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2023 AMC 12B problems

All 25 problems from the 2023 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 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…Algebra
  5. 5Problem 5You 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 you…Algebra
  6. 6Problem 6When the roots of the polynomial P(x)=(x-1)^1(x-2)^2 · (x-3)^3… · (x-10)^10 are removed from the number line, what remains is the union of 11…Algebra
  7. 7Problem 7For how many integers n does the expression √(log (n^2)-(log n)^2/log n-3) represent a real number, where log denotes the base 10 logarithm?Algebra
  8. 8Problem 8How many nonempty subsets B of {0,1,2,3,…,12} have the property that the number of elements in B is equal to the least element of B? For example…Counting & Probability
  9. 9Problem 9What is the area of the region in the coordinate plane defined by ||x|-1|+||y|-1|≤ 1?Algebra
  10. 10Problem 10In the xy-plane, a circle of radius 4 with center on the positive x-axis is tangent to the y-axis at the origin, and a circle with radius 10 with…Geometry
  11. 11Problem 11What is the maximum area of an isosceles trapezoid that has legs of length 1 and one base twice as long as the other?Algebra
  12. 12Problem 12For complex numbers u=a+bi and v=c+di, define the binary operation otimes by uotimes v=ac+bdi. Suppose z is a complex number such that zotimes…Algebra
  13. 13Problem 13A rectangular box P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edges of P is 13, the sum of the areas of all 6 faces of…Algebra
  14. 14Problem 14For how many ordered pairs (a,b) of integers does the polynomial x^3+ax^2+bx+6 have 3 distinct integer roots?Algebra
  15. 15Problem 15Suppose 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)=1 or…Number Theory
  16. 16Problem 16In Coinland, there are three types of coins, each worth 6, 10, and 15. What is the sum of the digits of the maximum amount of money that is…Number Theory
  17. 17Problem 17Triangle ABC has side lengths in arithmetic progression, and the smallest side has length 6. If the triangle has an angle of 120°, what is the area…Geometry
  18. 18Problem 18Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two…Algebra
  19. 19Problem 19Each of 2023 balls is placed in one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number…Algebra
  20. 20Problem 20Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his…Geometry
  21. 21Problem 21A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is 3√(3) inches, its top…Geometry
  22. 22Problem 22A real-valued function f has the property that for all real numbers a and b, f(a+b)+f(a-b) =2f(a)f(b). Which one of the following cannot be the value…Algebra
  23. 23Problem 23When n standard six-sided dice are rolled, the product of the numbers rolled can be any of 936 possible values. What is n?Number Theory
  24. 24Problem 24Suppose that a, b, c, and d are positive integers satisfying all of the following relations. abcd = 2^6· 3^9· 5^7 lcm(a,b) = 2^3· 3^2· 5^3 lcm(a,c) =…Number Theory
  25. 25Problem 25A regular pentagon with area √(5)+1 is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon…Geometry

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