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2022 AMC 10A problems

All 25 problems from the 2022 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 3+1/3+1/3+1/3?Algebra
  2. 2Problem 2Mike cycled 15 laps in 57 minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first 27…Algebra
  3. 3Problem 3The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the second number. What is the…Algebra
  4. 4Problem 4In some countries, automobile fuel efficiency is measured in liters per 100 kilometers while other countries use miles per gallon. Suppose that 1…Algebra
  5. 5Problem 5Square ABCD has side length 1. Points P, Q, R, and S each lie on a side of ABCD such that APQCRS is an equilateral convex hexagon with side length s…Geometry
  6. 6Problem 6Which expression is equal to |a-2-√((a-1)^2)| for a < 0?Algebra
  7. 7Problem 7The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What is the sum of the digits of…Number Theory
  8. 8Problem 8A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and X. The average (arithmetic mean) of the 6 numbers equals a value in the…Algebra
  9. 9Problem 9A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color—red, orange, yellow, blue, or green—so that regions…Counting & Probability
  10. 10Problem 10Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite…Geometry
  11. 11Problem 11Ted mistakenly wrote 2^m·√(1/4096) as 2·√(1/4096). What is the sum of all real numbers m for which these two expressions have the same value?Algebra
  12. 12Problem 12On Halloween 31 children walked into the principal’s office asking for candy. They can be classified into three types: Some always lie; some always…Counting & Probability
  13. 13Problem 13Let △ ABC be a scalene triangle. Point P lies on BC so that AP bisects ∠ BAC. The line through B perpendicular to AP intersects the line through A…Geometry
  14. 14Problem 14How many ways are there to split the integers 1 through 14 into 7 pairs such that in each pair, the greater number is at least 2 times the lesser…Counting & Probability
  15. 15Problem 15Quadrilateral ABCD with side lengths AB = 7, BC = 24, CD = 20, DA = 15 is inscribed in a circle. The area interior to the circle but exterior to the…Geometry
  16. 16Problem 16The roots of the polynomial 10x^3 - 39x^2 + 29x - 6 are the height, length, and width of a rectangular box (right rectangular prism). A new…Algebra
  17. 17Problem 17How many three-digit positive integers a b c are there whose nonzero digits a, b, and c satisfy 0.a~b~c = 1/3 (0.a + 0.b + 0.c)? (The bar indicates…Number Theory
  18. 18Problem 18Let T_k be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the origin and then reflects the…Geometry
  19. 19Problem 19Let L_n denote the least common multiple of the numbers 1, 2, 3, …, n, and let h be the unique positive integer such that 1/1 + 1/2 + 1/3 + … + 1/17…Number Theory
  20. 20Problem 20A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term…Algebra
  21. 21Problem 21A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of the adjacent hexagons coincide, as shown in the…Geometry
  22. 22Problem 22Suppose that 13 cards numbered 1, 2, 3, …, 13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly…Counting & Probability
  23. 23Problem 23Isosceles trapezoid ABCD has parallel sides AD and BC, with BC < AD and AB = CD. There is a point P in the plane such that PA=1, PB=2, PC=3, and…Geometry
  24. 24Problem 24How many strings of length 5 formed from the digits 0, 1, 2, 3, 4, are there such that for each j ∈ {1,2,3,4}, at least j of the digits are less than…Counting & Probability
  25. 25Problem 25Let R, S, and T be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together…Algebra

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