ΒΆ 2004 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2004 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2004 AMC 12A math contest by visiting Random Math AMC 12A 2004 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2004 AMC 12A problems, please refer below:
Problem 1: Alicia earns 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: On the AMC , each correct answer is worth points, each incorrect answer is worth points, and each problem left unanswered is worth points. If Charlyn leaves of the problems unanswered, how many of the remaining problems must she answer correctly in order to score at least
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: For how many ordered pairs of positive integers is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: Bertha has daughters and no sons. Some of her daughters have daughters, and the rest have none. Bertha has a total of daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: The graph of a line is shown. Which of the following is true?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: Let , and . Which of the following is largest?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: 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 , and start with , and tokens, respectively. How many rounds will there be in the game?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: In the Figure, and are right angles, , and and intersect at . What is the difference between the areas of and
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 9: 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: The sum of consecutive integers is . What is their median?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: 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 value would be cents. How many dimes does she have in her purse?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: Let and . Points and are on the line , and and intersect at . What is the length of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: Let be the set of points in the coordinate plane, where each of and may be , or . How many distinct lines pass through at least two members of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: A sequence of three real numbers forms an arithmetic progression with a first term of . If is added to the second term and 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They rst meet after Brenda has run meters. They next meet after Sally has run meters past their rst meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: The set of all real numbers for which
is defined as . What is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: Let be a function with the following properties:
(i) , and
(ii) for any positive integer .
What is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: 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 ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: Circles , and are externally tangent to each other and internally tangent to circle . Circles and are congruent. Circle has radius 1 and passes through the center of . What is the radius of circle ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: Select numbers and between and independently and at random, and let be their sum. Let , and be the results when , and , respectively, are rounded to the nearest integer. What is the probability that ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: If , what is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: Three mutually tangent spheres of radius rest on a horizontal plane. A sphere of radius rests on them. What is the distance from the plane to the top of the larger sphere?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: A polynomial
has real coefficients with and 2004 distinct complex zeros , with and real, , and
Which of the following quantities can be a nonzero number?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: A plane contains points and with . Let be the union of all disks of radius in the plane that cover . What is the area of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: For each integer , let denote the base- number . The product can be expressed as , where and are positive integers and is as small as possible. What is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions