2004 AMC 12A Problems

2004 AMC 12A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. 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). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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Problem 1

Alicia earns $20$ dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?

$\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D) } 2.9 \qquad \text{(E) } 29$

Solution

Problem 2

On the AMC 12, each correct answer is worth $6$ points, each incorrect answer is worth $0$ points, and each problem left unanswered is worth $2.5$ points. If Charlyn leaves $8$ of the $25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $100$ ?

$\text{(A) } 11 \qquad \text{(B) } 13 \qquad \text{(C) } 14 \qquad \text{(D) } 16 \qquad \text{(E) } 17$

Solution

Problem 3

For how many ordered pairs of positive integers $(x,y)$ is $x+2y=100$ ?

$\text{(A) } 33 \qquad \text{(B) } 49 \qquad \text{(C) } 50 \qquad \text{(D) } 99 \qquad \text{(E) } 100$

Solution

Problem 4

Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?

$\text{(A) } 22 \qquad \text{(B) } 23 \qquad \text{(C) } 24 \qquad \text{(D) } 25 \qquad \text{(E) } 26$

Solution

Problem 5

The graph of the line $y=mx+b$ is shown. Which of the following is true?

$\text{(A) } mb<-1 \qquad \text{(B) } -1<mb<0 \qquad \text{(C) } mb=0 \qquad \text{(D) } 0<mb<1 \qquad \text{(E) } mb>1$

Solution

Problem 6

Let $U=2\cdot 2004^{2005}$ , $V=2004^{2005}$ , $W=2003\cdot 2004^{2004}$ , $X=2\cdot 2004^{2004}$ , $Y=2004^{2004}$ and $Z=2004^{2003}$ . Which of the following is the largest?

$\text{(A) } U-V \qquad \text{(B) } V-W \qquad \text{(C) } W-X \qquad \text{(D) } X-Y \qquad \text{(E) } Y-Z \qquad$

Solution

Problem 7

A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $A$ , $B$ and $C$ start with $15$ , $14$ and $13$ tokens, respectively. How many rounds will there be in the game?

$\text{(A) } 36 \qquad \text{(B) } 37 \qquad \text{(C) } 38 \qquad \text{(D) } 39 \qquad \text{(E) } 40 \qquad$

Solution

Problem 8

In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$ , $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB=4$ , $BC=6$ , $AE=8$ , and $\overline{AC}$ and $\overline{BE}$ intersect at $D$ . What is the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$ ?

$\text{(A) } 2 \qquad \text{(B) } 4 \qquad \text{(C) } 5 \qquad \text{(D) } 8 \qquad \text{(E) } 9 \qquad$ $[asy] size(150); defaultpen(linewidth(0.4)); //Variable Declarations pair A, B, C, D, E; //Variable Definitions A=(0, 0); B=(4, 0); C=(4, 6); E=(0, 8); D=extension(A,C,B,E); //Initial Diagram draw(A--B--C--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,3N); label("$E$",E,NW); //Side labels label("$4$",A--B,S); label("$8$",A--E,W); label("$6$",B--C,ENE); [/asy]$

Solution

Problem 9

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased?

$\text {(A) } 10\% \qquad \text {(B) } 25\% \qquad \text {(C) } 36\% \qquad \text {(D) } 50\% \qquad \text {(E) }60\%$

Solution

Problem 10

The sum of $49$ consecutive integers is $7^5$ . What is their median?

$\text {(A) } 7 \qquad \text {(B) } 7^2\qquad \text {(C) } 7^3\qquad \text {(D) } 7^4\qquad \text {(E) }7^5$

Solution

Problem 11

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?

$\text {(A) }0 \qquad \text {(B) } 1 \qquad \text {(C) } 2 \qquad \text {(D) } 3\qquad \text {(E) }4$

Solution

Problem 12

Let $A = (0,9)$ and $B = (0,12)$ . Points $A'$ and $B'$ are on the line $y = x$ , and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C = (2,8)$ . What is the length of $\overline{A'B'}$ ?

$\text {(A) } 2 \qquad \text {(B) } 2\sqrt2 \qquad \text {(C) } 3 \qquad \text {(D) } 2 + \sqrt 2\qquad \text {(E) }3\sqrt 2$

Solution

Problem 13

Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $- 1$ , $0$ , or $1$ . How many distinct lines pass through at least two members of $S$ ?

$\text {(A) } 8 \qquad \text {(B) } 20 \qquad \text {(C) } 24 \qquad \text {(D) } 27\qquad \text {(E) }36$

Solution

Problem 14

A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

$\text {(A) } 1 \qquad \text {(B) } 4 \qquad \text {(C) } 36 \qquad \text {(D) } 49 \qquad \text {(E) }81$

Solution

Problem 15

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $100$ meters. They next meet after Sally has run $150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?

$\text {(A) }250 \qquad \text {(B) }300 \qquad \text {(C) }350 \qquad \text {(D) } 400\qquad \text {(E) }500$

Solution

Problem 16

The set of all real numbers $x$ for which

$\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))$

is defined is $\{x\mid x > c\}$ . What is the value of $c$ ?

$\textbf {(A) } 0\qquad \textbf {(B) }2001^{2002} \qquad \textbf {(C) }2002^{2003} \qquad \textbf {(D) }2003^{2004} \qquad \textbf {(E) }2001^{2002^{2003}}$

Solution

Problem 17

Let $f$ be a function with the following properties:

(i) $f(1) = 1$ , and

(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$ .

What is the value of $f(2^{100})$ ?

$\text {(A)}\ 1 \qquad \text {(B)}\ 2^{99} \qquad \text {(C)}\ 2^{100} \qquad \text {(D)}\ 2^{4950} \qquad \text {(E)}\ 2^{9999}$

Solution

Problem 18

Square $ABCD$ has side length $2$ . A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$ . What is the length of $\overline{CE}$ ?

$[asy] size(100); defaultpen(fontsize(10)); pair A=(0,0), B=(2,0), C=(2,2), D=(0,2), E=(0,1/2); draw(A--B--C--D--cycle);draw(C--E); draw(Arc((1,0),1,0,180)); label("$A$",A,(-1,-1)); label("$B$",B,( 1,-1)); label("$C$",C,( 1, 1)); label("$D$",D,(-1, 1)); label("$E$",E,(-1, 0)); [/asy]$

$\text {(A) } \frac {2 + \sqrt5}{2} \qquad \text {(B) } \sqrt 5 \qquad \text {(C) } \sqrt 6 \qquad \text {(D) } \frac52 \qquad \text {(E) }5 - \sqrt5$

Solution

Problem 19

Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$ . Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$ . What is the radius of circle $B$ ?

$[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0); draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9)); label("$A$", A); label("$B$", B); label("$C$", C); label("$D$", (-1.2,1.8)); [/asy]$

$\text{(A) } \frac23 \qquad \text{(B) } \frac {\sqrt3}{2} \qquad \text{(C) } \frac78 \qquad \text{(D) } \frac89 \qquad \text{(E) } \frac {1 + \sqrt3}{3}$

Solution

Problem 20

Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$ , respectively, are rounded to the nearest integer. What is the probability that $A + B = C$ ?

$\text {(A) } \frac14 \qquad \text {(B) } \frac13 \qquad \text {(C) } \frac12 \qquad \text {(D) } \frac23 \qquad \text {(E) }\frac34$

Solution

Problem 21

If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5$ , what is the value of $\cos{2\theta}$ ?

$\text {(A) } \frac15 \qquad \text {(B) } \frac25 \qquad \text {(C) } \frac {\sqrt5}{5}\qquad \text {(D) } \frac35 \qquad \text {(E) }\frac45$

Solution

Problem 22

Three mutually tangent spheres of radius $1$ rest on a horizontal plane. A sphere of radius $2$ rests on them. What is the distance from the plane to the top of the larger sphere?

$\text {(A) } 3 + \frac {\sqrt {30}}{2} \qquad \text {(B) } 3 + \frac {\sqrt {69}}{3} \qquad \text {(C) } 3 + \frac {\sqrt {123}}{4}\qquad \text {(D) } \frac {52}{9}\qquad \text {(E) }3 + 2\sqrt2$

Solution

Problem 23

A polynomial

$P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0$

has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$ , $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$ , and

$\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.$

Which of the following quantities can be a nonzero number?

$\text {(A) } c_0 \qquad \text {(B) } c_{2003} \qquad \text {(C) } b_2b_3...b_{2004} \qquad \text {(D) } \sum_{k = 1}^{2004}{a_k} \qquad \text {(E) }\sum_{k = 1}^{2004}{c_k}$

Solution

Problem 24

A plane contains points $A$ and $B$ with $AB = 1$ . Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}$ . What is the area of $S$ ?

$\text {(A) } 2\pi + \sqrt3 \qquad \text {(B) } \frac {8\pi}{3} \qquad \text {(C) } 3\pi - \frac {\sqrt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \text {(E) }4\pi - 2\sqrt3$

Solution

Problem 25

For each integer $n\geq 4$ , let $a_n$ denote the base- $n$ number $0.\overline{133}_n$ . The product $a_4a_5...a_{99}$ can be expressed as $\frac {m}{n!}$ , where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$ ?

$\text {(A) } 98 \qquad \text {(B) } 101 \qquad \text {(C) } 132\qquad \text {(D) } 798\qquad \text {(E) }962$

Solution

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by 2003 AMC 12B Problems	Followed by 2004 AMC 12B Problems
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All AMC 12 Problems and Solutions

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2004 AMC 12A Problems

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

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also