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2020 AMC 12A problems

All 25 problems from the 2020 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 1Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?Algebra
  2. 2Problem 2The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line…Geometry
  3. 3Problem 3A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her…Algebra
  4. 4Problem 4How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?Number Theory
  5. 5Problem 5The 25 integers from -10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in each row, the sum of the…Algebra
  6. 6Problem 6In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so…
  7. 7Problem 7Seven cubes, whose volumes are 1, 8, 27, 64, 125, 216, and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes…Algebra
  8. 8Problem 8What is the median of the following list of 4040 numbers? 1, 2, 3, …, 2020, 1^2, 2^2, 3^2, …, 2020^2Algebra
  9. 9Problem 9How many solutions does the equation tan (2x) = cos (x/2) have on the interval [0, 2π]?Geometry
  10. 10Problem 10There is a unique positive integer n such that log _2(log _16 n) = log _4(log _4 n). What is the sum of the digits of n?Algebra
  11. 11Problem 11A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the…Algebra
  12. 12Problem 12Line ℓ in the coordinate plane has the equation 3x - 5y + 40 = 0. This line is rotated 45° counterclockwise about the point (20, 20) to obtain line…Geometry
  13. 13Problem 13There are integers a, b, and c, each greater than 1, such that √(N √(N √(N))) = sqrt [36]N^25 for all N > 1. What is b?Algebra
  14. 14Problem 14Regular octagon ABCDEFGH has area n. Let m be the area of quadrilateral ACEG. What is m/n?Geometry
  15. 15Problem 15In the complex plane, let A be the set of solutions to z^3 - 8 = 0 and let B be the set of solutions to z^3 - 8z^2 - 8z + 64 = 0. What is the…Algebra
  16. 16Problem 16A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (2020, 0), (2020, 2020), and (0, 2020). The…Geometry
  17. 17Problem 17The vertices of a quadrilateral lie on the graph of y = ln x, and the x-coordinates of these vertices are consecutive positive integers. The area of…Algebra
  18. 18Problem 18Quadrilateral ABCD satisfies ∠ ABC = ∠ ACD = 90°, AC = 20, and CD = 30. Diagonals AC and BD intersect at point E, and AE = 5. What is the area of…Geometry
  19. 19Problem 19There exists a unique strictly increasing sequence of nonnegative integers a_1 < a_2 < … < a_k such that frac 2^289 + 12^17 + 1 = 2^a_1 + 2^a_2 + … +…Number Theory
  20. 20Problem 20Let T be the triangle in the coordinate plane with vertices (0, 0), (4, 0), and (0, 3). Consider the following five isometries (rigid…Geometry
  21. 21Problem 21How many positive integers n are there such that n is a multiple of 5, and the least common multiple of 5! and n equals 5 times the greatest common…Number Theory
  22. 22Problem 22Let (a_n) and (b_n) be the sequences of real numbers such that (2 + i)^n = a_n + b_n i for all integers n ≥ 0, where i = √(-1). What is sum _n=0^∞…Algebra
  23. 23Problem 23Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice)…Algebra
  24. 24Problem 24Suppose that △ ABC is an equilateral triangle of side length s, with the property that there is a unique point P inside the triangle such that AP =…Algebra
  25. 25Problem 25The number a = p/q, where p and q are relatively prime positive integers, has the property that the sum of all real numbers x satisfying ⌊ x ⌋ · {x}…Algebra

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