Skip to main content

2012 AMC 12A problems

All 25 problems from the 2012 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 bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?Algebra
  2. 2Problem 2Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5…Algebra
  3. 3Problem 3A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box with twice the height, three times the…Algebra
  4. 4Problem 4In a bag of marbles, 3/5 of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays…Algebra
  5. 5Problem 5A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 280 pieces of fruit. There are twice as many…Algebra
  6. 6Problem 6The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?Algebra
  7. 7Problem 7Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic…Algebra
  8. 8Problem 8An iterative average of the numbers 1, 2, 3, 4, and 5 is computed in the following way. Arrange the five numbers in some order. Find the mean of the…Algebra
  9. 9Problem 9A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not by 100 (such as 2012). The 200th…Algebra
  10. 10Problem 10A triangle has area 30, one side of length 10, and the median to that side of length 9. Let θ be the acute angle formed by that side and the median…Geometry
  11. 11Problem 11Alex, Mel, and Chelsea play a game that has 6 rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For…Counting & Probability
  12. 12Problem 12A square region ABCD is externally tangent to the circle with equation x^2 + y^2 = 1 at the point (0, 1) on the side CD. Vertices A and B are on the…Algebra
  13. 13Problem 13Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM and all three always take the same…Algebra
  14. 14Problem 14The closed curve in the figure is made up of 9 congruent circular arcs each of length 2π/3, where each of the centers of the corresponding circles is…Geometry
  15. 15Problem 15A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen…Counting & Probability
  16. 16Problem 16Circle C_1 has its center O lying on circle C_2. The two circles meet at X and Y. Point Z in the exterior of C_1 lies on circle C_2 and XZ = 13, OZ =…Geometry
  17. 17Problem 17Let S be a subset of {1, 2, 3, …, 30} with the property that no pair of distinct elements in S has a sum divisible by 5. What is the largest possible…Algebra
  18. 18Problem 18Triangle ABC has AB = 27, AC = 26, and BC = 25. Let I denote the intersection of the internal angle bisectors of △ ABC. What is BI?Geometry
  19. 19Problem 19Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of…Counting & Probability
  20. 20Problem 20Consider the polynomial P(x) = prod _k=0^10(x^2^k + 2^k) = (x+1)(x^2+2)(x^4+4) … (x^1024+1024). The coefficient of x^2012 is equal to 2^a. What is a?Number Theory
  21. 21Problem 21Let a, b, and c be positive integers with a ≥ b ≥ c such that a^2 - b^2 - c^2 + ab = 2011 and a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997. What is a?Algebra
  22. 22Problem 22Distinct planes p_1, p_2, …, p_k intersect the interior of a cube Q. Let S be the union of the faces of Q and let P = bigcup _j=1^k p_j. The…Geometry
  23. 23Problem 23Let S be the square one of whose diagonals has endpoints (0.1, 0.7) and (-0.1, -0.7). A point v = (x, y) is chosen uniformly at random over all pairs…Geometry
  24. 24Problem 24Let {a_k}_k=1^2011 be the sequence of real numbers defined by a_1 = 0.201, a_2 = (0.2011)^a_1, a_3 = (0.20101)^a_2, and a_4 = (0.201011)^a_3, and…Algebra
  25. 25Problem 25Let f(x) = |2{x} - 1| where {x} denotes the fractional part of x. The number n is the smallest positive integer such that the equation nf(xf(x)) = x…Algebra

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