AMC 10A 2004 Question Paper

Question 1

You and five friends need to raise $$ 1500$ in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?

(a) 250

(b) 300

(d) 7500

(e) 9000

Question 2

For any three real numbers $a, b$, and $c$, with $b \neq c$, the operation ⊗ is defined by: \[ \otimes(a, b, c)=\frac{a}{b-c} \] What is $\otimes(\otimes(1,2,3), \otimes(2,3,1), \otimes(3,1,2))$ ?

(a) $-\frac{1}{2}$

(b) $-\frac{1}{4}$

(d) $\frac{1}{4}$

(e) $\frac{1}{2}$

Question 3

Alicia earns $$ 20$ 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?

(a) 0.0029

(b) 0.029

(d) 2.9

(e) 29

Question 4

What is the value of $x$ if $|x-1|=|x-2|$ ?

(a) $-\frac{1}{2}$

(b) $\frac{1}{2}$

(d) $\frac{3}{2}$

(e) 2

Question 5

A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

(a) $\frac{1}{21}$

(b) $\frac{1}{14}$

(d) $\frac{1}{7}$

(e) $\frac{2}{7}$

Question 6

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-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?

(a) 22

(b) 23

(d) 25

(e) 26 AMC 10 2004

Question 7

A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?

(a) 96

(b) 98

(d) 101

(e) 134

Question 8

A game is played with tokens according to the following rule. 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?

(a) 36

(b) 37

(d) 39

(e) 40

Question 9

In the figure, $\angle E A B$ and $\angle A B C$ are right angles. $A B=4, B C=6, A E=8$, and $\overline{A C}$ and $\overline{B E}$ intersect at $D$. What is the difference between the areas of $\triangle A D E$ and $\triangle B D C$ ?

(a) 2

(b) 4

(d) 8

(e) 9

Question 10

Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

(a) $\frac{19}{128}$

(b) $\frac{23}{128}$

(d) $\frac{35}{128}$

(e) $\frac{1}{2}$

Question 11

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

(a) 10

(b) 25

(d) 50

(e) 60

Question 12

Henry's Hamburger Heaven orders its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose 2004 one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?

(a) 24

(b) 256

(d) 40,320

(e) 120,960

Question 13

At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?

(a) 8

(b) 12

(d) 18

(e) 24

Question 14

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?

(a) 0

(b) 1

(d) 3

(e) 4

Question 15

Given that $-4 \leq x \leq-2$ and $2 \leq y \leq 4$, what is the largest possible value of $(x+y) / x$ ?

(a) -1

(b) $-\frac{1}{2}$

(d) $\frac{1}{2}$

(e) 1

Question 16

The $5 \times 5$ grid shown contains a collection of squares with sizes from $1 \times 1$ to $5 \times 5$. How many of these squares contain the black center square?

(a) 12

(b) 15

(d) 19

(e) 20

Question 17

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?

(a) 250

(b) 300

(d) 400

(e) 500

Question 18

(a) 1

(b) 4

(d) 49

(e) 81

Question 19

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

(a) 120

(b) 180

(d) 360

(e) 480

Question 20

Points $E$ and $F$ are located on square $A B C D$ so that $\triangle B E F$ is equilateral. What is the ratio of the area of $\triangle D E F$ to that of $\triangle A B E$ ? AMC 10 2004

(a) $\frac{4}{3}$

(b) $\frac{3}{2}$

(d) 2

(e) $1+\sqrt{3}$

Question 21

Two distinct lines pass through the center of three concentric circles of radii 3,2 , and 1 . The area of the shaded region in the diagram is 8/13 of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is 180 degrees.)

(a) $\frac{\pi}{8}$

(b) $\frac{\pi}{7}$

(d) $\frac{\pi}{5}$

(e) $\frac{\pi}{4}$

Question 22

Square $A B C D$ has side length 2. A semicircle with diameter $A B$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $A D$ at $E$. What is the length of $C E$ ? AMC 10 2004

(a) $\frac{2+\sqrt{5}}{2}$

(b) $\sqrt{5}$

(d) $\frac{5}{2}$

(e) $5-\sqrt{5}$

Question 23

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 $B$ ?

(a) $\frac{2}{3}$

(b) $\frac{\sqrt{3}}{2}$

(d) $\frac{8}{9}$

(e) $\frac{1+\sqrt{3}}{3}$

Question 24

Let $a_{1}, a_{2}, \cdots$, be a sequence with the following properties.

(i) $a_{1}=1$, and II. $a_{2 n}=n \cdot a_{n}$ for any positive integer $n$. What is the value of $a_{2^{100}}$ ?

(a) 1

(b) $2^{99}$

(d) $2^{4950}$

(e) $2^{9999}$

Question 25

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?

(a) $3+\frac{\sqrt{30}}{2}$

(b) $3+\frac{\sqrt{69}}{3}$

(d) $\frac{52}{9}$

(e) $3+2 \sqrt{2}$