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2014 AMC 12B problems

All 25 problems from the 2014 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 1Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies…Algebra
  2. 2Problem 2Orvin went to the store with just enough money to buy 30 balloons. When he arrived he discovered that the store had a special sale on balloons: buy 1…Algebra
  3. 3Problem 3Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how…Algebra
  4. 4Problem 4Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a…Algebra
  5. 5Problem 5Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5: 2, and the borders…Algebra
  6. 6Problem 6Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After…Algebra
  7. 7Problem 7For how many positive integers n is n/30-n also a positive integer?Number Theory
  8. 8Problem 8In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? cccccc A B B C B + B C A D A hline D B…Number Theory
  9. 9Problem 9Convex quadrilateral ABCD has AB = 3, BC = 4, CD = 13, AD = 12, and ∠ ABC = 90°, as shown. What is the area of the quadrilateral?Geometry
  10. 10Problem 10Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles was displayed on…Number Theory
  11. 11Problem 11A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?Algebra
  12. 12Problem 12A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest…Geometry
  13. 13Problem 13Real numbers a and b are chosen with 1 < a < b such that no triangle with positive area has side lengths 1, a, and b or tfrac 1b, tfrac 1a, and 1…Algebra
  14. 14Problem 14A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths…Geometry
  15. 15Problem 15When p = sum _k=1^6 k ln k, the number e^p is an integer. What is the largest power of 2 that is a factor of e^p?Algebra
  16. 16Problem 16Let P be a cubic polynomial with P(0) = k, P(1) = 2k, and P(-1) = 3k. What is P(2) + P(-2)?Algebra
  17. 17Problem 17Let P be the parabola with equation y = x^2 and let Q = (20, 14). There are real numbers r and s such that the line through Q with slope m does not…Algebra
  18. 18Problem 18The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can find a subset…Counting & Probability
  19. 19Problem 19A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of…Geometry
  20. 20Problem 20For how many positive integers x is log _10(x - 40) + log _10(60 - x) < 2?Algebra
  21. 21Problem 21In the figure, ABCD is a square of side length 1. The rectangles JKHG and EBCF are congruent. What is BE?Geometry
  22. 22Problem 22In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad N, 0 < N < 10, it will…Counting & Probability
  23. 23Problem 23The number 2017 is prime. Let S = sum _k=0^62 binom 2014k. What is the remainder when S is divided by 2017?Algebra
  24. 24Problem 24Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3, BC = DE = 10, and AE = 14. The sum of the lengths of all diagonals of ABCDE is…Geometry
  25. 25Problem 25What is the sum of all positive real solutions x to the equation small 2cos (2x)(cos (2x) - cos (2014π^2/x)) = cos (4x) - 1?Geometry

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