2004 AMC 10A Problems
2004 AMC 10A (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
You and five friends need to raise dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
Problem 2
For any three real numbers , , and , with , the operation is defined by: What is ?
Problem 3
Alicia earns 20 dollars per hour, of which is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
Problem 4
What is the value of if ?
Problem 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?
Problem 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-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
Problem 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 below. The stack is completed by a single row of oranges. How many oranges are in the stack?
Problem 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 in the discard pile. The game ends when some player runs out of tokens. Players , , and start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?
Problem 9
In the figure, and are right angles. , and and intersect at . What is the difference between the areas of and ?
Problem 10
Coin is flipped three times and coin is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?
Problem 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 without altering the volume, by what percent must the height be decreased?
Problem 12
Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?
Problem 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?
Problem 14
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is cents. If she had one more quarter, the average would be cents. How many dimes does she have in her purse?
Problem 15
Given that and , what is the largest possible value of ?
Problem 16
The grid shown contains a collection of squares with sizes from to . How many of these squares contain the black center square?
Problem 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?
Problem 18
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 of the geometric progression?
Problem 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?
Problem 20
Points and are located on square so that is equilateral. What is the ratio of the area of to that of ?
Problem 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 of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: radians is degrees.)
Problem 22
Square has side length . A semicircle with diameter is constructed inside the square, and the tangent to the semicircle from intersects side at . What is the length of ?
Problem 23
Circles , , and are externally tangent to each other and internally tangent to circle . Circles and are congruent. Circle has radius and passes through the center of . What is the radius of circle ?
Problem 24
Let , be a sequence with the following properties.
- (i) , and
- (ii) for any positive integer .
What is the value of ?
Problem 25
Three pairwise-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?
See also
2004 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2003 AMC 10B Problems |
Followed by 2004 AMC 10B 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.