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2009 AMC 12A problems

All 25 problems from the 2009 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 1Kim’s flight took off from Newark at 10:34 am and landed in Miami at 1:18 pm. Both cities are in the same time zone. If her flight took h hours and m…Algebra
  2. 2Problem 2Which of the following is equal to 1 + 1/1 + 1/1 + 1?Algebra
  3. 3Problem 3What number is one third of the way from 1/4 to 3/4?Algebra
  4. 4Problem 4Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be…Algebra
  5. 5Problem 5One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5…Algebra
  6. 6Problem 6Suppose that P = 2^m and Q = 3^n. Which of the following is equal to 12^mn for every pair of integers (m, n)?Algebra
  7. 7Problem 7The first three terms of an arithmetic sequence are 2x - 3, 5x - 11, and 3x + 1 respectively. The nth term of the sequence is 2009. What is n?Algebra
  8. 8Problem 8Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of…Geometry
  9. 9Problem 9Suppose that f(x + 3) = 3x^2 + 7x + 4 and f(x) = ax^2 + bx + c. What is a + b + c?Algebra
  10. 10Problem 10In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an integer. What is BD?Algebra
  11. 11Problem 11The figures F_1, F_2, F_3, and F_4 shown are the first in a sequence of figures. For n ≥ 3, F_n is constructed from F_n-1 by surrounding it with a…Algebra
  12. 12Problem 12How many positive integers less than 1000 are 6 times the sum of their digits?Algebra
  13. 13Problem 13A ship sails 10 miles in a straight line from A to B, turns through an angle between 45° and 60°, and then sails another 20 miles to C. Let AC be…Algebra
  14. 14Problem 14A triangle has vertices (0, 0), (1, 1), and (6m, 0), and the line y = mx divides the triangle into two triangles of equal area. What is the sum of…Geometry
  15. 15Problem 15For what value of n is i + 2i^2 + 3i^3 + … + ni^n = 48 + 49i? Note: here i = √(-1).Algebra
  16. 16Problem 16A circle with center C is tangent to the positive x- and y-axes and externally tangent to the circle centered at (3, 0) with radius 1. What is the…Geometry
  17. 17Problem 17Let a + ar_1 + ar_1^2 + ar_1^3 + … and a + ar_2 + ar_2^2 + ar_2^3 + … be two different infinite geometric series of positive numbers with the same…Algebra
  18. 18Problem 18For k > 0, let I_k = 10ldots 064, where there are k zeros between the 1 and the 6. Let N(k) be the number of factors of 2 in the prime factorization…Algebra
  19. 19Problem 19Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the…Geometry
  20. 20Problem 20Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and △ AED and △ BEC have equal areas. What is AE?Geometry
  21. 21Problem 21Let p(x) = x^3 + ax^2 + bx + c, where a, b, and c are complex numbers. Suppose that p(2009 + 9002π i) = p(2009) = p(9002) = 0. What is the number of…Algebra
  22. 22Problem 22A regular octahedron has side length 1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon…Geometry
  23. 23Problem 23Functions f and g are quadratic, g(x) = -f(100 - x), and the graph of g contains the vertex of the graph of f. The four x-intercepts on the two…Algebra
  24. 24Problem 24The tower function of twos is defined recursively as follows: T(1) = 2 and T(n + 1) = 2^T(n) for n ≥ 1. Let A = (T(2009))^T(2009) and B =…Algebra
  25. 25Problem 25The first two terms of a sequence are a_1 = 1 and a_2 = 1/√(3). For n ≥ 1, a_n+2 = dfrac a_n + a_n+11 - a_n a_n+1. What is |a_2009|?Algebra

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