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2004 AMC 10A problems

All 25 problems from the 2004 AMC 10A, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1You 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…Algebra
  2. 2Problem 2For any three real numbers a, b, and c, with b ≠ c, the operation ◆ is defined by ◆(a, b, c) = a/b - c. What is ◆(◆(1, 2, 3), ◆(2, 3, 1), ◆(3, 1, 2))?Algebra
  3. 3Problem 3Alicia 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?Algebra
  4. 4Problem 4What is the value of x if |x - 1| = |x - 2|?Algebra
  5. 5Problem 5A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability…Counting & Probability
  6. 6Problem 6Bertha 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…Counting & Probability
  7. 7Problem 7A 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…Algebra
  8. 8Problem 8A 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…
  9. 9Problem 9In the figure, ∠ EAB and ∠ ABC are right angles, AB = 4, BC = 6, AE = 8, and AC and BE intersect at D. What is the difference between the areas of △…Geometry
  10. 10Problem 10Coin 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…Counting & Probability
  11. 11Problem 11A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars…Geometry
  12. 12Problem 12Henry’s Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and…Counting & Probability
  13. 13Problem 13At 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…Counting & Probability
  14. 14Problem 14The 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…Algebra
  15. 15Problem 15Given that -4 ≤ x ≤ -2 and 2 ≤ y ≤ 4, what is the largest possible value of x + y/x?Algebra
  16. 16Problem 16The 5 × 5 grid shown contains a collection of squares with sizes from 1 × 1 to 5 × 5. How many of these squares contain the shaded center square?Counting & Probability
  17. 17Problem 17Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100…Algebra
  18. 18Problem 18A 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…Algebra
  19. 19Problem 19A 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…Geometry
  20. 20Problem 20Points E and F are located on square ABCD so that △ BEF is equilateral. What is the ratio of the area of △ DEF to that of △ ABE?Geometry
  21. 21Problem 21Two 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…Geometry
  22. 22Problem 22Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects…Geometry
  23. 23Problem 23Circles 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…Geometry
  24. 24Problem 24Let a_1, a_2, … be a sequence with the following properties: a_1 = 1, and a_2n = n · a_n for any positive integer n. What is the value of a_2^100?Algebra
  25. 25Problem 25Three 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…Geometry

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.