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2014 AMC 10B problems

All 25 problems from the 2014 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 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 2What is dfrac 2^3 + 2^32^-3 + 2^-3?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 6Orvin 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…Algebra
  7. 7Problem 7Suppose A > B > 0 and A is x% greater than B. What is x?Algebra
  8. 8Problem 8A truck travels b/6 feet every t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?Algebra
  9. 9Problem 9For real numbers w and z, cfrac 1/w + 1/z1/w - 1/z = 2014. What is w+z/w-z?Algebra
  10. 10Problem 10In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? r ABBCB + BCADA hline DBDDDNumber Theory
  11. 11Problem 11For the consumer, a single discount of n% is more advantageous than any of the following discounts: (1) Two successive 15% discounts. (2) Three…Algebra
  12. 12Problem 12The largest divisor of 2,014,000,000 is itself. What is its fifth-largest divisor?Number Theory
  13. 13Problem 13Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of △ABC?Geometry
  14. 14Problem 14Danica 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
  15. 15Problem 15In rectangle ABCD, DC = 2 · CB and points E and F lie on AB so that ED and FD trisect ∠ ADC as shown. What is the ratio of the area of △ DEF to the…Geometry
  16. 16Problem 16Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?Counting & Probability
  17. 17Problem 17What is the greatest power of 2 that is a factor of 10^1002 - 4^501?Algebra
  18. 18Problem 18A 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
  19. 19Problem 19Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability…Geometry
  20. 20Problem 20For how many integers x is the number x^4-51x^2+50 negative?Algebra
  21. 21Problem 21Trapezoid ABCD has parallel sides AB of length 33 and CD of length 21. The other two sides are of lengths 10 and 14. The angles A and B are acute…Geometry
  22. 22Problem 22Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?Geometry
  23. 23Problem 23A 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
  24. 24Problem 24The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is textit bad if it is not true that for every n from 1 to 15 one can find a…Counting & Probability
  25. 25Problem 25In 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, where 0 < N < 10, it…Counting & Probability

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