Difference between revisions of "2005 AIME II Problems"
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== Problem 1 == | == Problem 1 == | ||
− | + | A game uses a deck of <math> n </math> different cards, where <math> n </math> is an integer and <math> n \geq 6. </math> The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> | |
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+ | [[2005 AIME II Problems/Problem 1|Solution]] | ||
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== Problem 2 == | == Problem 2 == | ||
− | + | A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> | |
[[2005 AIME II Problems/Problem 2|Solution]] | [[2005 AIME II Problems/Problem 2|Solution]] | ||
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== Problem 3 == | == Problem 3 == | ||
− | + | An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is <math> \frac mn </math> where <math> m </math> and <math> n </math> are relatively prime integers. Find <math> m+n. </math> | |
[[2005 AIME II Problems/Problem 3|Solution]] | [[2005 AIME II Problems/Problem 3|Solution]] | ||
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== Problem 4 == | == Problem 4 == | ||
− | + | Find the number of positive integers that are divisors of at least one of <math> 10^{10},15^7,18^{11}. </math> | |
[[2005 AIME II Problems/Problem 4|Solution]] | [[2005 AIME II Problems/Problem 4|Solution]] | ||
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== Problem 5 == | == Problem 5 == | ||
− | + | Determine the number of ordered pairs <math> (a,b) </math> of integers such that <math> \log_a b + 6\log_b a=5, 2 \leq a \leq 2005, </math> and <math> 2 \leq b \leq 2005. </math> | |
[[2005 AIME II Problems/Problem 5|Solution]] | [[2005 AIME II Problems/Problem 5|Solution]] | ||
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== Problem 6 == | == Problem 6 == | ||
− | + | The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position. | |
[[2005 AIME II Problems/Problem 6|Solution]] | [[2005 AIME II Problems/Problem 6|Solution]] | ||
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== Problem 7 == | == Problem 7 == | ||
− | + | Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math> (x+1)^{48}. </math> | |
[[2005 AIME II Problems/Problem 7|Solution]] | [[2005 AIME II Problems/Problem 7|Solution]] | ||
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== Problem 8 == | == Problem 8 == | ||
− | The | + | Circles <math> C_1 </math> and <math> C_2 </math> are externally tangent, and they are both internally tangent to circle <math> C_3. </math> The radii of <math> C_1 </math> and <math> C_2 </math> are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of <math> C_3 </math> is also a common external tangent of <math> C_1 </math> and <math> C_2. </math> Given that the length of the chord is <math> \frac{m\sqrt{n}}p </math> where <math> m,n, </math> and <math> p </math> are positive integers, <math> m </math> and <math> p </math> are relatively prime, and <math> n </math> is not divisible by the square of any prime, find <math> m+n+p. </math> |
[[2005 AIME II Problems/Problem 8|Solution]] | [[2005 AIME II Problems/Problem 8|Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
− | + | For how many positive integers <math> n </math> less than or equal to 1000 is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>? | |
[[2005 AIME II Problems/Problem 9|Solution]] | [[2005 AIME II Problems/Problem 9|Solution]] | ||
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== Problem 10 == | == Problem 10 == | ||
− | + | Given that <math> O </math> is a regular octahedron, that <math> C </math> is the cube whose vertices are the centers of the faces of <math> O, </math> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> | |
[[2005 AIME II Problems/Problem 10|Solution]] | [[2005 AIME II Problems/Problem 10|Solution]] | ||
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== Problem 11 == | == Problem 11 == | ||
− | A | + | Let <math> m </math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of reals such that <math> a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math> m. </math> |
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+ | [[2005 AIME II Problems/Problem 11|Solution]] | ||
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+ | == Problem 12 == | ||
+ | Square <math> ABCD </math> has center <math> O, AB=900, E </math> and <math> F </math> are on <math> AB </math> with <math> AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </math> Given that <math> BF=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> | ||
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+ | [[2005 AIME II Problems/Problem 12|Solution]] | ||
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+ | == Problem 13 == | ||
+ | Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_2, </math> find the product <math> n_1\cdot n_2. </math> | ||
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+ | [[2005 AIME II Problems/Problem 13|Solution]] | ||
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+ | == Problem 14 == | ||
+ | In triangle <math> ABC, AB=13, BC=15, </math> and <math>CA = 14. </math> Point <math> D </math> is on <math> \overline{BC} </math> with <math> CD=6. </math> Point <math> E </math> is on <math> \overline{BC} </math> such that <math> \angle BAE\cong \angle CAD. </math> Given that <math> BE=\frac pq </math> where <math> p </math> and <math> q </math> are relatively prime positive integers, find <math> q. </math> | ||
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+ | [[2005 AIME II Problems/Problem 14|Solution]] | ||
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+ | == Problem 15 == | ||
+ | Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest positive value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> | ||
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+ | [[2005 AIME II Problems/Problem 15|Solution]] | ||
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+ | == See also == | ||
+ | |||
+ | {{AIME box|year = 2005|n=II|before=[[2005 AIME I Problems]]|after=[[2006 AIME I Problems]]}} | ||
+ | |||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=51 2005 AIME II Math Jam Transcript] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 20:12, 28 February 2020
2005 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
A game uses a deck of different cards, where is an integer and The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find
Problem 2
A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is where and are relatively prime integers, find
Problem 3
An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is where and are relatively prime integers. Find
Problem 4
Find the number of positive integers that are divisors of at least one of
Problem 5
Determine the number of ordered pairs of integers such that and
Problem 6
The cards in a stack of cards are numbered consecutively from 1 through from top to bottom. The top cards are removed, kept in order, and form pile The remaining cards form pile The cards are then restacked by taking cards alternately from the tops of pile and respectively. In this process, card number becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles and are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.
Problem 7
Let Find
Problem 8
Circles and are externally tangent, and they are both internally tangent to circle The radii of and are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of is also a common external tangent of and Given that the length of the chord is where and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find
Problem 9
For how many positive integers less than or equal to 1000 is true for all real ?
Problem 10
Given that is a regular octahedron, that is the cube whose vertices are the centers of the faces of and that the ratio of the volume of to that of is where and are relatively prime integers, find
Problem 11
Let be a positive integer, and let be a sequence of reals such that and for Find
Problem 12
Square has center and are on with and between and and Given that where and are positive integers and is not divisible by the square of any prime, find
Problem 13
Let be a polynomial with integer coefficients that satisfies and Given that has two distinct integer solutions and find the product
Problem 14
In triangle and Point is on with Point is on such that Given that where and are relatively prime positive integers, find
Problem 15
Let and denote the circles and respectively. Let be the smallest positive value of for which the line contains the center of a circle that is externally tangent to and internally tangent to Given that where and are relatively prime integers, find
See also
2005 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2005 AIME I Problems |
Followed by 2006 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
- 2005 AIME II Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.