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2020 AMC 10A problems

All 25 problems from the 2020 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 value of x satisfies x- 3/4 = 5/12 - 1/3?Algebra
  2. 2Problem 2The numbers 3, 5, 7, a, and b have an average (arithmetic mean) of 15. What is the average of a and b?Algebra
  3. 3Problem 3Assuming aneq 3, bneq 4, and cneq 5, what is the value in simplest form of the following expression? a-3/5-c · b-4/3-a · c-5/4-bAlgebra
  4. 4Problem 4A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her…Algebra
  5. 5Problem 5What is the sum of all real numbers x for which |x^2-12x+34|=2?Algebra
  6. 6Problem 6How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?Number Theory
  7. 7Problem 7The 25 integers from -10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in each row, the sum of the…Algebra
  8. 8Problem 8What is the value of 1+2+3-4+5+6+7-8 +… +197+198+199-200?Algebra
  9. 9Problem 9A single bench section at a school event can hold either 7 adults or 11 children. When N bench sections are connected end to end, an equal number of…Number Theory
  10. 10Problem 10Seven cubes, whose volumes are 1, 8, 27, 64, 125, 216, and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes…Geometry
  11. 11Problem 11What is the median of the following list of 4040 numbers? 1,2,3,…,2020, 1^2,2^2,3^2,…,2020^2Algebra
  12. 12Problem 12Triangle AMC is isosceles with AM = AC. Medians MV and CU are perpendicular to each other, and MV=CU=12. What is the area of △ AMC?Geometry
  13. 13Problem 13A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the…Algebra
  14. 14Problem 14Real numbers x and y satisfy x + y = 4 and x · y = -2. What is the value of x + x^3/y^2 + y^3/x^2 + y?Algebra
  15. 15Problem 15A positive integer divisor of 12! is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as m/n, where m…Number Theory
  16. 16Problem 16A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (2020, 0), (2020, 2020), and (0, 2020). The…Geometry
  17. 17Problem 17Define P(x)= (x-1^2)(x-2^2) …(x-100^2). How many integers n are there such that P(n)≤ 0?Algebra
  18. 18Problem 18Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3}. For how many such quadruples is it…Algebra
  19. 19Problem 19As shown in the figure below, a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two…Counting & Probability
  20. 20Problem 20Quadrilateral ABCD satisfies ∠ ABC = ∠ ACD = 90^°, AC=20, and CD=30. Diagonals AC and BD intersect at point E, and AE=5. What is the area of…Geometry
  21. 21Problem 21There exists a unique strictly increasing sequence of nonnegative integers a_1<a_2<…<a_k such that frac 2^289+12^17+1=2^a_1+2^a_2+…+2^a_k. What is k?Algebra
  22. 22Problem 22For how many positive integers n ≤ 1000 is⌊ 998/n ⌋+⌊ 999/n ⌋+⌊ 1000/n ⌋not divisible by 3? (Recall that ⌊ x ⌋ is the greatest integer less than or…Number Theory
  23. 23Problem 23Let T be the triangle in the coordinate plane with vertices (0,0), (4,0), and (0,3). Consider the following five isometries (rigid transformations)…Geometry
  24. 24Problem 24Let n be the least positive integer greater than 1000 for which gcd (63,n+120)=21 and gcd (n+63,120)=60. What is the sum of the digits of n?Number Theory
  25. 25Problem 25Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice)…Algebra

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