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2021 Fall AMC 10A problems

All 25 problems from the 2021 Fall AMC 10A, 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 the following expression? (2112-2021)^2/169Algebra
  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 3What is the maximum number of balls of clay with radius 2 that can completely fit inside a cube of side length 6 assuming that the balls can be…Geometry
  4. 4Problem 4Mr. 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
  5. 5Problem 5The six-digit number 2 0 2 1 0 A is prime for only one digit A. What is A?Number Theory
  6. 6Problem 6Elmer 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
  7. 7Problem 7As shown in the figure below, point E lies in the opposite half-plane determined by line CD from point A so that ∠ CDE = 110°. Point F lies on AD so…Geometry
  8. 8Problem 8A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many…Number Theory
  9. 9Problem 9When a certain unfair die is rolled, an even number is 3 times as likely to appear as an odd number. The die is rolled twice. What is the probability…Counting & Probability
  10. 10Problem 10A 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
  11. 11Problem 11Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster…Algebra
  12. 12Problem 12The 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
  13. 13Problem 13Each of 6 balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is…Counting & Probability
  14. 14Problem 14How many ordered pairs (x,y) of real numbers satisfy the following system of equations? x^2+3y =9, (|x|+|y|-4)^2 =1.Algebra
  15. 15Problem 15Isosceles triangle ABC has AB = AC = 3sqrt 6, and a circle with radius 5sqrt 2 is tangent to line AB at B and to line AC at C. What is the area of…Geometry
  16. 16Problem 16Consider the function f(x) = |⌊ x ⌋| - |⌊ 1 - x ⌋| Its graph is symmetric about which of the following? (Here ⌊ x ⌋ is the greatest integer not…Algebra
  17. 17Problem 17An architect is building a structure that will place vertical pillars at the vertices of regular hexagon ABCDEF, which is lying horizontally on the…Geometry
  18. 18Problem 18A farmer’s rectangular field is partitioned into a 2 by 2 grid of 4 rectangular sections as shown in the figure. In each section the farmer will…Counting & Probability
  19. 19Problem 19A disk of radius 1 rolls all the way around the inside of a square of side length s > 4 and sweeps out a region of area A. A second disk of radius 1…Geometry
  20. 20Problem 20For 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
  21. 21Problem 21Each of 20 balls is tossed independently and at random into one of the 5 bins. Let p be the probability that some bin ends up with 3 balls, another…Counting & Probability
  22. 22Problem 22Inside a right circular cone with base radius 5 and height 12 are three congruent spheres each with radius r. Each sphere is tangent to the other two…Geometry
  23. 23Problem 23For 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
  24. 24Problem 24Each of the 12 edges of a cube is labeled 0 or 1. Two labelings are considered different even if one can be obtained from the other by a sequence of…Geometry
  25. 25Problem 25A quadratic polynomial p(x) with real coefficients and leading coefficient 1 is called disrespectful if the equation p(p(x))=0 is satisfied by…Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.