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2012 AMC 10B problems

All 25 problems from the 2012 AMC 10B, 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 3The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of the reflected point?Geometry
  4. 4Problem 4When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3…Number Theory
  5. 5Problem 5Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the…Algebra
  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 7For 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
  8. 8Problem 8What is the sum of all integer solutions to the following inequality? 1 < (x-2)^2 < 25Algebra
  9. 9Problem 9Two 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
  10. 10Problem 10How many ordered pairs of positive integers (M,N) satisfy the equation M/6=6/NNumber Theory
  11. 11Problem 11A 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
  12. 12Problem 12Point B is due east of point A. Point C is due north of point B. The distance between points A and C is 10sqrt 2 meters, and ∠ BAC = 45°. Point D is…Geometry
  13. 13Problem 13It 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
  14. 14Problem 14Two equilateral triangles are contained in a square whose side length is 2sqrt 3. The bases of these triangles are opposite sides of the square, and…Geometry
  15. 15Problem 15In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team…Algebra
  16. 16Problem 16Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?Geometry
  17. 17Problem 17Jesse 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
  18. 18Problem 18Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for…Counting & Probability
  19. 19Problem 19In rectangle ABCD, AB=6, AD=30, and G is the midpoint of AD. Segment AB is extended 2 units beyond B to point E, and F is the intersection of ED and…Geometry
  20. 20Problem 20Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a…Algebra
  21. 21Problem 21Four distinct points are arranged in a plane so that the segments connecting them have lengths a, a, a, a, 2a, and b. What is the ratio of b to a?Geometry
  22. 22Problem 22Let (a_1,a_2,…,a_10) be a list of the first 10 positive integers such that for each 2≤ ile 10 either a_i + 1 or a_i-1 or both appear somewhere before…Counting & Probability
  23. 23Problem 23A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the…Geometry
  24. 24Problem 24Amy, 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
  25. 25Problem 25A 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

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.