# Difference between revisions of "2006 AIME I Problems"

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In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD=14. </math> Find the perimeter of <math> ABCD. </math> | In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD=14. </math> Find the perimeter of <math> ABCD. </math> | ||

− | [[2006 AIME I Problem 1|Solution]] | + | [[2006 AIME I Problems/Problem 1|Solution]] |

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

Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math> | Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math> | ||

− | [[2006 AIME I Problem 2|Solution]] | + | [[2006 AIME I Problems/Problem 2|Solution]] |

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

Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | ||

− | [[2006 AIME I Problem 3|Solution]] | + | [[2006 AIME I Problems/Problem 3|Solution]] |

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

Let <math> N </math> be the number of consecutive 0's at the right end of the decimal representation of the product <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by 1000. | Let <math> N </math> be the number of consecutive 0's at the right end of the decimal representation of the product <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by 1000. | ||

− | [[2006 AIME I Problem 4|Solution]] | + | [[2006 AIME I Problems/Problem 4|Solution]] |

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

The number <center><math> \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math></center> can be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are positive integers. Find <math> a\cdot b\cdot c. </math> | The number <center><math> \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math></center> can be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are positive integers. Find <math> a\cdot b\cdot c. </math> | ||

− | [[2006 AIME I Problem 5|Solution]] | + | [[2006 AIME I Problems/Problem 5|Solution]] |

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

Let <math> \mathcal{S} </math> be the set of real numbers that can be represented as repeating decimals of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct digits. Find the sum of the elements of <math> \mathcal{S}. </math> | Let <math> \mathcal{S} </math> be the set of real numbers that can be represented as repeating decimals of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct digits. Find the sum of the elements of <math> \mathcal{S}. </math> | ||

− | [[2006 AIME I Problem 6|Solution]] | + | [[2006 AIME I Problems/Problem 6|Solution]] |

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

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region <math> \mathcal{C} </math> to the area of shaded region <math> \mathcal{B} </math> is 11/5. Find the ratio of shaded region <math> \mathcal{D} </math> to the area of shaded region <math> \mathcal{A}. </math> | An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region <math> \mathcal{C} </math> to the area of shaded region <math> \mathcal{B} </math> is 11/5. Find the ratio of shaded region <math> \mathcal{D} </math> to the area of shaded region <math> \mathcal{A}. </math> | ||

− | [[2006 AIME I Problem 7|Solution]] | + | [[2006 AIME I Problems/Problem 7|Solution]] |

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

Hexagon <math> ABCDEF </math> is divided into four rhombuses, <math> \mathcal{P, Q, R, S,} </math> and <math> \mathcal{T,} </math> as shown. Rhombuses <math> \mathcal{P, Q, R,} </math> and <math> \mathcal{S} </math> are congruent, and each has area <math> \sqrt{2006}. </math> Let <math> K </math> be the area of rhombus <math> \mathcal{T}. </math> Given that <math> K </math> is a positive integer, find the number of possible values for <math> K. </math> | Hexagon <math> ABCDEF </math> is divided into four rhombuses, <math> \mathcal{P, Q, R, S,} </math> and <math> \mathcal{T,} </math> as shown. Rhombuses <math> \mathcal{P, Q, R,} </math> and <math> \mathcal{S} </math> are congruent, and each has area <math> \sqrt{2006}. </math> Let <math> K </math> be the area of rhombus <math> \mathcal{T}. </math> Given that <math> K </math> is a positive integer, find the number of possible values for <math> K. </math> | ||

− | [[2006 AIME I Problem 8|Solution]] | + | [[2006 AIME I Problems/Problem 8|Solution]] |

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

The sequence <math> a_1, a_2, \ldots </math> is geometric with <math> a_1=a </math> and common ratio <math> r, </math> where <math> a </math> and <math> r </math> are positive integers. Given that <math> \log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> | The sequence <math> a_1, a_2, \ldots </math> is geometric with <math> a_1=a </math> and common ratio <math> r, </math> where <math> a </math> and <math> r </math> are positive integers. Given that <math> \log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> | ||

− | [[2006 AIME I Problem 9|Solution]] | + | [[2006 AIME I Problems/Problem 9|Solution]] |

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

Eight circles of diameter 1 are packed in the first quadrant of the coordinte plane as shown. Let region <math> \mathcal{R} </math> be the union of the eight circular regions. Line <math> l, </math> with slope 3, divides <math> \mathcal{R} </math> into two regions of equal area. Line <math> l </math>'s equation can be expressed in the form <math> ax=by+c, </math> where <math> a, b, </math> and <math> c </math> are positive integers whose greatest common divisor is 1. Find <math> a^2+b^2+c^2. </math> | Eight circles of diameter 1 are packed in the first quadrant of the coordinte plane as shown. Let region <math> \mathcal{R} </math> be the union of the eight circular regions. Line <math> l, </math> with slope 3, divides <math> \mathcal{R} </math> into two regions of equal area. Line <math> l </math>'s equation can be expressed in the form <math> ax=by+c, </math> where <math> a, b, </math> and <math> c </math> are positive integers whose greatest common divisor is 1. Find <math> a^2+b^2+c^2. </math> | ||

− | [[2006 AIME I Problem 10|Solution]] | + | [[2006 AIME I Problems/Problem 10|Solution]] |

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

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Let <math> T </math> be the number of different towers than can be constructed. What is the remainder when <math> T </math> is divided by 1000? | Let <math> T </math> be the number of different towers than can be constructed. What is the remainder when <math> T </math> is divided by 1000? | ||

− | [[2006 AIME I Problem 11|Solution]] | + | [[2006 AIME I Problems/Problem 11|Solution]] |

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

Find the sum of the values of <math> x </math> such that <math> \cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x, </math> where <math> x </math> is measured in degrees and <math> 100< x< 200. </math> | Find the sum of the values of <math> x </math> such that <math> \cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x, </math> where <math> x </math> is measured in degrees and <math> 100< x< 200. </math> | ||

− | [[2006 AIME I Problem 12|Solution]] | + | [[2006 AIME I Problems/Problem 12|Solution]] |

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

For each even positive integer <math> x, </math> let <math> g(x) </math> denote the greatest power of 2 that divides <math> x. </math> For example, <math> g(20)=4 </math> and <math> g(16)=16. </math> For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a perfect square. | For each even positive integer <math> x, </math> let <math> g(x) </math> denote the greatest power of 2 that divides <math> x. </math> For example, <math> g(20)=4 </math> and <math> g(16)=16. </math> For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a perfect square. | ||

− | [[2006 AIME I Problem 13|Solution]] | + | [[2006 AIME I Problems/Problem 13|Solution]] |

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

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let <math> h </math> be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then <math> h </math> can be written in the form <math> \frac m{\sqrt{n}}, </math> where <math> m </math> and <math> n </math> are positive integers and <math> n </math> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) | A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let <math> h </math> be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then <math> h </math> can be written in the form <math> \frac m{\sqrt{n}}, </math> where <math> m </math> and <math> n </math> are positive integers and <math> n </math> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) | ||

− | [[2006 AIME I Problem 14|Solution]] | + | [[2006 AIME I Problems/Problem 14|Solution]] |

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

Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006}|. </math> | Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006}|. </math> | ||

− | [[2006 AIME I Problem 15|Solution]] | + | [[2006 AIME I Problems/Problem 15|Solution]] |

== See also == | == See also == |

## Revision as of 21:23, 4 July 2006

## Contents

## Problem 1

In quadrilateral is a right angle, diagonal is perpendicular to and Find the perimeter of

## Problem 2

Let set be a 90-element subset of and let be the sum of the elements of Find the number of possible values of

## Problem 3

Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer.

## Problem 4

Let be the number of consecutive 0's at the right end of the decimal representation of the product Find the remainder when is divided by 1000.

## Problem 5

The number

can be written as where and are positive integers. Find

## Problem 6

Let be the set of real numbers that can be represented as repeating decimals of the form where are distinct digits. Find the sum of the elements of

## Problem 7

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region is 11/5. Find the ratio of shaded region to the area of shaded region

## Problem 8

Hexagon is divided into four rhombuses, and as shown. Rhombuses and are congruent, and each has area Let be the area of rhombus Given that is a positive integer, find the number of possible values for

## Problem 9

The sequence is geometric with and common ratio where and are positive integers. Given that find the number of possible ordered pairs

## Problem 10

Eight circles of diameter 1 are packed in the first quadrant of the coordinte plane as shown. Let region be the union of the eight circular regions. Line with slope 3, divides into two regions of equal area. Line 's equation can be expressed in the form where and are positive integers whose greatest common divisor is 1. Find

## Problem 11

A collection of 8 cubes consists of one cube with edge-length for each integer A tower is to be built using all 8 cubes according to the rules:

- Any cube may be the bottom cube in the tower.
- The cube immediately on top of a cube with edge-length must have edge-length at most

Let be the number of different towers than can be constructed. What is the remainder when is divided by 1000?

## Problem 12

Find the sum of the values of such that where is measured in degrees and

## Problem 13

For each even positive integer let denote the greatest power of 2 that divides For example, and For each positive integer let Find the greatest integer less than 1000 such that is a perfect square.

## Problem 14

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then can be written in the form where and are positive integers and is not divisible by the square of any prime. Find (The notation denotes the greatest integer that is less than or equal to )

## Problem 15

Given that a sequence satisfies and for all integers find the minimum possible value of