# Difference between revisions of "1985 AIME Problems"

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Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>. | Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>. | ||

− | [[ | + | [[1985 AIME Problems/Problem 1 | Solution]] |

==Problem 2== | ==Problem 2== | ||

− | When a right triangle is rotated about one leg, the volume of the cone produced is <math>800\pi \ | + | When a right triangle is rotated about one leg, the volume of the cone produced is <math>800\pi \;\textrm{cm}^3</math>. When the triangle is rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the triangle? |

− | [[ | + | [[1985 AIME Problems/Problem 2 | Solution]] |

==Problem 3== | ==Problem 3== | ||

− | [[ | + | [[1985 AIME Problems/Problem 3 | Solution]] |

==Problem 4== | ==Problem 4== | ||

− | [[ | + | [[1985 AIME Problems/Problem 4 | Solution]] |

==Problem 5== | ==Problem 5== | ||

− | [[ | + | [[1985 AIME Problems/Problem 5 | Solution]] |

==Problem 6== | ==Problem 6== | ||

− | [[ | + | [[1985 AIME Problems/Problem 6 | Solution]] |

==Problem 7== | ==Problem 7== | ||

− | [[ | + | [[1985 AIME Problems/Problem 7 | Solution]] |

==Problem 8== | ==Problem 8== | ||

− | [[ | + | [[1985 AIME Problems/Problem 8 | Solution]] |

==Problem 9== | ==Problem 9== | ||

− | [[ | + | [[1985 AIME Problems/Problem 9 | Solution]] |

==Problem 10== | ==Problem 10== | ||

− | [[ | + | [[1985 AIME Problems/Problem 10 | Solution]] |

==Problem 11== | ==Problem 11== | ||

− | [[ | + | [[1985 AIME Problems/Problem 11 | Solution]] |

==Problem 12== | ==Problem 12== | ||

− | [[ | + | [[1985 AIME Problems/Problem 12 | Solution]] |

==Problem 13== | ==Problem 13== | ||

− | [[ | + | [[1985 AIME Problems/Problem 13 | Solution]] |

==Problem 14== | ==Problem 14== | ||

− | [[ | + | [[1985 AIME Problems/Problem 14 | Solution]] |

==Problem 15== | ==Problem 15== | ||

− | [[ | + | [[1985 AIME Problems/Problem 15 | Solution]] |

− | + | ==See also== | |

− | * [[ | + | * [[1985 AIME]] |

+ | * [[American Invitational Mathematics Examination]] | ||

+ | * [[AIME Problems and Solutions]] | ||

+ | * [[Mathematics competition resources]] |

## Revision as of 22:25, 7 November 2006

## Contents

## Problem 1

Let , and for let. Calculate the product .

## Problem 2

When a right triangle is rotated about one leg, the volume of the cone produced is . When the triangle is rotated about the other leg, the volume of the cone produced is . What is the length (in cm) of the hypotenuse of the triangle?