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2021 AMC 12A problems

All 25 problems from the 2021 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 2^1+2+3 - (2^1 + 2^2 + 2^3)?Algebra
  2. 2Problem 2Under what conditions is √(a^2 + b^2) = a + b true, where a and b are real numbers?Algebra
  3. 3Problem 3The sum of two natural numbers is 17,402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is…Algebra
  4. 4Problem 4Tom has a collection of 13 snakes, 4 of which are purple and 5 of which are happy. He observes that • all of his happy snakes can add, • none of his…
  5. 5Problem 5When a student multiplied the number 66 by the repeating decimal 1.ab = 1.ababab…, where a and b are digits, he did not notice the notation and just…Algebra
  6. 6Problem 6A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is dfrac 13. When 4 black cards are added to…Algebra
  7. 7Problem 7What is the least possible value of (xy - 1)^2 + (x + y)^2 for real numbers x and y?Algebra
  8. 8Problem 8A sequence of numbers is defined by D_0 = 0, D_1 = 0, D_2 = 1, and D_n = D_n-1 + D_n-3 for n ≥ 3. What are the parities (evenness or oddness) of the…Algebra
  9. 9Problem 9Which of the following is equivalent to (2 + 3)(2^2 + 3^2)(2^4 + 3^4) · (2^8 + 3^8)(2^16 + 3^16) · (2^32 + 3^32)(2^64 + 3^64)?Algebra
  10. 10Problem 10Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the…Geometry
  11. 11Problem 11A laser is placed at the point (3, 5). The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the y-axis, then hit and…Geometry
  12. 12Problem 12All the roots of the polynomial z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16 are positive integers, possibly repeated. What is the value of B?Algebra
  13. 13Problem 13Of the following complex numbers z, which one has the property that z^5 has the greatest real part?Algebra
  14. 14Problem 14What is the value of (sum _k=1^20 log _5^k 3^k^2) · (sum _k=1^100 log _9^k 25^k)?Algebra
  15. 15Problem 15A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the…Algebra
  16. 16Problem 16In the following list of numbers, the integer n appears n times in the list for 1 ≤ n ≤ 200. 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, 200, 200, …, 200 What…Algebra
  17. 17Problem 17Trapezoid ABCD has AB ∥ CD, BC = CD = 43, and AD ⊥ BD. Let O be the intersection of the diagonals AC and BD, and let P be the midpoint of BD. Given…Geometry
  18. 18Problem 18Let f be a function defined on the set of positive rational numbers with the property that f(a· b) = f(a) + f(b) for all positive rational numbers a…Algebra
  19. 19Problem 19How many solutions does the equation sin (π/2cos x) = cos (π/2sin x) have in the closed interval [0, π]?Geometry
  20. 20Problem 20Suppose that on a parabola with vertex V and a focus F there exists a point A such that AF = 20 and AV = 21. What is the sum of all possible values…Geometry
  21. 21Problem 21The five solutions to the equation (z - 1)(z^2 + 2z + 4) · (z^2 + 4z + 6) = 0 may be written in the form x_k + y_k i for 1 ≤ k ≤ 5, where x_k and y_k…Algebra
  22. 22Problem 22Suppose that the roots of the polynomial P(x) = x^3 + ax^2 + bx + c are cos 2π/7, cos 4π/7, and cos 6π/7, where angles are in radians. What is abc?Algebra
  23. 23Problem 23Frieda the frog begins a sequence of hops on a 3× 3 grid of squares, moving one square on each hop and choosing at random the direction of each hop…Counting & Probability
  24. 24Problem 24Semicircle Gamma has diameter AB of length 14. Circle Omega lies tangent to AB at a point P and intersects Gamma at points Q and R. If QR = 3sqrt 3…Geometry
  25. 25Problem 25Let d(n) denote the number of positive integers that divide n, including 1 and n. For example, d(1) = 1, d(2) = 2, and d(12) = 6. (This function is…Number Theory

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