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2011 AMC 12A problems

All 25 problems from the 2011 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 cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text…Algebra
  2. 2Problem 2There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
  3. 3Problem 3A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the…Algebra
  4. 4Problem 4At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively…Algebra
  5. 5Problem 5Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were…Algebra
  6. 6Problem 6The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with…Algebra
  7. 7Problem 7A majority of the 30 students in Ms. Demeanor’s class bought pencils at the school bookstore. Each of these students bought the same number of…Number Theory
  8. 8Problem 8In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?Algebra
  9. 9Problem 9At a twins and triplets convention, there were 9 sets of twins and 6 sets of triplets, all from different families. Each twin shook hands with all…Counting & Probability
  10. 10Problem 10A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that…Algebra
  11. 11Problem 11Circles A, B, and C each have radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What…Geometry
  12. 12Problem 12A power boat and a raft both left dock A on a river and headed downstream. The raft drifted at the speed of the river current. The power boat…Algebra
  13. 13Problem 13Triangle ABC has side-lengths AB = 12, BC = 24, and AC = 18. The line through the incenter of △ ABC parallel to BC intersects AB at M and AC at N…Geometry
  14. 14Problem 14Suppose a and b are single-digit positive integers chosen independently and at random. What is the probability that the point (a, b) lies above the…Geometry
  15. 15Problem 15The circular base of a hemisphere of radius 2 rests on the base of a square pyramid of height 6. The hemisphere is tangent to the other four faces of…Geometry
  16. 16Problem 16Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different…Counting & Probability
  17. 17Problem 17Circles with radii 1, 2, and 3 are mutually externally tangent. What is the area of the triangle determined by the points of tangency?Geometry
  18. 18Problem 18Suppose that |x + y| + |x - y| = 2. What is the maximum possible value of x^2 - 6x + y^2?Algebra
  19. 19Problem 19At a competition with N players, the number of players given elite status is equal to 2^1 + ⌊ log _2 (N - 1) ⌋ - N. Suppose that 19 players are given…Algebra
  20. 20Problem 20Let f(x) = ax^2 + bx + c, where a, b, and c are integers. Suppose that f(1) = 0, 50 < f(7) < 60, 70 < f(8) < 80, and 5000k < f(100) < 5000(k+1) for…Algebra
  21. 21Problem 21Let f_1(x) = √(1 - x), and for integers n ≥ 2, let f_n(x) = f_n-1 (√(n^2 - x)). If N is the largest value of n for which the domain of f_n is…Algebra
  22. 22Problem 22Let R be a square region and n ≥ 4 an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that…Geometry
  23. 23Problem 23Let f(z) = z + a/z + b and g(z) = f(f(z)), where a and b are complex numbers. Suppose that |a| = 1 and g(g(z)) = z for all z for which g(g(z)) is…Algebra
  24. 24Problem 24Consider all quadrilaterals ABCD such that AB = 14, BC = 9, CD = 7, and DA = 12. What is the radius of the largest possible circle that fits inside…Geometry
  25. 25Problem 25Triangle ABC has ∠ BAC = 60°, ∠ CBA ≤ 90°, BC = 1, and AC ≥ AB. Let H, I, and O be the orthocenter, incenter, and circumcenter of △ ABC…Geometry

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