# 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?