# Difference between revisions of "1990 IMO Problems/Problem 5"

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5. Given an initial integer <math>n_{0}\textgreater1</math>, two players, <math>\mathbb{A}</math> and <math>\mathbb{B}</math>, choose integers <math>n_{1}, n_{2},n_{3}</math>, . . . alternately according to the following rules: | 5. Given an initial integer <math>n_{0}\textgreater1</math>, two players, <math>\mathbb{A}</math> and <math>\mathbb{B}</math>, choose integers <math>n_{1}, n_{2},n_{3}</math>, . . . alternately according to the following rules: | ||

− | Knowing <math>n_{2k}, < | + | Knowing <math>n_{2k}</math>, <math>\mathbb{A}</math> chooses any integer <math>n_{2k+1}</math> such that |

− | < | + | <math>n_{2k}\leq n_{2k+1}\leq n_{2k}^2</math>. |

− | Knowing < | + | Knowing <math>n_{2k+1}</math>, <math>\mathbb{B}</math> chooses any integer <math>n_{2k+2}</math> such that |

− | < | + | <math>\frac{n_{2k+1}}{n_{2k+2}}</math> |

is a prime raised to a positive integer power. | is a prime raised to a positive integer power. | ||

− | Player < | + | Player <math>\mathbb{A}</math> wins the game by choosing the number 1990; player <math>\mathbb{B}</math> wins by choosing |

− | the number 1. For which < | + | the number 1. For which <math>n_{0}</math> does: |

− | (a) < | + | (a) <math>\mathbb{A}</math> have a winning strategy? |

− | (b) < | + | (b) <math>\mathbb{B}</math> have a winning strategy? |

(c) Neither player have a winning strategy? | (c) Neither player have a winning strategy? |

## Revision as of 05:57, 5 July 2016

5. Given an initial integer , two players, and , choose integers , . . . alternately according to the following rules: Knowing , chooses any integer such that . Knowing , chooses any integer such that is a prime raised to a positive integer power. Player wins the game by choosing the number 1990; player wins by choosing the number 1. For which does: (a) have a winning strategy? (b) have a winning strategy? (c) Neither player have a winning strategy?