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2021 Fall AMC 10B problems

All 25 problems from the 2021 Fall AMC 10B, 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 1234 + 2341 + 3412 + 4123?Number Theory
  2. 2Problem 2What is the area of the shaded figure shown below?Geometry
  3. 3Problem 3The expression 2021/2020 - 2020/2021 is equal to the fraction p/q in which p and q are positive integers whose greatest common divisor is 1. What is…Algebra
  4. 4Problem 4At noon on a certain day, Minneapolis is N degrees warmer than St. Louis. At 4:00 the temperature in Minneapolis has fallen by 5 degrees while the…Algebra
  5. 5Problem 5Let n=8^2022. Which of the following is equal to n/4?Algebra
  6. 6Problem 6The least positive integer with exactly 2021 distinct positive divisors can be written in the form m · 6^k, where m and k are integers and 6 is not a…Number Theory
  7. 7Problem 7Call a fraction a/b, not necessarily in simplest form, special if a and b are positive integers whose sum is 15. How many distinct integers can be…Algebra
  8. 8Problem 8The greatest prime number that is a divisor of 16,384 is 2 because 16,384 = 2^14. What is the sum of the digits of the greatest prime number that is…Algebra
  9. 9Problem 9The knights in a certain kingdom come in two colors. 2/7 of them are red, and the rest are blue. Furthermore, 1/6 of the knights are magical, and the…Algebra
  10. 10Problem 10Forty slips of paper numbered 1 to 40 are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers…Number Theory
  11. 11Problem 11A regular hexagon of side length 1 is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that…Geometry
  12. 12Problem 12Which of the following conditions is sufficient to guarantee that integers x, y, and z satisfy the equation x(x-y)+y(y-z)+z(z-x)= 1?Algebra
  13. 13Problem 13A square with side length 3 is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side…Geometry
  14. 14Problem 14Una rolls 6 standard 6-sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the probability that the product is…Counting & Probability
  15. 15Problem 15In square ABCD, points P and Q lie on AD and AB, respectively. Segments BP and CQ intersect at right angles at R, with BR = 6 and PR = 7. What is the…Geometry
  16. 16Problem 16Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice…Counting & Probability
  17. 17Problem 17Distinct lines ℓ and m lie in the xy-plane. They intersect at the origin. Point P(-1, 4) is reflected about line ℓ to point P', and then P' is…Geometry
  18. 18Problem 18Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30° about its…Geometry
  19. 19Problem 19Let N be the positive integer 7777ldots 777, a 313-digit number where each digit is a 7. Let f(r) be the leading digit of the rth root of N. What is…Algebra
  20. 20Problem 20In a particular game, each of 4 players rolls a standard 6-sided die. The winner is the player who rolls the highest number. If there is a tie for…Counting & Probability
  21. 21Problem 21Regular polygons with 5, 6, 7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides…Counting & Probability
  22. 22Problem 22For each integer nge 2, let S_n be the sum of all products jk, where j and k are integers and 1≤ j<k≤ n. What is the sum of the 10 least values of n…Algebra
  23. 23Problem 23Each of the 5 sides and the 5 diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the…Counting & Probability
  24. 24Problem 24A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the 2 × 2 × 2 cube using these…Geometry
  25. 25Problem 25A rectangle with side lengths 1 and 3, a square with side length 1, and a rectangle R are inscribed inside a larger square as shown. The sum of all…Geometry

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