Difference between revisions of "2009 AMC 12A Problems"

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{{AMC12 Problems|year=2009|ab=A}}
 
== Problem 1 ==
 
== Problem 1 ==
 +
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took <math>h</math> hours and <math>m</math> minutes, with <math>0 < m < 60</math>, what is <math>h + m</math>?
 +
 +
<math>\textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 53 \qquad \textbf{(E)}\ 54</math>
  
 
[[2009 AMC 12A Problems/Problem 1|Solution]]
 
[[2009 AMC 12A Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
Which of the following is equal to <math>1 + \frac {1}{1 + \frac {1}{1 + 1}}</math>?
 +
 +
<math>\textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {3}{2} \qquad \textbf{(C)}\ \frac {5}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3</math>
  
 
[[2009 AMC 12A Problems/Problem 2|Solution]]
 
[[2009 AMC 12A Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
What number is one third of the way from <math>\frac14</math> to <math>\frac34</math>?
 +
 +
<math>\textbf{(A)}\ \frac {1}{3} \qquad \textbf{(B)}\ \frac {5}{12} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)}\ \frac {7}{12} \qquad \textbf{(E)}\ \frac {2}{3}</math>
  
 
[[2009 AMC 12A Problems/Problem 3|Solution]]
 
[[2009 AMC 12A Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could <em>not</em> be the total value of the four coins, in cents?
 +
 +
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55</math>
  
 
[[2009 AMC 12A Problems/Problem 4|Solution]]
 
[[2009 AMC 12A Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
One dimension of a cube is increased by <math>1</math>, another is decreased by <math>1</math>, and the third is left unchanged. The volume of the new rectangular solid is <math>5</math> less than that of the cube. What was the volume of the cube?
 +
 +
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216</math>
  
 
[[2009 AMC 12A Problems/Problem 5|Solution]]
 
[[2009 AMC 12A Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>?
 +
 +
<math>\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m</math>
  
 
[[2009 AMC 12A Problems/Problem 6|Solution]]
 
[[2009 AMC 12A Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
The first three terms of an arithmetic sequence are <math>2x - 3</math>, <math>5x - 11</math>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>?
 +
 +
<math>\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037</math>
  
 
[[2009 AMC 12A Problems/Problem 7|Solution]]
 
[[2009 AMC 12A Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Four congruent rectangles are placed as shown. The area of the outer square is <math>4</math> times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
 +
<center><asy>
 +
unitsize(6mm);
 +
defaultpen(linewidth(.8pt));
 +
 +
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
 +
draw(p);
 +
draw(rotate(90)*p);
 +
draw(rotate(180)*p);
 +
draw(rotate(270)*p);
 +
</asy></center>
 +
 +
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 + \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4</math>
  
 
[[2009 AMC 12A Problems/Problem 8|Solution]]
 
[[2009 AMC 12A Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
 +
Suppose that <math>f(x+3)=3x^2 + 7x + 4</math> and <math>f(x)=ax^2 + bx + c</math>. What is <math>a+b+c</math>?
 +
 +
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3</math>
  
 
[[2009 AMC 12A Problems/Problem 9|Solution]]
 
[[2009 AMC 12A Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
In quadrilateral <math>ABCD</math>, <math>AB = 5</math>, <math>BC = 17</math>, <math>CD = 5</math>, <math>DA = 9</math>, and <math>BD</math> is an integer. What is <math>BD</math>?
 +
<center><asy>
 +
unitsize(4mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
dotfactor=4;
 +
 +
pair C=(0,0), B=(17,0);
 +
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
 +
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
 +
pair[] dotted={A,B,C,D};
 +
 +
draw(D--A--B--C--D--B);
 +
dot(dotted);
 +
label("$D$",D,NW);
 +
label("$C$",C,W);
 +
label("$B$",B,E);
 +
label("$A$",A,NE);
 +
</asy></center><math>\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15</math>
  
 
[[2009 AMC 12A Problems/Problem 10|Solution]]
 
[[2009 AMC 12A Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
The figures <math>F_1</math>, <math>F_2</math>, <math>F_3</math>, and <math>F_4</math> shown are the first in a sequence of figures. For <math>n\ge3</math>, <math>F_n</math> is constructed from <math>F_{n - 1}</math> by surrounding it with a square and placing one more diamond on each side of the new square than <math>F_{n - 1}</math> had on each side of its outside square. For example, figure <math>F_3</math> has <math>13</math> diamonds. How many diamonds are there in figure <math>F_{20}</math>?
 +
<center><asy>
 +
unitsize(3mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
 +
path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
 +
marker m=marker(scale(5)*d,Fill);
 +
path f1=(0,0);
 +
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
 +
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
 +
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
 +
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
 +
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
 +
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
 +
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
 +
(-2,2)--(-3,3);
 +
 +
draw(f1,m);
 +
draw(shift(5,0)*f2,m);
 +
draw(shift(5,0)*g2);
 +
draw(shift(12,0)*f3,m);
 +
draw(shift(12,0)*g3);
 +
draw(shift(21,0)*f4,m);
 +
draw(shift(21,0)*g4);
 +
label("$F_1$",(0,-4));
 +
label("$F_2$",(5,-4));
 +
label("$F_3$",(12,-4));
 +
label("$F_4$",(21,-4));
 +
</asy></center><math>\textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761</math>
  
 
[[2009 AMC 12A Problems/Problem 11|Solution]]
 
[[2009 AMC 12A Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
How many positive integers less than <math>1000</math> are <math>6</math> times the sum of their digits?
 +
 +
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12</math>
  
 
[[2009 AMC 12A Problems/Problem 12|Solution]]
 
[[2009 AMC 12A Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
A ship sails <math>10</math> miles in a straight line from <math>A</math> to <math>B</math>, turns through an angle between <math>45^{\circ}</math> and <math>60^{\circ}</math>, and then sails another <math>20</math> miles to <math>C</math>. Let <math>AC</math> be measured in miles. Which of the following intervals contains <math>AC^2</math>?
 +
<asy>
 +
unitsize(2mm);
 +
defaultpen(linewidth(.8pt)+fontsize(10pt));
 +
dotfactor=4;
 +
 +
pair B=(0,0), A=(-10,0), C=20*dir(50);
 +
 +
draw(A--B--C);
 +
draw(A--C,linetype("4 4"));
 +
 +
dot(A);
 +
dot(B);
 +
dot(C);
 +
label("$10$",midpoint(A--B),S);
 +
label("$20$",midpoint(B--C),SE);
 +
label("$A$",A,SW);
 +
label("$B$",B,SE);
 +
label("$C$",C,NE);
 +
</asy>
 +
 +
<math>\textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800] \qquad \textbf{(E)}\ [800,900]</math>
  
 
[[2009 AMC 12A Problems/Problem 13|Solution]]
 
[[2009 AMC 12A Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
A triangle has vertices <math>(0,0)</math>, <math>(1,1)</math>, and <math>(6m,0)</math>, and the line <math>y = mx</math> divides the triangle into two triangles of equal area. What is the sum of all possible values of <math>m</math>?
 +
 +
<math>\textbf{(A)} - \!\frac {1}{3} \qquad \textbf{(B)} - \!\frac {1}{6} \qquad \textbf{(C)}\ \frac {1}{6} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}</math>
  
 
[[2009 AMC 12A Problems/Problem 14|Solution]]
 
[[2009 AMC 12A Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
For what value of <math>n</math> is <math>i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i</math>?
 +
 +
Note: here <math>i = \sqrt { - 1}</math>.
 +
 +
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 97 \qquad \textbf{(E)}\ 98</math>
  
 
[[2009 AMC 12A Problems/Problem 15|Solution]]
 
[[2009 AMC 12A Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
A circle with center <math>C</math> is tangent to the positive <math>x</math> and <math>y</math>-axes and externally tangent to the circle centered at <math>(3,0)</math> with radius <math>1</math>. What is the sum of all possible radii of the circle with center <math>C</math>?
 +
 +
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9</math>
  
 
[[2009 AMC 12A Problems/Problem 16|Solution]]
 
[[2009 AMC 12A Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
Let <math>a + ar_1 + ar_1^2 + ar_1^3 + \cdots</math> and <math>a + ar_2 + ar_2^2 + ar_2^3 + \cdots</math> be two different infinite geometric series of positive numbers with the same first term.  The sum of the first series is <math>r_1</math>, and the sum of the second series is <math>r_2</math>.  What is <math>r_1 + r_2</math>?
 +
 +
<math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {1 + \sqrt {5}}{2}\qquad \textbf{(E)}\ 2</math>
  
 
[[2009 AMC 12A Problems/Problem 17|Solution]]
 
[[2009 AMC 12A Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
For <math>k > 0</math>, let <math>I_k = 10\ldots 064</math>, where there are <math>k</math> zeros between the <math>1</math> and the <math>6</math>.  Let <math>N(k)</math> be the number of factors of <math>2</math> in the prime factorization of <math>I_k</math>.  What is the maximum value of <math>N(k)</math>?
 +
 +
<math>\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10</math>
  
 
[[2009 AMC 12A Problems/Problem 18|Solution]]
 
[[2009 AMC 12A Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles.  Bethany did the same with a regular heptagon (7 sides).  The areas of the two regions were <math>A</math> and <math>B</math>, respectively.  Each polygon had a side length of <math>2</math>.  Which of the following is true?
 +
 +
<math>\textbf{(A)}\ A = \frac {25}{49}B\qquad \textbf{(B)}\ A = \frac {5}{7}B\qquad \textbf{(C)}\ A = B\qquad \textbf{(D)}\ A </math> <math>= \frac {7}{5}B\qquad \textbf{(E)}\ A = \frac {49}{25}B</math>
  
 
[[2009 AMC 12A Problems/Problem 19|Solution]]
 
[[2009 AMC 12A Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
Convex quadrilateral <math>ABCD</math> has <math>AB = 9</math> and <math>CD = 12</math>.  Diagonals <math>AC</math> and <math>BD</math> intersect at <math>E</math>, <math>AC = 14</math>, and <math>\triangle AED</math> and <math>\triangle BEC</math> have equal areas.  What is <math>AE</math>?
 +
 +
<math>\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad \textbf{(E)}\ 6</math>
  
 
[[2009 AMC 12A Problems/Problem 20|Solution]]
 
[[2009 AMC 12A Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
Let <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are complex numbers.  Suppose that
 +
 +
<center><math>p(2009 + 9002\pi i) = p(2009) = p(9002) = 0</math></center>
 +
 +
What is the number of nonreal zeros of <math>x^{12} + ax^8 + bx^4 + c</math>?
 +
 +
<math>\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12</math>
  
 
[[2009 AMC 12A Problems/Problem 21|Solution]]
 
[[2009 AMC 12A Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
A regular octahedron has side length <math>1</math>. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area <math>\frac {a\sqrt {b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>a</math> and <math>c</math> are relatively prime, and <math>b</math> is not divisible by the square of any prime. What is <math>a + b + c</math>?
 +
 +
<math>\textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14</math>
  
 
[[2009 AMC 12A Problems/Problem 22|Solution]]
 
[[2009 AMC 12A Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. Then <math>x_4 - x_1 = m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?
 +
 +
<math>\textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752 \qquad \textbf{(E)}\ 802</math>
  
 
[[2009 AMC 12A Problems/Problem 23|Solution]]
 
[[2009 AMC 12A Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
The ''tower function of twos'' is defined recursively as follows: <math>T(1) = 2</math> and <math>T(n + 1) = 2^{T(n)}</math> for <math>n\ge1</math>. Let <math>A = (T(2009))^{T(2009)}</math> and <math>B = (T(2009))^A</math>. What is the largest integer <math>k</math> such that
 +
 +
<cmath>\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}</cmath>
 +
 +
is defined?
 +
 +
<math>\textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013</math>
  
 
[[2009 AMC 12A Problems/Problem 24|Solution]]
 
[[2009 AMC 12A Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
The first two terms of a sequence are <math>a_1 = 1</math> and <math>a_2 = \frac {1}{\sqrt3}</math>. For <math>n\ge1</math>,
 +
 +
<center><cmath>a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.</cmath></center>
 +
 +
What is <math>|a_{2009}|</math>?
 +
 +
<math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 - \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 + \sqrt3</math>
  
 
[[2009 AMC 12A Problems/Problem 25|Solution]]
 
[[2009 AMC 12A Problems/Problem 25|Solution]]
 +
 +
==See also==
 +
 +
{{AMC12 box|year=2009|ab=A|before=[[2008 AMC 12B Problems]]|after=[[2009 AMC 12B Problems]]}}
 +
 +
{{MAA Notice}}

Latest revision as of 12:13, 12 August 2020

2009 AMC 12A (Answer Key)
Printable version: Wiki | AoPS ResourcesPDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Problem 1

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is $h + m$?

$\textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 53 \qquad \textbf{(E)}\ 54$

Solution

Problem 2

Which of the following is equal to $1 + \frac {1}{1 + \frac {1}{1 + 1}}$?

$\textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {3}{2} \qquad \textbf{(C)}\ \frac {5}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Solution

Problem 3

What number is one third of the way from $\frac14$ to $\frac34$?

$\textbf{(A)}\ \frac {1}{3} \qquad \textbf{(B)}\ \frac {5}{12} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)}\ \frac {7}{12} \qquad \textbf{(E)}\ \frac {2}{3}$

Solution

Problem 4

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Solution

Problem 5

One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?

$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

Solution

Problem 6

Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?

$\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m$

Solution

Problem 7

The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$?

$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Solution

Problem 8

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

[asy] unitsize(6mm); defaultpen(linewidth(.8pt));  path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy]

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 + \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4$

Solution

Problem 9

Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$?

$\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Solution

Problem 10

In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?

[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4;  pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D};  draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE); [/asy]

$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Solution

Problem 11

The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?

[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt));  path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3);  draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label("$F_1$",(0,-4)); label("$F_2$",(5,-4)); label("$F_3$",(12,-4)); label("$F_4$",(21,-4)); [/asy]

$\textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$

Solution

Problem 12

How many positive integers less than $1000$ are $6$ times the sum of their digits?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$

Solution

Problem 13

A ship sails $10$ miles in a straight line from $A$ to $B$, turns through an angle between $45^{\circ}$ and $60^{\circ}$, and then sails another $20$ miles to $C$. Let $AC$ be measured in miles. Which of the following intervals contains $AC^2$? [asy] unitsize(2mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4;  pair B=(0,0), A=(-10,0), C=20*dir(50);  draw(A--B--C); draw(A--C,linetype("4 4"));  dot(A); dot(B); dot(C); label("$10$",midpoint(A--B),S); label("$20$",midpoint(B--C),SE); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); [/asy]

$\textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800] \qquad \textbf{(E)}\ [800,900]$

Solution

Problem 14

A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?

$\textbf{(A)} - \!\frac {1}{3} \qquad \textbf{(B)} - \!\frac {1}{6} \qquad \textbf{(C)}\ \frac {1}{6} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$

Solution

Problem 15

For what value of $n$ is $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$?

Note: here $i = \sqrt { - 1}$.

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 97 \qquad \textbf{(E)}\ 98$

Solution

Problem 16

A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Solution

Problem 17

Let $a + ar_1 + ar_1^2 + ar_1^3 + \cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1$, and the sum of the second series is $r_2$. What is $r_1 + r_2$?

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {1 + \sqrt {5}}{2}\qquad \textbf{(E)}\ 2$

Solution

Problem 18

For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?

$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

Solution

Problem 19

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$, respectively. Each polygon had a side length of $2$. Which of the following is true?

$\textbf{(A)}\ A = \frac {25}{49}B\qquad \textbf{(B)}\ A = \frac {5}{7}B\qquad \textbf{(C)}\ A = B\qquad \textbf{(D)}\ A$ $= \frac {7}{5}B\qquad \textbf{(E)}\ A = \frac {49}{25}B$

Solution

Problem 20

Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad \textbf{(E)}\ 6$

Solution

Problem 21

Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that

$p(2009 + 9002\pi i) = p(2009) = p(9002) = 0$

What is the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$?

$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Solution

Problem 22

A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?

$\textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$

Solution

Problem 23

Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$?

$\textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752 \qquad \textbf{(E)}\ 802$

Solution

Problem 24

The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ such that

\[\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}\]

is defined?

$\textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

Solution

Problem 25

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,

\[a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.\]

What is $|a_{2009}|$?

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 - \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 + \sqrt3$

Solution

See also

2009 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
2008 AMC 12B Problems
Followed by
2009 AMC 12B Problems
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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