Difference between revisions of "2006 AIME II Problems"
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+ | {{AIME Problems|year=2006|n=II}} | ||
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== Problem 1 == | == Problem 1 == | ||
In convex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \angle C, \angle E,</math> and <math>\angle F</math> are congruent. The area of the hexagonal region is <math>2116(\sqrt{2}+1).</math> Find <math>AB</math>. | In convex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \angle C, \angle E,</math> and <math>\angle F</math> are congruent. The area of the hexagonal region is <math>2116(\sqrt{2}+1).</math> Find <math>AB</math>. | ||
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The lengths of the sides of a triangle with positive area are <math>\log_{10} 12</math>, <math>\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>. | The lengths of the sides of a triangle with positive area are <math>\log_{10} 12</math>, <math>\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>. | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 2|Solution]] |
== Problem 3 == | == Problem 3 == | ||
− | Let <math> P </math> be the product of the first 100 positive odd integers. Find the largest integer <math> k </math> such that <math> P </math> is divisible by <math> 3^k </math> | + | Let <math> P </math> be the product of the first 100 positive odd integers. Find the largest integer <math> k </math> such that <math> P </math> is divisible by <math> 3^k </math>. |
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 3|Solution]] |
== Problem 4 == | == Problem 4 == | ||
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An example of such a permutation is <math> (6,5,4,3,2,1,7,8,9,10,11,12). </math> Find the number of such permutations. | An example of such a permutation is <math> (6,5,4,3,2,1,7,8,9,10,11,12). </math> Find the number of such permutations. | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 4|Solution]] |
== Problem 5 == | == Problem 5 == | ||
− | When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face <math> F </math> is greater than 1/ | + | When rolling a certain unfair six-sided die with faces numbered <math>1, 2, 3, 4, 5</math>, and <math>6</math>, the probability of obtaining face <math> F </math> is greater than <math>\frac{1}{6}</math>, the probability of obtaining the face opposite is less than <math>\frac{1}{6}</math>, the probability of obtaining any one of the other four faces is <math>\frac{1}{6}</math>, and the sum of the numbers on opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is <math>\frac{47}{288}</math>. Given that the probability of obtaining face <math> F </math> is <math> \frac{m}{n}, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> |
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 5|Solution]] |
== Problem 6 == | == Problem 6 == | ||
Square <math> ABCD </math> has sides of length 1. Points <math> E </math> and <math> F </math> are on <math> \overline{BC} </math> and <math> \overline{CD}, </math> respectively, so that <math> \triangle AEF </math> is equilateral. A square with vertex <math> B </math> has sides that are parallel to those of <math> ABCD </math> and a vertex on <math> \overline{AE}. </math> The length of a side of this smaller square is <math>\frac{a-\sqrt{b}}{c}, </math> where <math> a, b, </math> and <math> c </math> are positive integers and <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> | Square <math> ABCD </math> has sides of length 1. Points <math> E </math> and <math> F </math> are on <math> \overline{BC} </math> and <math> \overline{CD}, </math> respectively, so that <math> \triangle AEF </math> is equilateral. A square with vertex <math> B </math> has sides that are parallel to those of <math> ABCD </math> and a vertex on <math> \overline{AE}. </math> The length of a side of this smaller square is <math>\frac{a-\sqrt{b}}{c}, </math> where <math> a, b, </math> and <math> c </math> are positive integers and <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 6|Solution]] |
== Problem 7 == | == Problem 7 == | ||
Find the number of ordered pairs of positive integers <math> (a,b) </math> such that <math> a+b=1000 </math> and neither <math> a </math> nor <math> b </math> has a zero digit. | Find the number of ordered pairs of positive integers <math> (a,b) </math> such that <math> a+b=1000 </math> and neither <math> a </math> nor <math> b </math> has a zero digit. | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 7|Solution]] |
== Problem 8 == | == Problem 8 == | ||
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Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed? | Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed? | ||
+ | <asy> | ||
+ | size(50); | ||
+ | pair A,B; | ||
+ | A=(0,0); B=(2,0); | ||
+ | pair C=rotate(60,A)*B; | ||
+ | pair D, E, F; | ||
+ | D = (1,0); | ||
+ | E=rotate(60,A)*D; | ||
+ | F=rotate(60,C)*E; | ||
+ | draw(C--A--B--cycle); draw(D--E--F--cycle); | ||
+ | </asy> | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 8|Solution]] |
== Problem 9 == | == Problem 9 == | ||
Circles <math> \mathcal{C}_1, \mathcal{C}_2, </math> and <math> \mathcal{C}_3 </math> have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line <math> t_1 </math> is a common internal tangent to <math>\mathcal{C}_1</math> and <math>\mathcal{C}_2</math> and has a positive slope, and line <math>t_2</math> is a common internal tangent to <math>\mathcal{C}_2</math> and <math>\mathcal{C}_3</math> and has a negative slope. Given that lines <math>t_1</math> and <math>t_2</math> intersect at <math>(x,y),</math> and that <math>x=p-q\sqrt{r},</math> where <math>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime, find <math>p+q+r.</math> | Circles <math> \mathcal{C}_1, \mathcal{C}_2, </math> and <math> \mathcal{C}_3 </math> have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line <math> t_1 </math> is a common internal tangent to <math>\mathcal{C}_1</math> and <math>\mathcal{C}_2</math> and has a positive slope, and line <math>t_2</math> is a common internal tangent to <math>\mathcal{C}_2</math> and <math>\mathcal{C}_3</math> and has a negative slope. Given that lines <math>t_1</math> and <math>t_2</math> intersect at <math>(x,y),</math> and that <math>x=p-q\sqrt{r},</math> where <math>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime, find <math>p+q+r.</math> | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 9|Solution]] |
== Problem 10 == | == Problem 10 == | ||
− | Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a <math> 50\% </math> chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are | + | Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a <math> 50\% </math> chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team <math> A </math> beats team <math> B. </math> The probability that team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> |
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 10|Solution]] |
== Problem 11 == | == Problem 11 == | ||
A sequence is defined as follows <math> a_1=a_2=a_3=1, </math> and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n. </math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find the remainder when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000. | A sequence is defined as follows <math> a_1=a_2=a_3=1, </math> and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n. </math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find the remainder when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000. | ||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 11|Solution]] |
== Problem 12 == | == Problem 12 == | ||
Equilateral <math> \triangle ABC </math> is inscribed in a circle of radius 2. Extend <math> \overline{AB} </math> through <math> B </math> to point <math> D </math> so that <math> AD=13, </math> and extend <math> \overline{AC} </math> through <math> C </math> to point <math> E </math> so that <math> AE = 11. </math> Through <math> D, </math> draw a line <math> l_1 </math> parallel to <math> \overline{AE}, </math> and through <math> E, </math> draw a line <math> l_2 </math> parallel to <math> \overline{AD}. </math> Let <math> F </math> be the intersection of <math> l_1 </math> and <math> l_2. </math> Let <math> G </math> be the point on the circle that is collinear with <math> A </math> and <math> F </math> and distinct from <math> A. </math> Given that the area of <math> \triangle CBG </math> can be expressed in the form <math> \frac{p\sqrt{q}}{r}, </math> where <math> p, q, </math> and <math> r </math> are positive integers, <math> p </math> and <math> r</math> are relatively prime, and <math> q </math> is not divisible by the square of any prime, find <math> p+q+r. </math> | Equilateral <math> \triangle ABC </math> is inscribed in a circle of radius 2. Extend <math> \overline{AB} </math> through <math> B </math> to point <math> D </math> so that <math> AD=13, </math> and extend <math> \overline{AC} </math> through <math> C </math> to point <math> E </math> so that <math> AE = 11. </math> Through <math> D, </math> draw a line <math> l_1 </math> parallel to <math> \overline{AE}, </math> and through <math> E, </math> draw a line <math> l_2 </math> parallel to <math> \overline{AD}. </math> Let <math> F </math> be the intersection of <math> l_1 </math> and <math> l_2. </math> Let <math> G </math> be the point on the circle that is collinear with <math> A </math> and <math> F </math> and distinct from <math> A. </math> Given that the area of <math> \triangle CBG </math> can be expressed in the form <math> \frac{p\sqrt{q}}{r}, </math> where <math> p, q, </math> and <math> r </math> are positive integers, <math> p </math> and <math> r</math> are relatively prime, and <math> q </math> is not divisible by the square of any prime, find <math> p+q+r. </math> | ||
− | |||
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 12|Solution]] |
== Problem 13 == | == Problem 13 == | ||
− | How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 | + | How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>? |
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 13|Solution]] |
== Problem 14 == | == Problem 14 == | ||
− | Let <math> S_n </math> be the sum of the reciprocals of the non-zero digits of the integers from 1 to <math> 10^n </math> inclusive. Find the smallest positive integer n for which <math> S_n </math> is an integer. | + | Let <math> S_n </math> be the sum of the reciprocals of the non-zero digits of the integers from <math>1</math> to <math> 10^n </math> inclusive. Find the smallest positive integer <math>n</math> for which <math> S_n </math> is an integer. |
− | [[2006 AIME | + | [[2006 AIME II Problems/Problem 14|Solution]] |
== Problem 15 == | == Problem 15 == | ||
Given that <math> x, y, </math> and <math>z</math> are real numbers that satisfy: | Given that <math> x, y, </math> and <math>z</math> are real numbers that satisfy: | ||
− | + | <cmath>\begin{align*} | |
− | < | + | x &= \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}, \\ |
− | + | y &= \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}, \\ | |
− | + | z &= \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}, | |
− | + | \end{align*}</cmath> | |
and that <math> x+y+z = \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> m+n.</math> | and that <math> x+y+z = \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> m+n.</math> | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2006|n=II|before=[[2006 AIME I Problems]]|after=[[2007 AIME I Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[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]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 17:18, 27 June 2021
2006 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
In convex hexagon , all six sides are congruent, and are right angles, and and are congruent. The area of the hexagonal region is Find .
Problem 2
The lengths of the sides of a triangle with positive area are , , and , where is a positive integer. Find the number of possible values for .
Problem 3
Let be the product of the first 100 positive odd integers. Find the largest integer such that is divisible by .
Problem 4
Let be a permutation of for which
An example of such a permutation is Find the number of such permutations.
Problem 5
When rolling a certain unfair six-sided die with faces numbered , and , the probability of obtaining face is greater than , the probability of obtaining the face opposite is less than , the probability of obtaining any one of the other four faces is , and the sum of the numbers on opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is . Given that the probability of obtaining face is where and are relatively prime positive integers, find
Problem 6
Square has sides of length 1. Points and are on and respectively, so that is equilateral. A square with vertex has sides that are parallel to those of and a vertex on The length of a side of this smaller square is where and are positive integers and is not divisible by the square of any prime. Find
Problem 7
Find the number of ordered pairs of positive integers such that and neither nor has a zero digit.
Problem 8
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
Problem 9
Circles and have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line is a common internal tangent to and and has a positive slope, and line is a common internal tangent to and and has a negative slope. Given that lines and intersect at and that where and are positive integers and is not divisible by the square of any prime, find
Problem 10
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team beats team The probability that team finishes with more points than team is where and are relatively prime positive integers. Find
Problem 11
A sequence is defined as follows and, for all positive integers Given that and find the remainder when is divided by 1000.
Problem 12
Equilateral is inscribed in a circle of radius 2. Extend through to point so that and extend through to point so that Through draw a line parallel to and through draw a line parallel to Let be the intersection of and Let be the point on the circle that is collinear with and and distinct from Given that the area of can be expressed in the form where and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find
Problem 13
How many integers less than 1000 can be written as the sum of consecutive positive odd integers from exactly 5 values of ?
Problem 14
Let be the sum of the reciprocals of the non-zero digits of the integers from to inclusive. Find the smallest positive integer for which is an integer.
Problem 15
Given that and are real numbers that satisfy: and that where and are positive integers and is not divisible by the square of any prime, find
See also
2006 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2006 AIME I Problems |
Followed by 2007 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- 2006 AIME I Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.