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2020 AMC 12B problems

All 25 problems from the 2020 AMC 12B, 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 in simplest form of the following expression? √(1) + √(1+3) + √(1+3+5) + √(1+3+5+7)Algebra
  2. 2Problem 2What is the value of the following expression? 100^2 - 7^2/70^2 - 11^2 · (70 - 11)(70 + 11)/(100 - 7)(100 + 7)Algebra
  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 5Teams A and B are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team A has won tfrac 23…Algebra
  6. 6Problem 6For all integers n ≥ 9, the value of (n + 2)! - (n + 1)!/n! is always which of the following?Algebra
  7. 7Problem 7Two nonhorizontal, non-vertical lines in the xy-coordinate plane intersect to form a 45° angle. One line has slope equal to 6 times the slope of the…Algebra
  8. 8Problem 8How many ordered pairs of integers (x, y) satisfy the equation x^2020 + y^2 = 2y?Algebra
  9. 9Problem 9A 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
  10. 10Problem 10In unit square ABCD, the inscribed circle ω intersects CD at M, and AM intersects ω at a point P different from M. What is AP?Geometry
  11. 11Problem 11As 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
  12. 12Problem 12Let AB be a diameter in a circle of radius 5sqrt 2. Let CD be a chord in the circle that intersects AB at a point E such that BE = 2sqrt 5 and ∠ AEC…Geometry
  13. 13Problem 13Which of the following is the value of √(log _2 6 + log _3 6)?Algebra
  14. 14Problem 14Bela 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
  15. 15Problem 15There 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
  16. 16Problem 16An 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
  17. 17Problem 17How many polynomials of the form x^5 + ax^4 + bx^3 + cx^2 + dx + 2020, where a, b, c, and d are real numbers, have the property that whenever r is a…Algebra
  18. 18Problem 18In 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
  19. 19Problem 19Square 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…Geometry
  20. 20Problem 20Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or…Counting & Probability
  21. 21Problem 21How many positive integers n satisfy n + 1000/70 = ⌊ √(n) ⌋? (Recall that ⌊ x ⌋ is the greatest integer not exceeding x.)Algebra
  22. 22Problem 22What is the maximum value of (2^t - 3t) t/4^t for real values of t?Algebra
  23. 23Problem 23How many integers n ≥ 2 are there such that whenever z_1, z_2, …, z_n are complex numbers such that |z_1| = |z_2| = … = |z_n| = 1 and z_1 + z_2 + … +…Algebra
  24. 24Problem 24Let D(n) denote the number of ways of writing the positive integer n as a product n = f_1 · f_2 … f_k, where k ≥ 1, the f_i are integers strictly…Number Theory
  25. 25Problem 25For each real number a with 0 ≤ a ≤ 1, let numbers x and y be chosen independently at random from the intervals [0, a] and [0, 1], respectively, and…Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.