{x_1}{y_1}$ the first player has a winning strategy.