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2021 Fall AMC 12A problems

All 25 problems from the 2021 Fall 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 (2112 - 2021)^2/169?Algebra
  2. 2Problem 2Menkara has a 4 × 6 index card. If she shortens the length of one side of this card by 1 inch, the card would have area 18 square inches. What would…Geometry
  3. 3Problem 3Mr. Lopez has a choice of two routes to get to work. Route A is 6 miles long, and his average speed along this route is 30 miles per hour. Route B is…Algebra
  4. 4Problem 4The six-digit number 2 0 2 1 0 A is prime for only one digit A. What is A?Number Theory
  5. 5Problem 5Elmer the emu takes 44 equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in 12…Algebra
  6. 6Problem 6As shown in the figure below, point E lies on the opposite half-plane determined by line CD from point A so that ∠ CDE = 110°. Point F lies on AD so…Geometry
  7. 7Problem 7A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The…Algebra
  8. 8Problem 8Let M be the least common multiple of all the integers 10 through 30, inclusive. Let N be the least common multiple of M, 32, 33, 34, 35, 36, 37, 38…Number Theory
  9. 9Problem 9A right rectangular prism whose surface area and volume are numerically equal has edge lengths log _2 x, log _3 x, and log _4 x. What is x?Geometry
  10. 10Problem 10The base-nine representation of the number N is 27,006,000,052_9. What is the remainder when N is divided by 5?Number Theory
  11. 11Problem 11Consider two concentric circles of radius 17 and 19. The larger circle has a chord, half of which lies inside the smaller circle. What is the length…Geometry
  12. 12Problem 12What is the number of terms with rational coefficients among the 1001 terms in the expansion of (x√(2) + y√(3))^1000?Algebra
  13. 13Problem 13The angle bisector of the acute angle formed at the origin by the graphs of the lines y = x and y = 3x has equation y = kx. What is k?Geometry
  14. 14Problem 14In the figure, equilateral hexagon ABCDEF has three nonadjacent acute interior angles that each measure 30°. The enclosed area of the hexagon is…Geometry
  15. 15Problem 15Recall that the conjugate of the complex number w = a + bi, where a and b are real numbers and i = √(-1), is the complex number w = a - bi. For any…Algebra
  16. 16Problem 16An organization has 30 employees, 20 of whom have a brand A computer while the other 10 have a brand B computer. For security, the computers can only…Algebra
  17. 17Problem 17For how many ordered pairs (b, c) of positive integers does neither x^2 + bx + c = 0 nor x^2 + cx + b = 0 have two distinct real solutions?Algebra
  18. 18Problem 18Each of 20 balls is tossed independently and at random into one of 5 bins. Let p be the probability that some bin ends up with 3 balls, another with…Counting & Probability
  19. 19Problem 19Let x be the least real number greater than 1 such that sin (x) = sin (x^2), where the arguments are in degrees. What is x rounded up to the closest…Algebra
  20. 20Problem 20For each positive integer n, let f_1(n) be twice the number of positive integer divisors of n, and for j ≥ 2, let f_j(n) = f_1(f_j-1(n)). For how…Algebra
  21. 21Problem 21Let ABCD be an isosceles trapezoid with BC ∥ AD and AB = CD. Points X and Y lie on diagonal AC with X between A and Y, as shown in the figure…Geometry
  22. 22Problem 22Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in a 3-by-3 array of boxes, then Carl places an O in one of the…Counting & Probability
  23. 23Problem 23A quadratic polynomial with real coefficients and leading coefficient 1 is called disrespectful if the equation p(p(x)) = 0 is satisfied by exactly…Algebra
  24. 24Problem 24Convex quadrilateral ABCD has AB = 18, ∠ A = 60°, and AB ∥ CD. In some order, the lengths of the four sides form an arithmetic progression, and side…Algebra
  25. 25Problem 25Let m ≥ 5 be an odd integer, and let D(m) denote the number of quadruples (a_1, a_2, a_3, a_4) of distinct integers with 1 ≤ a_i ≤ m for all i such…Algebra

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