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2009 AMC 12B problems

All 25 problems from the 2009 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 morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of…Algebra
  2. 2Problem 2Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so…Algebra
  3. 3Problem 3Twenty percent less than 60 is one-third more than what number?Algebra
  4. 4Problem 4A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape…Geometry
  5. 5Problem 5Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?Number Theory
  6. 6Problem 6By inserting parentheses, it is possible to give the expression 2 × 3 + 4 × 5 several values. How many different values can be obtained?Algebra
  7. 7Problem 7In a certain year the price of gasoline rose by 20% during January, fell by 20% during February, rose by 25% during March, and fell by x% during…Algebra
  8. 8Problem 8When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is b…Algebra
  9. 9Problem 9Triangle ABC has vertices A = (3, 0), B = (0, 3), and C, where C is on the line x + y = 7. What is the area of △ ABC?Geometry
  10. 10Problem 10A particular 12-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1, it mistakenly…Algebra
  11. 11Problem 11On Monday, Millie puts a quart of seeds, 25% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix…Algebra
  12. 12Problem 12The fifth and eighth terms of a geometric sequence of real numbers are 7! and 8! respectively. What is the first term?Algebra
  13. 13Problem 13Triangle ABC has AB = 13 and AC = 15, and the altitude to BC has length 12. What is the sum of the two possible values of BC?Geometry
  14. 14Problem 14Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from (a, 0) to…Geometry
  15. 15Problem 15Assume 0 < r < 3. Below are five equations for x. Which equation has the largest solution x?Algebra
  16. 16Problem 16Trapezoid ABCD has AD ∥ BC, BD = 1, ∠ DBA = 23°, and ∠ BDC = 46°. The ratio BC: AD is 9: 5. What is CD?Geometry
  17. 17Problem 17Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge…Geometry
  18. 18Problem 18Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes…Algebra
  19. 19Problem 19For each positive integer n, let f(n) = n^4 - 360n^2 + 400. What is the sum of all values of f(n) that are prime numbers?Algebra
  20. 20Problem 20A convex polyhedron Q has vertices V_1, V_2, …, V_n, and 100 edges. The polyhedron is cut by planes P_1, P_2, …, P_n in such a way that plane P_k…Geometry
  21. 21Problem 21Ten women sit in 10 seats in a line. All of the 10 get up and then reseat themselves using all 10 seats, each sitting in the seat she was in before…Algebra
  22. 22Problem 22Parallelogram ABCD has area 1,000,000. Vertex A is at (0, 0) and all other vertices are in the first quadrant. Vertices B and D are lattice points on…Geometry
  23. 23Problem 23A region S in the complex plane is defined by scriptsize S = {x + iy: -1 ≤ x ≤ 1, -1 ≤ y ≤ 1}. A complex number z = x + iy is chosen uniformly at…Algebra
  24. 24Problem 24For how many values of x in [0, π] is sin ^-1(sin 6x) = cos ^-1(cos x)? Note: The functions sin ^-1 = arcsin and cos ^-1 = arccos denote inverse…Geometry
  25. 25Problem 25The set G is defined by the points (x, y) with integer coordinates, 3 ≤ |x| ≤ 7, and 3 ≤ |y| ≤ 7. How many squares of side at least 6 have their four…Counting & Probability

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