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2018 AMC 12B problems

All 25 problems from the 2018 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 1Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread…Geometry
  2. 2Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second…Algebra
  3. 3Problem 3A line with slope 2 intersects a line with slope 6 at the point (40, 30). What is the distance between the x-intercepts of these two lines?Geometry
  4. 4Problem 4A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?Geometry
  5. 5Problem 5How many subsets of {2, 3, 4, 5, 6, 7, 8, 9} contain at least one prime number?Counting & Probability
  6. 6Problem 6Suppose S cans of soda can be purchased from a vending machine for Q quarters. Which of the following expressions describes the number of cans of…Algebra
  7. 7Problem 7What is the value of log _3 7·log _5 9·log _7 11 ·log _9 13…log _21 25·log _23 27?Algebra
  8. 8Problem 8Line segment AB is a diameter of a circle with AB=24. Point C, not equal to A or B, lies on the circle. As point C moves around the circle, the…Geometry
  9. 9Problem 9What is sum _i=1^100sum _j=1^100(i+j)?Algebra
  10. 10Problem 10A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the…Algebra
  11. 11Problem 11A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices…Geometry
  12. 12Problem 12Side AB of △ ABC has length 10. The bisector of angle A meets BC at D, and CD=3. The set of all possible values of AC is an open interval (m, n)…Geometry
  13. 13Problem 13Square ABCD has side length 30. Point P lies inside the square so that AP=12 and BP=26. The centroids of △ ABP, △ BCP, △ CDP, and △ DAP are the…Geometry
  14. 14Problem 14Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the…Number Theory
  15. 15Problem 15How many 3-digit positive odd multiples of 3 do not include the digit 3?Number Theory
  16. 16Problem 16The solutions to the equation (z+6)^8=81 are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled A…Geometry
  17. 17Problem 17Let p and q be positive integers such that 5/9ltp /qlt 4/7 and q is as small as possible. What is q-p?Algebra
  18. 18Problem 18A function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+n for all integers nge 3. What is f(2018)?Algebra
  19. 19Problem 19Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: 1, 2, …, n/2, n. At some moment…Number Theory
  20. 20Problem 20Let ABCDEF be a regular hexagon with side length 1. Denote by X, Y, and Z the midpoints of sides AB, CD, and EF, respectively. What is the area of…Geometry
  21. 21Problem 21In △ ABC with side lengths AB=13, AC=12, and BC=5, let O and I denote the circumcenter and incenter, respectively. A circle with center M is tangent…Geometry
  22. 22Problem 22Consider polynomials P(x) of degree at most 3, each of whose coefficients is an element of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. How many such polynomials…Algebra
  23. 23Problem 23Ajay is standing at point A near Pontianak, Indonesia, 0° latitude and 110° E longitude. Billy is standing at point B near Big Baldy Mountain, Idaho…Geometry
  24. 24Problem 24Let ⌊ x⌋ denote the greatest integer less than or equal to x. How many real numbers x satisfy the equation x^2+10,000⌊ x⌋=10,000x?Algebra
  25. 25Problem 25Circles omega _1, omega _2, and omega _3 each have radius 4 and are placed in the plane so that each circle is externally tangent to the other two…Geometry

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