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2020 AMC 10B problems

All 25 problems from the 2020 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 1 - (-2) - 3 - (-4) - 5 - (-6)?Algebra
  2. 2Problem 2Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes?Geometry
  3. 3Problem 3The ratio of w to x is 4:3, the ratio of y to z is 3:2, and the ratio of z to x is 1:6. What is the ratio of w to y?Algebra
  4. 4Problem 4The acute angles of a right triangle are a^° and b^°, where a>b and both a and b are prime numbers. What is the least possible value of b?Geometry
  5. 5Problem 5How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles…Counting & Probability
  6. 6Problem 6Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome—it reads the same forward and backward…Algebra
  7. 7Problem 7How many positive even multiples of 3 less than 2020 are perfect squares?Number Theory
  8. 8Problem 8Points P and Q lie in a plane with PQ=8. How many locations for point R in this plane are there such that the triangle with vertices P, Q, and R is a…Geometry
  9. 9Problem 9How many ordered pairs of integers (x, y) satisfy the equation x^2020+y^2=2y?Algebra
  10. 10Problem 10A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface of a right circular…Geometry
  11. 11Problem 11Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same…Counting & Probability
  12. 12Problem 12The decimal representation of frac 120^20 consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How…Algebra
  13. 13Problem 13Andy the Ant lives on a coordinate plane and is currently at (-20, 20) facing east (that is, in the positive x-direction). Andy moves 1 unit and then…Geometry
  14. 14Problem 14As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles…Geometry
  15. 15Problem 15Steve wrote the digits 1, 2, 3, 4, and 5 in order repeatedly from left to right, forming a list of 10,000 digits, beginning 123451234512… He then…Number Theory
  16. 16Problem 16Bela and Jenn play the following game on the closed interval [0, n] of the real number line, where n is a fixed integer greater than 4. They take…Counting & Probability
  17. 17Problem 17There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or…Counting & Probability
  18. 18Problem 18An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he…Counting & Probability
  19. 19Problem 19In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be…Number Theory
  20. 20Problem 20Let B be a right rectangular prism (box) with edge lengths 1, 3, and 4, together with its interior. For real rgeq 0, let S(r) be the set of points in…Geometry
  21. 21Problem 21In square ABCD, points E and H lie on AB and DA, respectively, so that AE=AH. Points F and G lie on BC and CD, respectively, and points I and J lie…Geometry
  22. 22Problem 22What is the remainder when 2^202 +202 is divided by 2^101+2^51+1?Algebra
  23. 23Problem 23Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the following four transformations: •…Geometry
  24. 24Problem 24How many positive integers n satisfy n+1000/70 = ⌊ √(n) ⌋? (Recall that ⌊ x⌋ is the greatest integer not exceeding x.)Algebra
  25. 25Problem 25Let D(n) denote the number of ways of writing the positive integer n as a product n = f_1· f_2… f_k, where kge 1, the f_i are integers strictly…Number Theory

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