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2021 AMC 10B problems

All 25 problems from the 2021 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 1How many integer values of x satisfy |x| < 3π?Algebra
  2. 2Problem 2What is the value of √((3-2√(3))^2) +√((3+2√(3))^2)?Algebra
  3. 3Problem 3In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the 28…Algebra
  4. 4Problem 4At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66…Counting & Probability
  5. 5Problem 5The ages of Jonie’s four cousins are distinct single-digit positive integers. Two of the cousins’ ages multiplied together give 24, while the other…Algebra
  6. 6Problem 6Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84, and the afternoon class’s mean score…Algebra
  7. 7Problem 7In a plane, four circles with radii 1, 3, 5, and 7 are tangent to line ℓ at the same point A, but they may be on either side of ℓ. Region S consists…Geometry
  8. 8Problem 8Mr. Zhou places all the integers from 1 to 225 into a 15 by 15 grid. He places 1 in the middle square (eighth row and eighth column) and places other…Number Theory
  9. 9Problem 9The point P(a,b) in the xy-plane is first rotated counterclockwise by 90° around the point (1,5) and then reflected about the line y = -x. The image…Geometry
  10. 10Problem 10An inverted cone with base radius 12 cm and height 18 cm is full of water. The water is poured into a tall cylinder whose horizontal base has a…Geometry
  11. 11Problem 11Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with…Algebra
  12. 12Problem 12Let N = 34 · 34 · 63 · 270. What is the ratio of the sum of the odd divisors of N to the sum of the even divisors of N?Number Theory
  13. 13Problem 13Let n be a positive integer and d be a digit such that the value of the numeral 32d in base n equals 263, and the value of the numeral 324 in base n…Algebra
  14. 14Problem 14Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent…Geometry
  15. 15Problem 15The real number x satisfies the equation x+1/x = √(5). What is the value of x^11-7x^7+x^3?Algebra
  16. 16Problem 16Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, 1357, 89, and 5 are all uphill…Number Theory
  17. 17Problem 17Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered 1, 2, 3, …, 10. The score of a…
  18. 18Problem 18A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the…Counting & Probability
  19. 19Problem 19Suppose that S is a finite set of positive integers. If the greatest integer in S is removed from S, then the average value (arithmetic mean) of the…Algebra
  20. 20Problem 20The figure below is constructed from 11 line segments, each of which has length 2. The area of pentagon ABCDE can be written as √(m) + √(n), where m…Geometry
  21. 21Problem 21A square piece of paper has side length 1 and vertices A, B, C, and D in that order. As shown in the figure, the paper is folded so that vertex C…Geometry
  22. 22Problem 22Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and…Counting & Probability
  23. 23Problem 23A square with side length 8 is colored white except for 4 black isosceles right triangular regions with legs of length 2 in each corner of the square…Geometry
  24. 24Problem 24Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one “wall” among a set of several walls of bricks…Counting & Probability
  25. 25Problem 25Let S be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30, inclusive. Exactly 300 points in…Algebra

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