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 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 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. | ||
| − | [ | + | [[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 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> | ||
| − | [ | + | [[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 Problems/Problem 6|Solution]] |
== Problem 7 == | == Problem 7 == | ||
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[[Image:2006AimeA7.PNG]] | [[Image:2006AimeA7.PNG]] | ||
| − | [ | + | [[2006 AIME I Problems/Problem 7|Solution]] |
== Problem 8 == | == Problem 8 == | ||
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[[Image:2006AimeA8.PNG]] | [[Image:2006AimeA8.PNG]] | ||
| − | [ | + | [[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 Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
| − | Eight circles of diameter 1 are packed in the first quadrant of the | + | 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> | ||
| − | [ | + | [[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 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 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 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 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 Problems/Problem 15|Solution]] |
== See also == | == See also == | ||
| − | * [ | + | * [[American Invitational Mathematics Examination]] |
| − | * [ | + | * [[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] | ||
| − | * [ | + | * [[Mathematics competition resources]] |
Revision as of 15:39, 12 May 2011
| 2006 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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 
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 five 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 coordinate 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 
![[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]](http://latex.artofproblemsolving.com/4/2/0/420456ae501936e7d4f9073b71122dd6cea147d6.png)
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 
