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2021 Fall AMC 12B problems

All 25 problems from the 2021 Fall 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 the value of 1234 + 2341 + 3412 + 4123?Number Theory
  2. 2Problem 2What is the area of the shaded figure shown below?Geometry
  3. 3Problem 3At noon on a certain day, Minneapolis is N degrees warmer than St. Louis. At 4:00 the temperature in Minneapolis has fallen by 5 degrees while the…Algebra
  4. 4Problem 4Let n = 8^2022. Which of the following is equal to n/4?Algebra
  5. 5Problem 5Call a fraction a/b, not necessarily in the simplest form, special if a and b are positive integers whose sum is 15. How many distinct integers can…Algebra
  6. 6Problem 6The greatest prime number that is a divisor of 16,384 is 2 because 16,384 = 2^14. What is the sum of the digits of the greatest prime number that is…Algebra
  7. 7Problem 7Which of the following conditions is sufficient to guarantee that integers x, y, and z satisfy the equation x(x - y) + y(y - z) + z(z - x) = 1?Algebra
  8. 8Problem 8The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s…Geometry
  9. 9Problem 9Triangle ABC is equilateral with side length 6. Suppose that O is the center of the inscribed circle of this triangle. What is the area of the circle…Geometry
  10. 10Problem 10What is the sum of all possible values of t between 0 and 360 such that the triangle in the coordinate plane whose vertices are (cos 40°, sin 40°)…Geometry
  11. 11Problem 11Una rolls 6 standard 6-sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the probability that the product is…Counting & Probability
  12. 12Problem 12For n a positive integer, let f(n) be the quotient obtained when the sum of all positive divisors of n is divided by n. For example, f(14) = (1 + 2 +…Number Theory
  13. 13Problem 13Let c = 2π/11. What is the value of small sin 3c · sin 6c · sin 9c · sin 12c · sin 15c/sin c · sin 2c · sin 3c · sin 4c · sin 5c?Geometry
  14. 14Problem 14Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and constant terms 1, 2, and 3…Algebra
  15. 15Problem 15Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30° about its…Geometry
  16. 16Problem 16Suppose a, b, c are positive integers such that a + b + c = 23 and gcd (a, b) + gcd (b, c) + gcd (c, a) = 9. What is the sum of all possible distinct…Number Theory
  17. 17Problem 17A bug starts at a vertex of a grid made of equilateral triangles of side length 1. At each step the bug moves in one of the 6 possible directions…Counting & Probability
  18. 18Problem 18Set u_0 = 1/4, and for k ≥ 0 let u_k+1 be determined by the recurrence u_k+1 = 2u_k - 2u_k^2. This sequence tends to a limit; call it L. What is the…Algebra
  19. 19Problem 19Regular polygons with 5, 6, 7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides…Geometry
  20. 20Problem 20A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the 2 × 2 × 2 cube using these…Geometry
  21. 21Problem 21For real numbers x, let P(x) = 1 + cos (x) + isin (x) - cos (2x) - isin (2x) + cos (3x) + isin (3x) where i = √(-1). For how many values of x with 0…Algebra
  22. 22Problem 22Right triangle ABC has side lengths BC = 6, AC = 8, and AB = 10. A circle centered at O is tangent to line BC at B and passes through A. A circle…Geometry
  23. 23Problem 23What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set {1, 2, 3, ……Counting & Probability
  24. 24Problem 24Triangle ABC has side lengths AB = 11, BC = 24, and CA = 20. The bisector of ∠ BAC intersects BC in point D, and intersects the circumcircle of △ ABC…Geometry
  25. 25Problem 25For n a positive integer, let R(n) be the sum of the remainders when n is divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10. For example, R(15) = 1 + 0 + 3 +…Number Theory

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