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2004 AMC 12A problems

All 25 problems from the 2004 AMC 12A, 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 1Alicia 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
  2. 2Problem 2On the AMC 12, each correct answer is worth 6 points, each incorrect answer is worth 0 points, and each problem left unanswered is worth 2.5 points…Algebra
  3. 3Problem 3For how many ordered pairs of positive integers (x, y) is x + 2y = 100?Number Theory
  4. 4Problem 4Bertha 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
  5. 5Problem 5The graph of a line y = mx + b is shown. Which of the following is true?Geometry
  6. 6Problem 6Let U = 2 · 2004^2005, V = 2004^2005, W = 2003 · 2004^2004, X = 2 · 2004^2004, Y = 2004^2004 and Z = 2004^2003. Which of the following is largest?Algebra
  7. 7Problem 7A 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…
  8. 8Problem 8In 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
  9. 9Problem 9A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars…Geometry
  10. 10Problem 10The sum of 49 consecutive integers is 7^5. What is their median?Algebra
  11. 11Problem 11The 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
  12. 12Problem 12Let A = (0, 9) and B = (0, 12). Points A' and B' are on the line y = x, and AA' and BB' intersect at C = (2, 8). What is the length of A'B'?Geometry
  13. 13Problem 13Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least…Geometry
  14. 14Problem 14A 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
  15. 15Problem 15Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100…Algebra
  16. 16Problem 16The set of all real numbers x for which small log _2004(log _2003(log _2002(log _2001 x))) is defined is {x mid x > c}. What is the value of c?Algebra
  17. 17Problem 17Let f be a function with the following properties: (i) f(1) = 1, and (ii) f(2n) = n · f(n) for any positive integer n. What is the value of f(2^100)?Algebra
  18. 18Problem 18Square 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
  19. 19Problem 19Circles 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
  20. 20Problem 20Select numbers a and b between 0 and 1 independently and at random, and let c be their sum. Let A, B, and C be the results when a, b, and c…Counting & Probability
  21. 21Problem 21If sum _n=0^∞ cos ^2n θ = 5, what is the value of cos 2θ?Algebra
  22. 22Problem 22Three 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
  23. 23Problem 23A polynomial P(x) = c_2004 x^2004 + c_2003 x^2003 + … + c_1 x + c_0 has real coefficients with c_2004 ≠ 0 and 2004 distinct complex zeros z_k = a_k +…Algebra
  24. 24Problem 24A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover AB. What is the area of S?Geometry
  25. 25Problem 25For each integer n ≥ 4, let a_n denote the base-n number 0.133_n. The product a_4 a_5 … a_99 can be expressed as m/n!, where m and n are positive…Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.