1990 IMO Problems/Problem 5

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5. Given an initial integer $n_{0}\textgreater1$, two players, $\mathbb{A}$ and $\mathbb{B}$, choose integers $n_{1}, n_{2},n_{3}$, . . . alternately according to the following rules: Knowing $n_{2k}$, $\mathbb{A}$ chooses any integer $n_{2k+1}$ such that $n_{2k}\leq n_{2k+1}\leq n_{2k}^2$. Knowing $n_{2k+1}$, $\mathbb{B}$ chooses any integer $n_{2k+2}$ such that $\frac{n_{2k+1}}{n_{2k+2}}$ is a prime raised to a positive integer power. Player $\mathbb{A}$ wins the game by choosing the number 1990; player $\mathbb{B}$ wins by choosing the number 1. For which $n_{0}$ does: (a) $\mathbb{A}$ have a winning strategy? (b) $\mathbb{B}$ have a winning strategy? (c) Neither player have a winning strategy?

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