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2011 AMC 12B problems

All 25 problems from the 2011 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 1What is 2+4+6/1+3+5 - 1+3+5/2+4+6?Algebra
  2. 2Problem 2Josanna’s test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the…Algebra
  3. 3Problem 3LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint…Algebra
  4. 4Problem 4In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct…Number Theory
  5. 5Problem 5Let N be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of N?Number Theory
  6. 6Problem 6Two tangents to a circle are drawn from a point A. The points of contact B and C divide the circle into arcs with lengths in the ratio 2:3. What is…Geometry
  7. 7Problem 7Let x and y be two-digit positive integers with mean 60. What is the maximum value of the ratio x/y?Algebra
  8. 8Problem 8Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The…Algebra
  9. 9Problem 9Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is…Counting & Probability
  10. 10Problem 10Rectangle ABCD has AB=6 and BC=3. Point M is chosen on side AB so that ∠ AMD=∠ CMD. What is the degree measure of ∠ AMD?Geometry
  11. 11Problem 11A frog located at (x, y), with both x and y integers, makes successive jumps of length 5 and always lands on points with integer coordinates. Suppose…Geometry
  12. 12Problem 12A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the…Geometry
  13. 13Problem 13Brian writes down four integers w > x > y > z whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is…Algebra
  14. 14Problem 14A segment through the focus F of a parabola with vertex V is perpendicular to FV and intersects the parabola in points A and B. What is cos (∠ AVB)?Geometry
  15. 15Problem 15How many positive two-digit integers are factors of 2^24-1?Number Theory
  16. 16Problem 16Rhombus ABCD has side length 2 and ∠ B=120°. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other…Geometry
  17. 17Problem 17Let f(x)=10^10x, g(x)=log _10 (x/10), h_1(x)=g(f(x)), and h_n(x)=h_1(h_n-1(x)) for integers nge 2. What is the sum of the digits of h_2011(1)?Algebra
  18. 18Problem 18A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that…Geometry
  19. 19Problem 19A lattice point in an xy-coordinate system is any point (x, y) where both x and y are integers. The graph of y=mx+2 passes through no lattice point…Geometry
  20. 20Problem 20Triangle ABC has AB=13, BC=14, and AC=15. The points D, E, and F are the midpoints of AB, BC, and AC respectively. Let X≠ E be the intersection of…Geometry
  21. 21Problem 21The arithmetic mean of two distinct positive integers x and y is a two-digit integer. The geometric mean of x and y is obtained by reversing the…Number Theory
  22. 22Problem 22Let T_1 be a triangle with sides 2011, 2012, and 2013. For nge 1, if T_n=△ ABC and D, E, and F are the points of tangency of the incircle of △ ABC to…Geometry
  23. 23Problem 23A bug travels in the coordinate plane, moving only along the lines that are parallel to the x-axis or y-axis. Let A=(-3, 2) and B=(3, -2). Consider…Geometry
  24. 24Problem 24Let P(z)=z^8+(4√(3)+6)z^4 -(4√(3)+7). What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely…Algebra
  25. 25Problem 25For every m and k integers with k odd, denote by [m/k] the integer closest to m/k. For every odd integer k, let P(k) be the probability that…Algebra

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