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2013 AMC 12B problems

All 25 problems from the 2013 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 1On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high…Algebra
  2. 2Problem 2Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr. Green’s steps is 2 feet…Algebra
  3. 3Problem 3When counting from 3 to 201, 53 is the 51st number counted. When counting backwards from 201 to 3, 53 is the nth number counted. What is n?Counting & Probability
  4. 4Problem 4Ray’s car averages 40 miles per gallon of gasoline, and Tom’s car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of…Algebra
  5. 5Problem 5The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and…Algebra
  6. 6Problem 6Real numbers x and y satisfy the equation x^2 + y^2 = 10x - 6y - 34. What is x+y?Algebra
  7. 7Problem 7Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying “1”, so Blair follows by…Algebra
  8. 8Problem 8Line ell _1 has equation 3x - 2y = 1 and goes through A = (-1, -2). Line ell _2 has equation y = 1 and meets line ell _1 at point B. Line ell _3 has…Geometry
  9. 9Problem 9What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!?Number Theory
  10. 10Problem 10Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token…Algebra
  11. 11Problem 11Two bees start at the same spot and fly at the same rate in the following directions. Bee A travels 1 foot north, then 1 foot east, then 1 foot…Geometry
  12. 12Problem 12Cities A, B, C, D, and E are connected by roads AB, AD, AE, BC, BD, CD, and DE. How many different routes are there from A to B that use each road…Counting & Probability
  13. 13Problem 13The internal angles of quadrilateral ABCD form an arithmetic progression. Triangles ABD and DCB are similar with ∠ DBA = ∠ DCB and ∠ ADB = ∠ CBD…Algebra
  14. 14Problem 14Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the…Number Theory
  15. 15Problem 15The number 2013 is expressed in the form 2013 = a_1! a_2!… a_m!/b_1! b_2!… b_n!, where a_1 ≥ a_2 ≥ … ≥ a_m and b_1 ≥ b_2 ≥ … ≥ b_n are positive…Algebra
  16. 16Problem 16Let ABCDE be an equiangular convex pentagon of perimeter 1. The pairwise intersections of the lines that extend the sides of the pentagon determine a…Geometry
  17. 17Problem 17Let a, b, and c be real numbers such that a + b + c = 2 and a^2 + b^2 + c^2 = 12. What is the difference between the maximum and minimum possible…Algebra
  18. 18Problem 18Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove 2…Number Theory
  19. 19Problem 19In triangle ABC, AB = 13, BC = 14, and CA = 15. Distinct points D, E, and F lie on segments BC, CA, and DE, respectively, such that AD ⊥ BC, DE ⊥ AC…Geometry
  20. 20Problem 20For 135° < x < 180°, points P = (cos x, cos ^2 x), Q = (cot x, cot ^2 x), R = (sin x, sin ^2 x), and S = (tan x, tan ^2 x) are the vertices of a…Geometry
  21. 21Problem 21Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0, 0) and the directrix lines have the form y = ax + b…Counting & Probability
  22. 22Problem 22Let m > 1 and n > 1 be integers. Suppose that the product of the solutions for x of the equation 8(log _n x)(log _m x) - 7log _n x - 6log _m x - 2013…Algebra
  23. 23Problem 23Bernardo chooses a three-digit positive integer N and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two…Number Theory
  24. 24Problem 24Let ABC be a triangle where M is the midpoint of AC, and CN is the angle bisector of ∠ ACB with N on AB. Let X be the intersection of the median BM…Geometry
  25. 25Problem 25Let G be the set of polynomials of the form P(z) = z^n + c_n-1z^n-1 + … + c_2 z^2 + c_1 z + 50, where c_1, c_2, …, c_n-1 are integers and P(z) has n…Algebra

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