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

(Created page with "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>,...")
 
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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}, </math>\mathbb{A}<math> chooses any integer </math>n_{2k+1}<math> such that
+
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>.
+
<math>n_{2k}\leq n_{2k+1}\leq n_{2k}^2</math>.
Knowing </math>n_{2k+1}<math>, </math>\mathbb{B}<math> chooses any integer </math>n_{2k+2}<math> such that
+
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>
+
<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 </math>\mathbb{A}<math> wins the game by choosing the number 1990; player </math>\mathbb{B}<math> wins by choosing
+
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 </math>n_{0}<math> does:
+
the number 1. For which <math>n_{0}</math> does:
(a) </math>\mathbb{A}<math> have a winning strategy?
+
(a) <math>\mathbb{A}</math> have a winning strategy?
(b) </math>\mathbb{B}$ have a winning strategy?
+
(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 $n_{0}\textgreater1$, 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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