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2023 AMC 10B problems

All 25 problems from the 2023 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 1Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice…Algebra
  2. 2Problem 2Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew…Algebra
  3. 3Problem 3A 3-4-5 right triangle is inscribed in circle A, and a 5-12-13 right triangle is inscribed in circle B. What is the ratio of the area of circle A to…Geometry
  4. 4Problem 4Jackson’s paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square…Geometry
  5. 5Problem 5Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 3 to each number in the list and finds that the sum of her new numbers is…Algebra
  6. 6Problem 6Let L_1 = 1, L_2 = 3, and L_n+2 = L_n+1 + L_n for n ≥ 1. How many terms in the sequence L_1, L_2, L_3, …, L_2023 are even?Algebra
  7. 7Problem 7Square ABCD is rotated 20° clockwise about its center to obtain square EFGH, as shown below. What is the degree measure of ∠ EAB?Geometry
  8. 8Problem 8What is the units digit of 2022^2023 + 2023^2022?Number Theory
  9. 9Problem 9The numbers 16 and 25 are a pair of consecutive positive perfect squares whose difference is 9. How many pairs of consecutive positive perfect…Algebra
  10. 10Problem 10You are playing a game. A 2 × 1 rectangle covers two adjacent squares (oriented either horizontally or vertically) of a 3 × 3 grid of squares, but…Algebra
  11. 11Problem 11Suzanne went to the bank and withdrew $800. The teller gave her this amount using $20 bills, $50 bills, and $100 bills, with at least one of each…Number Theory
  12. 12Problem 12When the roots of the polynomial P(x) = (x-1)^1(x-2)^2(x-3)^3 … (x-10)^10 are removed from the real number line, what remains is the union of 11…Algebra
  13. 13Problem 13What is the area of the region in the coordinate plane defined by the inequality bigl ||x| - 1bigr | + bigl ||y| - 1bigr | ≤ 1?Geometry
  14. 14Problem 14How many ordered pairs of integers (m, n) satisfy the equation m^2 + mn + n^2 = m^2 n^2?Algebra
  15. 15Problem 15What is the least positive integer m such that m · 2! · 3! · 4! · 5! … 16! is a perfect square?Number Theory
  16. 16Problem 16Define an upno to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right. Similarly, define a downno…Number Theory
  17. 17Problem 17A rectangular box mathcal P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edges of mathcal P is 13, the sum of the areas of…Geometry
  18. 18Problem 18Suppose a, b, and c are positive integers such that a/14 + b/15 = c/210. Which of the following statements are necessarily true? I. If gcd (a, 14) =…Number Theory
  19. 19Problem 19Sonya the frog chooses a point uniformly at random lying within the square [0, 6] × [0, 6] in the coordinate plane and hops to that point. She then…Counting & Probability
  20. 20Problem 20Four congruent semicircles are drawn on the surface of a sphere with radius 2, as shown, creating a closed curve that divides the surface into two…Geometry
  21. 21Problem 21Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an…Algebra
  22. 22Problem 22How many distinct values of x satisfy ⌊ x ⌋^2 - 3x + 2 = 0, where ⌊ x ⌋ denotes the largest integer less than or equal to x?Algebra
  23. 23Problem 23An arithmetic sequence of positive integers has n ≥ 3 terms, initial term a, and common difference d > 1. Carl wrote down all the terms in this…Number Theory
  24. 24Problem 24What is the perimeter of the boundary of the region consisting of all points which can be expressed as (2u - 3w, v + 4w) with 0 ≤ u ≤ 1, 0 ≤ v ≤ 1…Geometry
  25. 25Problem 25A regular pentagon with area 1 + √(5) is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller…Geometry

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