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2021 AMC 10A problems

All 25 problems from the 2021 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 (2^2-2)-(3^2-3)+(4^2-4)?Algebra
  2. 2Problem 2Portia’s high school has 3 times as many students as Lara’s high school. The two high schools have a total of 2600 students. How many students does…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 4A cart rolls down a hill, traveling 5 inches the first second and accelerating so that during each successive 1-second time interval, it travels 7…Algebra
  5. 5Problem 5The quiz scores of a class with k > 12 students have a mean of 8. The mean of a collection of 12 of these quiz scores is 14. What is the mean of the…Algebra
  6. 6Problem 6Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at 4…Algebra
  7. 7Problem 7Tom 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…
  8. 8Problem 8When a student multiplied the number 66 by the repeating decimal, 1.a b a b…=1.a b, where a and b are digits, he did not notice the notation and just…Algebra
  9. 9Problem 9What is the least possible value of (xy-1)^2+(x+y)^2 for real numbers x and y?Algebra
  10. 10Problem 10Which 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
  11. 11Problem 11For which of the following integers b is the base-b number 2021_b - 221_b not divisible by 3?Number Theory
  12. 12Problem 12Two 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
  13. 13Problem 13What is the volume of tetrahedron ABCD with edge lengths AB = 2, AC = 3, AD = 4, BC = √(13), BD = 2√(5), and CD = 5?Geometry
  14. 14Problem 14All 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
  15. 15Problem 15Values for A, B, C, and D are to be selected from {1, 2, 3, 4, 5, 6} without replacement (i.e., no two letters have the same value). How many ways…Geometry
  16. 16Problem 16In the following list of numbers, the integer n appears n times in the list for 1≤ nleq 200. 1,2,2,3,3,3,4,4,4,4,…, 200,200,…,200 What is the median…Algebra
  17. 17Problem 17Trapezoid ABCD has ABparallelCD, BC=CD=43, and ADperpBD. 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 and…Algebra
  19. 19Problem 19The area of the region bounded by the graph of x^2+y^2 = 3|x-y| + 3|x+y| is m+nπ, where m and n are integers. What is m + n?Geometry
  20. 20Problem 20In how many ways can the sequence 1, 2, 3, 4, 5 be rearranged so that no three consecutive terms are increasing and no three consecutive terms are…Counting & Probability
  21. 21Problem 21Let ABCDEF be an equiangular hexagon. The lines AB, CD, and EF determine a triangle with area 192√(3), and the lines BC, DE, and FA determine a…Geometry
  22. 22Problem 22Hiram’s algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2, the second sheet contains…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…Counting & Probability
  24. 24Problem 24The interior of a quadrilateral is bounded by the graphs of (x+ay)^2=4a^2 and (ax-y)^2=a^2, where a is a positive real number. What is the area of…Geometry
  25. 25Problem 25How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable green chips in the squares of…Counting & Probability

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