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Difference between revisions of "2015 AIME II Problems"

(Problem 7)
(→‎Problem 8)
 
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Triangle <math>ABC</math> has side lengths <math>AB = 12</math>, <math>BC = 25</math>, and <math>CA = 17</math>. Rectangle <math>PQRS</math> has vertex <math>P</math> on <math>\overline{AB}</math>, vertex <math>Q</math> on <math>\overline{AC}</math>, and vertices <math>R</math> and <math>S</math> on <math>\overline{BC}</math>. In terms of the side length <math>PQ = w</math>, the area of <math>PQRS</math> can be expressed as the quadratic polynomial
 
Triangle <math>ABC</math> has side lengths <math>AB = 12</math>, <math>BC = 25</math>, and <math>CA = 17</math>. Rectangle <math>PQRS</math> has vertex <math>P</math> on <math>\overline{AB}</math>, vertex <math>Q</math> on <math>\overline{AC}</math>, and vertices <math>R</math> and <math>S</math> on <math>\overline{BC}</math>. In terms of the side length <math>PQ = w</math>, the area of <math>PQRS</math> can be expressed as the quadratic polynomial
  
−
Area(<math>PQRS</math>) = <math>\alpha w - \beta \cdot w^2</math>.
+
<cmath>\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.</cmath>
  
 
Then the coefficient <math>\beta = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
Then the coefficient <math>\beta = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
Line 62: Line 62:
  
 
==Problem 9==
 
==Problem 9==
 +
A cylindrical barrel with radius <math>4</math> feet and height <math>10</math> feet is full of water. A solid cube with side length <math>8</math> feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is <math>v</math> cubic feet. Find <math>v^2</math>.
 +
 +
<asy>
 +
import three; import solids;
 +
size(5cm);
 +
currentprojection=orthographic(1,-1/6,1/6);
 +
 +
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight);
 +
 +
triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2));
 +
 +
draw(X--X+A--X+A+B--X+A+B+C);
 +
draw(X--X+B--X+A+B);
 +
draw(X--X+C--X+A+C--X+A+B+C);
 +
draw(X+A--X+A+C);
 +
draw(X+C--X+C+B--X+A+B+C,linetype("2 4"));
 +
draw(X+B--X+C+B,linetype("2 4"));
 +
 +
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight);
 +
draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0));
 +
draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0));
 +
draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); </asy>
 +
 +
[[2015 AIME II Problems/Problem 9 | Solution]]
  
 
==Problem 10==
 
==Problem 10==
  
−
Call a permutation <math>a_1, a_2, \ldots, a_n</math> of the integers <math>1, 2, \ldots, n</math> ''quasi-increasing'' if <math>a_k \leq a_{k+1} + 2</math> for each <math>1 \leq k \leq n-1</math>. For example, 53421 and 14253 are quasi-increasing permutations of the integers <math>1, 2, 3, 4, 5</math>, but 45123 is not. Find the number of quasi-increasing permutations of the integers <math>1, 2, \ldots, 7</math>.
+
Call a permutation <math>a_1, a_2, \ldots, a_n</math> of the integers <math>1, 2, \ldots, n</math> ''quasi-increasing'' if <math>a_k \leq a_{k+1} + 2</math> for each <math>1 \leq k \leq n-1</math>. For example, <math>53421</math> and <math>14253</math> are quasi-increasing permutations of the integers <math>1, 2, 3, 4, 5</math>, but <math>45123</math> is not. Find the number of quasi-increasing permutations of the integers <math>1, 2, \ldots, 7</math>.
  
 
[[2015 AIME II Problems/Problem 10 | Solution]]
 
[[2015 AIME II Problems/Problem 10 | Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
 +
The circumcircle of acute <math>\triangle ABC</math> has center <math>O</math>. The line passing through point <math>O</math> perpendicular to <math>\overline{OB}</math> intersects lines <math>AB</math> and <math>BC</math> at <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4</math>, <math>BQ=4.5</math>, and <math>BP=\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 +
[[2015 AIME II Problems/Problem 11 | Solution]]
  
 
==Problem 12==
 
==Problem 12==
  
−
There are <math>2^{10} = 1024</math> possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.
+
There are <math>2^{10} = 1024</math> possible <math>10</math>-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than <math>3</math> adjacent letters that are identical.
  
 
[[2015 AIME II Problems/Problem 12 | Solution]]
 
[[2015 AIME II Problems/Problem 12 | Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
Define the sequence <math>a_1, a_2, a_3, \ldots</math> by <math>a_n = \sum\limits_{k=1}^n \sin{k}</math>, where <math>k</math> represents radian measure. Find the index of the 100th term for which <math>a_n < 0</math>.
 +
 +
[[2015 AIME II Problems/Problem 13 | Solution]]
  
 
==Problem 14==
 
==Problem 14==
Line 87: Line 119:
 
==Problem 15==
 
==Problem 15==
  
 +
Circles <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> have radii <math>1</math> and <math>4</math>, respectively, and are externally tangent at point <math>A</math>. Point <math>B</math> is on <math>\mathcal{P}</math> and point <math>C</math> is on <math>\mathcal{Q}</math> such that <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> again at <math>E</math>. Points <math>B</math> and <math>C</math> lie on the same side of <math>\ell</math>, and the areas of <math>\triangle DBA</math> and <math>\triangle ACE</math> are equal. This common area is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 +
<asy>
 +
import cse5;
 +
pathpen=black; pointpen=black;
 +
size(6cm);
 +
 +
pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
 +
 +
filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
 +
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
 +
D(CR((0,1),1)); D(CR((4,4),4,150,390));
 +
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
 +
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
 +
D(MP("E",E,N));
 +
</asy>
 +
 +
[[2015 AIME II Problems/Problem 15 | Solution]]
  
−
{{AIME box|year=2015|n=II|before=[[2014 AIME I]], [[2014 AIME II]]|after=[[2016 AIME]]}}
+
{{AIME box|year=2015|n=II|before=[[2015 AIME I Problems]]|after=[[2016 AIME I Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:15, 13 January 2024

2015 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

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

Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.

Solution

Problem 2

In a new school, $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 3

Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.

Solution

Problem 4

In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$, and the altitude to these bases has length $\log 16$. The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.

Solution

Problem 5

Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.

Solution

Problem 6

Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?"

After some calculations, Jon says, "There is more than one such polynomial."

Steve says, "You're right. Here is the value of $a$." He writes down a positive integer and asks, "Can you tell me the value of $c$?"

Jon says, "There are still two possible values of $c$."

Find the sum of the two possible values of $c$.

Solution

Problem 7

Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = w$, the area of $PQRS$ can be expressed as the quadratic polynomial

\[\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.\]

Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 8

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 9

A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.

[asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]

Solution

Problem 10

Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.

Solution

Problem 11

The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 12

There are $2^{10} = 1024$ possible $10$-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than $3$ adjacent letters that are identical.

Solution

Problem 13

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$.

Solution

Problem 14

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Solution

Problem 15

Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ such that $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]

Solution

2015 AIME II (Problems • Answer Key • Resources)
Preceded by
2015 AIME I Problems 		Followed by
2016 AIME I Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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