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2004 AMC 10A Problems
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History 1 Problem 1
2 Problem 2
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3 Problem 3
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4 Problem 4
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5 Problem 5
Help 6 Problem 6
What links here 7 Problem 7
Special pages 8 Problem 8
9 Problem 9
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10 Problem 10
Search 11 Problem 11
12 Problem 12
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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?
Solution
Problem 2
For any three real numbers , , and , with , the operation is defined by: What is
?
Solution
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?
Solution
Problem 4
What is the value of if ?
Solution
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?
Solution
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?
Solution
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?
Solution
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?
Solution
Problem 9
In the figure, and are right angles. , and and intersect at . What is the difference
between the areas of and ?
Solution
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?
Solution
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?
Solution
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?
Solution
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?
Solution
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?
Solution
Problem 15
Given that and , what is the largest possible value of ?
Solution
Problem 16
The grid shown contains a collection of squares with sizes from to . How many of these squares contain the black center
square?
Solution
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?
Solution
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?
Solution
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?
Solution
Problem 20
Points and are located on square so that is equilateral. What is the ratio of the area of to that of ?
Solution
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.)
Solution
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 ?
Solution
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 ?
Solution
Problem 24
Let , be a sequence with the following properties.
(i) , and
(ii) for any positive integer .
What is the value of ?
Solution
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?
Solution
See also
2004 AMC 10A (Problems • Answer Key • Resources)
Preceded by Followed by
2003 AMC 10B Problems 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
AMC 10
AMC 10 Problems and Solutions
AMC Problems and Solutions
Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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