Difference between revisions of "2006 AIME I Problems"

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(Undo revision 38424 by HyperSet (Talk) Links to solutions go to the solutions, not a youtube video of "Friday".)
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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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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 <math>1/29</math> of the original integer.
 
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is <math>1/29</math> of the original integer.
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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.
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[2006 AIME I Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 
The number <math> \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math> 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 <math> \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</math> 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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[2006 AIME I Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
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[[Image:2006AimeA7.PNG]]
 
[[Image:2006AimeA7.PNG]]
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[2006 AIME I Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
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[[Image:2006AimeA8.PNG]]
 
[[Image:2006AimeA8.PNG]]
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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 +
[[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 coordinate 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>  
  
 
<asy>
 
<asy>
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</asy>
 
</asy>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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?
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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.
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>)
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[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>
  
[http://www.youtube.com/watch?v=CD2LRROpph0 Solution]
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[[2006 AIME I Problems/Problem 15|Solution]]
  
 
== See also ==
 
== See also ==
* [http://www.youtube.com/watch?v=CD2LRROpph0 American Invitational Mathematics Examination]
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* [[American Invitational Mathematics Examination]]
* [http://www.youtube.com/watch?v=CD2LRROpph0 AIME Problems and Solutions]
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* [[AIME Problems and Solutions]]
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=144 2006 AIME I Math Jam Transcript]
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=144 2006 AIME I Math Jam Transcript]
* [http://www.youtube.com/watch?v=CD2LRROpph0 Mathematics competition resources]
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* [[Mathematics competition resources]]

Revision as of 16:39, 12 May 2011

2006 AIME I (Answer Key)
Printable version | AoPS Contest Collections

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Problem 1

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

Solution

Problem 2

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

Solution

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.

Solution

Problem 4

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

Solution

Problem 5

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

Solution

Problem 6

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

Solution

Problem 7

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

2006AimeA7.PNG

Solution

Problem 8

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

2006AimeA8.PNG

Solution

Problem 9

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

Solution

Problem 10

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

[asy] unitsize(0.50cm); draw((0,-1)--(0,6)); draw((-1,0)--(6,0)); draw(shift(1,1)*unitcircle); draw(shift(1,3)*unitcircle); draw(shift(1,5)*unitcircle); draw(shift(3,1)*unitcircle); draw(shift(3,3)*unitcircle); draw(shift(3,5)*unitcircle); draw(shift(5,1)*unitcircle); draw(shift(5,3)*unitcircle); [/asy]

Solution

Problem 11

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ 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 $k$ must have edge-length at most $k+2.$

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

Solution

Problem 12

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

Solution

Problem 13

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

Solution

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 $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$)

Solution

Problem 15

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

Solution

See also

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