# Difference between revisions of "2008 AIME II Problems"

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

− | + | Rudolph bikes at a [[constant]] rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the <math>50</math>-mile mark at exactly the same time. How many minutes has it taken them? | |

[[2008_AIME_II_Problems/Problem_2|Solution]] | [[2008_AIME_II_Problems/Problem_2|Solution]] | ||

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

− | + | A block of cheese in the shape of a rectangular solid measures <math>10</math> cm by <math>13</math> cm by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off? | |

[[2008_AIME_II_Problems/Problem_3|Solution]] | [[2008_AIME_II_Problems/Problem_3|Solution]] | ||

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

− | {{ | + | There exist <math>r</math> unique nonnegative integers <math>n_1 > n_2 > \cdots > n_r</math> and <math>r</math> unique integers <math>a_k</math> (<math>1\le k\le r</math>) with each <math>a_k</math> either <math>1</math> or <math>- 1</math> such that |

+ | <cmath> | ||

+ | a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008. | ||

+ | </cmath> | ||

+ | Find <math>n_1 + n_2 + \cdots + n_r</math>. | ||

[[2008_AIME_II_Problems/Problem_4|Solution]] | [[2008_AIME_II_Problems/Problem_4|Solution]] | ||

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

− | {{ | + | In [[trapezoid]] <math>ABCD</math> with <math>\overline{BC}\parallel\overline{AD}</math>, let <math>BC = 1000</math> and <math>AD = 2008</math>. Let <math>\angle A = 37^\circ</math>, <math>\angle D = 53^\circ</math>, and <math>M</math> and <math>N</math> be the [[midpoint]]s of <math>\overline{BC}</math> and <math>\overline{AD}</math>, respectively. Find the length <math>MN</math>. |

[[2008_AIME_II_Problems/Problem_5|Solution]] | [[2008_AIME_II_Problems/Problem_5|Solution]] | ||

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

− | {{ | + | The sequence <math>\{a_n\}</math> is defined by |

+ | <cmath> | ||

+ | a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2. | ||

+ | </cmath> | ||

+ | The sequence <math>\{b_n\}</math> is defined by | ||

+ | <cmath> | ||

+ | b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.</cmath> | ||

+ | Find <math>\frac {b_{32}}{a_{32}}</math>. | ||

[[2008_AIME_II_Problems/Problem_6|Solution]] | [[2008_AIME_II_Problems/Problem_6|Solution]] | ||

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

− | + | Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation | |

+ | <cmath> | ||

+ | 8x^3 + 1001x + 2008 = 0. | ||

+ | </cmath> | ||

+ | Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>. | ||

[[2008_AIME_II_Problems/Problem_7|Solution]] | [[2008_AIME_II_Problems/Problem_7|Solution]] |

## Revision as of 18:39, 3 April 2008

## Contents

## Problem 1

Let , where the additions and subtractions alternate in pairs. Find the remainder when is divided by .

## Problem 2

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?

## Problem 3

A block of cheese in the shape of a rectangular solid measures cm by cm by cm. Ten slices are cut from the cheese. Each slice has a width of cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

## Problem 4

There exist unique nonnegative integers and unique integers () with each either or such that Find .

## Problem 5

In trapezoid with , let and . Let , , and and be the midpoints of and , respectively. Find the length .

## Problem 6

The sequence is defined by The sequence is defined by Find .

## Problem 7

Let , , and be the three roots of the equation Find .

## Problem 8

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## Problem 9

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## Problem 10

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## Problem 11

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## Problem 12

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## Problem 13

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## Problem 14

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## Problem 15

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