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2008 AMC 12A problems

All 25 problems from the 2008 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 1A bakery owner turns on his doughnut machine at 8: 30 am. At 11: 10 am the machine has completed one third of the day’s job. At what time will the…Algebra
  2. 2Problem 2What is the reciprocal of 1/2 + 2/3?Algebra
  3. 3Problem 3Suppose that 2/3 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as 1/2 of 5 bananas?Algebra
  4. 4Problem 4Which of the following is equal to the product 8/4 · 12/8 · 16/12 … 4n + 4/4n … 2008/2004?Algebra
  5. 5Problem 5Suppose that 2x/3 - x/6 is an integer. Which of the following statements must be true about x?Algebra
  6. 6Problem 6Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a $90 rebate, and store B…Algebra
  7. 7Problem 7While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The…Algebra
  8. 8Problem 8What is the volume of a cube whose surface area is twice that of a cube with volume 1?Geometry
  9. 9Problem 9Older television screens have an aspect ratio of 4:3. That is, the ratio of the width to the height is 4:3. The aspect ratio of many movies is not…Algebra
  10. 10Problem 10Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch…Algebra
  11. 11Problem 11Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the 13 visible numbers have the…Algebra
  12. 12Problem 12A function f has domain [0, 2] and range [0, 1]. (The notation [a, b] denotes {x: a ≤ x ≤ b}.) What are the domain and range, respectively, of the…Algebra
  13. 13Problem 13Points A and B lie on a circle centered at O, and ∠ AOB = 60°. A second circle is internally tangent to the first and tangent to both OA and OB. What…Geometry
  14. 14Problem 14What is the area of the region defined by the inequality |3x - 18| + |2y + 7| ≤ 3?Geometry
  15. 15Problem 15Let k = 2008^2 + 2^2008. What is the units digit of k^2 + 2^k?Number Theory
  16. 16Problem 16The numbers log (a^3 b^7), log (a^5 b^12), and log (a^8 b^15) are the first three terms of an arithmetic sequence, and the 12th term of the sequence…Algebra
  17. 17Problem 17Let a_1, a_2, … be a sequence of integers determined by the rule a_n = frac a_n-12 if a_n-1 is even and a_n = 3a_n-1 + 1 if a_n-1 is odd. For how…Algebra
  18. 18Problem 18Triangle ABC, with sides of length 5, 6, and 7, has one vertex on the positive x-axis, one on the positive y-axis, and one on the positive z-axis…Geometry
  19. 19Problem 19In the expansion of (1 + x + x^2 + … + x^27) · (1 + x + x^2 + … + x^14)^2, what is the coefficient of x^28?Algebra
  20. 20Problem 20Triangle ABC has AC = 3, BC = 4, and AB = 5. Point D is on AB, and CD bisects the right angle. The inscribed circles of △ ADC and △ BCD have radii…Geometry
  21. 21Problem 21A permutation (a_1, a_2, a_3, a_4, a_5) of (1, 2, 3, 4, 5) is heavy-tailed if a_1 + a_2 < a_4 + a_5. What is the number of heavy-tailed permutations?Counting & Probability
  22. 22Problem 22A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned…Geometry
  23. 23Problem 23The solutions of the equation z^4 + 4z^3 i - 6z^2 - 4zi - i = 0 are the vertices of a convex polygon in the complex plane. What is the area of the…Algebra
  24. 24Problem 24Triangle ABC has ∠ C = 60° and BC = 4. Point D is the midpoint of BC. What is the largest possible value of tan (∠ BAD)?Geometry
  25. 25Problem 25A sequence (a_1, b_1), (a_2, b_2), (a_3, b_3), … of points in the coordinate plane satisfies (a_n+1, b_n+1) = (√(3) a_n - b_n, √(3) b_n + a_n) (n =…Algebra

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