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2012 AMC 12B problems

All 25 problems from the 2012 AMC 12B, 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 1Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the…Algebra
  2. 2Problem 2A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the…Geometry
  3. 3Problem 3For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The…Algebra
  4. 4Problem 4Suppose that the euro is worth 1.30 dollars. If Diana has 500 dollars and Étienne has 400 euros, by what percent is the value of Étienne’s money…Algebra
  5. 5Problem 5Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to…Number Theory
  6. 6Problem 6In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the…Algebra
  7. 7Problem 7Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this…Algebra
  8. 8Problem 8A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding…Counting & Probability
  9. 9Problem 9It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How…Algebra
  10. 10Problem 10What is the area of the polygon whose vertices are the points of intersection of the curves x^2 + y^2 = 25 and (x - 4)^2 + 9y^2 = 81?Algebra
  11. 11Problem 11In the equation below, A and B are consecutive positive integers, and A, B, and A + B represent number bases: 132_A + 43_B = 69_A+B. What is A + B?Algebra
  12. 12Problem 12How many sequences of zeros and/or ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?Counting & Probability
  13. 13Problem 13Two parabolas have equations y = x^2 + ax + b and y = x^2 + cx + d, where a, b, c, and d are integers (not necessarily different), each chosen…Counting & Probability
  14. 14Problem 14Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a…Algebra
  15. 15Problem 15Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two…Geometry
  16. 16Problem 16Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three…Counting & Probability
  17. 17Problem 17Square PQRS lies in the first quadrant. Points (3, 0), (5, 0), (7, 0), and (13, 0) lie on lines SP, RQ, PQ, and SR, respectively. What is the sum of…Geometry
  18. 18Problem 18Let (a_1, a_2, …, a_10) be a list of the first 10 positive integers such that for each 2 ≤ i ≤ 10 either a_i + 1 or a_i - 1 or both appear somewhere…Counting & Probability
  19. 19Problem 19A unit cube has vertices P_1, P_2, P_3, P_4, P_1', P_2', P_3', and P_4'. Vertices P_2, P_3, and P_4 are adjacent to P_1, and for 1 ≤ i ≤ 4, vertices…Geometry
  20. 20Problem 20A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of r_1√(n_1) + r_2√(n_2)…Geometry
  21. 21Problem 21Square AXYZ is inscribed in equiangular hexagon ABCDEF with X on BC, Y on DE, and Z on EF. Suppose that AB = 40 and EF = 41(√(3) - 1). What is the…Geometry
  22. 22Problem 22A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the…Counting & Probability
  23. 23Problem 23Consider all polynomials of a complex variable, P(z) = 4z^4 + az^3 + bz^2 + cz + d, where a, b, c, and d are integers, 0 ≤ d ≤ c ≤ b ≤ a ≤ 4, and the…Algebra
  24. 24Problem 24Define the function f_1 on the positive integers by setting f_1(1) = 1 and if n = p_1^e_1 p_2^e_2 … p_k^e_k is the prime factorization of n > 1, then…Number Theory
  25. 25Problem 25Let S = {(x, y): x ∈ {0, 1, 2, 3, 4}, y ∈ {0, 1, 2, 3, 4, 5}, and (x, y) ≠ (0, 0)}. Let T be the set of all right triangles whose vertices are in S…Algebra

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