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2022 AMC 12B problems

All 25 problems from the 2022 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 1Define x ◆ y to be |x - y| for all real numbers x and y. What is the value of (1 ◆ (2 ◆ 3)) - ((1 ◆ 2) ◆ 3)?Algebra
  2. 2Problem 2In rhombus ABCD, point P lies on segment AD so that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD? (Note: the figure is not drawn to scale.)Geometry
  3. 3Problem 3How many of the first ten numbers of the sequence 121, 11211, 1112111, … are prime numbers?Algebra
  4. 4Problem 4For how many values of the constant k will the polynomial x^2 + kx + 36 have two distinct integer roots?Algebra
  5. 5Problem 5The point (-1, -2) is rotated 270° counterclockwise about the point (3, 1). What are the coordinates of its new position?Geometry
  6. 6Problem 6Consider the following 100 sets of 10 elements each: {1,2,3,…,10}, {11,12,13,…,20}, {21,22,23,…,30}, vdots {991,992,993,…,1000}. How many of these…Number Theory
  7. 7Problem 7Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their…Algebra
  8. 8Problem 8What is the graph of y^4 + 1 = x^4 + 2y^2 in the coordinate plane?Geometry
  9. 9Problem 9The sequence a_0, a_1, a_2, … is a strictly increasing arithmetic sequence of positive integers such that 2^a_7 = 2^27 · a_7. What is the minimum…Algebra
  10. 10Problem 10Regular hexagon ABCDEF has side length 2. Let G be the midpoint of AB, and let H be the midpoint of DE. What is the perimeter of GCHF?Geometry
  11. 11Problem 11Let f(n) = (-1 + isqrt 3/2)^n + (-1 - isqrt 3/2)^n, where i = √(-1). What is f(2022)?Algebra
  12. 12Problem 12Kayla rolls four fair 6-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the…Counting & Probability
  13. 13Problem 13The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different…Geometry
  14. 14Problem 14The graph of y = x^2 + 2x - 15 intersects the x-axis at points A and C and the y-axis at point B. What is tan (∠ ABC)?Geometry
  15. 15Problem 15One of the following numbers is not divisible by any prime number less than 10. Which is it?Number Theory
  16. 16Problem 16Suppose x and y are positive real numbers such that x^y = 2^64 and (log _2 x)^log _2 y = 2^7. What is the greatest possible value of log _2 y?Algebra
  17. 17Problem 17How many 4 × 4 arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some…Counting & Probability
  18. 18Problem 18Each square in a 5 × 5 grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a…
  19. 19Problem 19In △ ABC medians AD and BE intersect at G and △ AGE is equilateral. Then cos (C) can be written as msqrt p/n, where m and n are relatively prime…Geometry
  20. 20Problem 20Let P(x) be a polynomial with rational coefficients such that when P(x) is divided by the polynomial x^2 + x + 1, the remainder is x + 2, and when…Algebra
  21. 21Problem 21Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations x^2 + y^2 = 4, x^2 + y^2 = 64, and…Geometry
  22. 22Problem 22Ant Amelia starts on the number line at 0 and crawls in the following manner. For n = 1, 2, 3, Amelia chooses a time duration t_n and an increment…Counting & Probability
  23. 23Problem 23Let x_0, x_1, x_2, … be a sequence of numbers, where each x_k is either 0 or 1. For each positive integer n, define S_n = sum _k=0^n-1 x_k 2^k…Number Theory
  24. 24Problem 24The figure below depicts a regular 7-gon inscribed in a unit circle. What is the sum of the 4th powers of the lengths of all 21 of its edges and…Geometry
  25. 25Problem 25Four regular hexagons surround a square with a side length 1, each one sharing an edge with the square, as shown in the figure below. The area of the…Geometry

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