Difference between revisions of "2005 AIME II Problems"

Latest revision as of 20:12, 28 February 2020

2005 AIME II (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions 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. 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

A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ 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 $n.$

Solution

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 $\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$

Solution

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 $\frac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n.$

Solution

Problem 4

Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$

Solution

Problem 5

Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5, 2 \leq a \leq 2005,$ and $2 \leq b \leq 2005.$

Solution

Problem 6

The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles $A$ and $B$ 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.

Solution

Problem 7

Let $x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$ Find $(x+1)^{48}.$

Solution

Problem 8

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord is $\frac{m\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

Solution

Problem 9

For how many positive integers $n$ less than or equal to 1000 is $(\sin t + i \cos t)^n = \sin nt + i \cos nt$ true for all real $t$ ?

Solution

Problem 10

Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$

Solution

Problem 11

Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0 = 37, a_1 = 72, a_m = 0,$ and $a_{k+1} = a_{k-1} - \frac 3{a_k}$ for $k = 1,2,\ldots, m-1.$ Find $m.$

Solution

Problem 12

Square $ABCD$ has center $O, AB=900, E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$

Solution

Problem 13

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2.$

Solution

Problem 14

In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$

Solution

Problem 15

Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and $q$ are relatively prime integers, find $p+q.$

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=2005|n=II}}
 == Problem 1 ==
-Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle <math>C</math> with radius <math>30</math>. Let <math>K</math> be the area of the region inside circle <math>C</math> and outside of the six circles in the ring. Find <math>\lfloor K \rfloor</math>.
+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>
+[[2005 AIME II Problems/Problem 1|Solution]]
-[[2005 AIME I Problems/Problem 1|Solution]]
 == Problem 2 ==
-For each positive integer ''k'', let <math>S_k</math> denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is ''k''. For example, <math>S_3</math> is the squence <math>1,4,7,10 ...</math>. For how many values of ''k'' does <math>S_k</math> contain the term 2005?
+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]]
 == Problem 3 ==
-How many positive integers have exactly three proper divisors, each of which is less than 50?
+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]]
 == Problem 4 ==
-The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.
+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]]
 == Problem 5 ==
-Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distunguishable arrangements of the 8 coins.
+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]]
 == Problem 6 ==
-Let <math>P</math> be the product of nonreal roots of <math>x^4-4x^3+6x^2-4x=2005</math>. Find <math>\lfloor P \rfloor</math>
+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]]
 == Problem 7 ==
-In quadrilateral <math>ABCD</math>, <math>BC=8</math>, <math>CD=12</math>, <math>AD=10</math> and <math>m\angle A=m\angle B=60\circ</math>. Given that <math>AB=p+\sqrt{q}</math>, where ''p'' and ''q'' are positive integers, find <math>p+q</math>.
+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]]
 == Problem 8 ==
-The equation '''<math>2^{333x-2}+2^{111x+2}=2^{222x+1}+1</math>''' has three real roots. Given that their sum is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
+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]]
 == Problem 9 ==
-Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The <math>27</math> cubes are randomly arranged to form a <math>3\times 3\times 3</math> cube. Given the probability of the entire surface area of the larger cube is orange is <math>\frac{p^a}{q^b r^c}</math> where <math>p,q,</math> and <math>r</math> are distinct primes and <math>a,b,</math> and <math>c</math> are positive integers, find <math>a+b+c+p+q+r</math>.
+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]]
 == Problem 10 ==
-Triangle <math>ABC</math> lies in the Cartesian Plane and has an area of 70. The coordinates of <math>B</math> and <math>C</math> are <math>(12,19)</math> and <math>(23,20),</math> respectively, and the coordinates of <math>A</math> are <math>(p,q).</math> The line containing the median to side <math>BC</math> has slope <math>-5</math>. Find the largest possible value of <math>p+q</math>.
+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]]
 == Problem 11 ==
-A semicircle with diameter <math>d</math> is contained in a square whose sides have length 8. Given the maximum value of <math>d</math> is <math>m-\sqrt{n}</math>, find <math>m+n</math>.
+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>
+[[2005 AIME II Problems/Problem 11|Solution]]
+== 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>
+[[2005 AIME II Problems/Problem 12|Solution]]
+== 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>
+[[2005 AIME II Problems/Problem 13|Solution]]
+== 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>
+[[2005 AIME II Problems/Problem 14|Solution]]
+== 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>
+[[2005 AIME II Problems/Problem 15|Solution]]
+== 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}}

2005 AIME II (Problems • Answer Key • Resources)
Preceded by 2005 AIME I Problems	Followed by 2006 AIME I Problems
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Difference between revisions of "2005 AIME II Problems"

Latest revision as of 20:12, 28 February 2020

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also