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2022 AMC 12A problems

All 25 problems from the 2022 AMC 12A, 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 3+cfrac 13+cfrac 13+frac 13?Algebra
  2. 2Problem 2The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the second number. What is the…Algebra
  3. 3Problem 3Five rectangles, A, B, C, D, and E, are arranged in a square as shown below. These rectangles have dimensions 1times 6, 2times 4, 5times 6, 2times 7…Geometry
  4. 4Problem 4The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What is the sum of the digits of…Number Theory
  5. 5Problem 5Let the taxicab distance between points (x_1,y_1) and (x_2,y_2) in the coordinate plane be given by |x_1-x_2|+|y_1-y_2|. For how many points P with…Algebra
  6. 6Problem 6A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and X. The average (arithmetic mean) of the 6 numbers equals a value in the…Algebra
  7. 7Problem 7A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that…Counting & Probability
  8. 8Problem 8The infinite product √(10)·√(√(10))·√(√(√(10)))… evaluates to a real number. What is that number?Algebra
  9. 9Problem 9On Halloween 31 children walked into the principal’s office asking for candy. They can be classified into three types: some always lie; some always…Counting & Probability
  10. 10Problem 10What is the number of ways the numbers from 1 to 14 can be split into 7 pairs such that for each pair, the greater number is at least 2 times the…Counting & Probability
  11. 11Problem 11What is the product of all real numbers x such that the distance on the number line between log _6 x and log _6 9 is twice the distance on the number…Algebra
  12. 12Problem 12Let M be the midpoint of AB in regular tetrahedron ABCD. What is cos (∠ CMD)?Geometry
  13. 13Problem 13Let mathcal R be the region in the complex plane consisting of all complex numbers z that can be written as the sum of complex numbers z_1 and z_2…Geometry
  14. 14Problem 14What is the value of (log 5)^3+(log 20)^3 +(log 8)(log 0.25) where log denotes the base-ten logarithm?Algebra
  15. 15Problem 15The roots of the polynomial 10x^3-39x^2+29x-6 are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box…Algebra
  16. 16Problem 16A triangular number is a positive integer that can be expressed in the form t_n=1+2+3+…+n, for some positive integer n. The three smallest triangular…Algebra
  17. 17Problem 17Suppose a is a real number such that the equation a·(sin x+sin (2x))=sin (3x) has more than one solution in the interval (0,π). The set of all such a…Geometry
  18. 18Problem 18Let T_k be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the origin and then reflects the…Geometry
  19. 19Problem 19Suppose that 13 cards numbered 1, 2, 3, …, 13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly…Counting & Probability
  20. 20Problem 20Isosceles trapezoid ABCD has parallel sides AD and BC, with BC< AD and AB=CD. There is a point P in the plane such that PA=1, PB=2, PC=3, and PD=4…Geometry
  21. 21Problem 21Let P(x)=x^2022+x^1011+1. Which of the following polynomials divides P(x)?Algebra
  22. 22Problem 22Let c be a real number, and let z_1, z_2 be the two complex numbers satisfying the quadratic z^2-cz+10=0. Points z_1, z_2, 1/z_1, and 1/z_2 are the…Algebra
  23. 23Problem 23Let h_n and k_n be the unique relatively prime positive integers such that frac 11+frac 12+frac 13+…+frac 1n=h_n/k_n. Let L_n denote the least common…Number Theory
  24. 24Problem 24How many strings of length 5 formed from the digits 0, 1, 2, 3, 4 are there such that for each j∈{1,2,3,4}, at least j of the digits are less than j?…Counting & Probability
  25. 25Problem 25A circle with integer radius r is centered at (r,r). Distinct line segments of length c_i connect points (0,a_i) to (b_i,0) for 1≤ ile 14 and are…Geometry

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