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2017 AMC 10B problems

All 25 problems from the 2017 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 1Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number…Number Theory
  2. 2Problem 2Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and…Algebra
  3. 3Problem 3Real numbers x, y, and z satisfy the inequalities 0 < x < 1, -1 < y < 0, and 1 < z < 2. Which of the following numbers is necessarily positive?Algebra
  4. 4Problem 4Suppose that x and y are nonzero real numbers such that 3x+y/x-3y=-2. What is the value of x+3y/3x-y?Algebra
  5. 5Problem 5Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry…Algebra
  6. 6Problem 6What is the largest number of solid 2-in × 2-in × 1-in blocks that can fit in a 3-in × 2-in × 3-in box?Geometry
  7. 7Problem 7Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her…Algebra
  8. 8Problem 8Points A(11, 9) and B(2, -3) are vertices of △ ABC with AB=AC. The altitude from A meets the opposite side at D(-1, 3). What are the coordinates of…Geometry
  9. 9Problem 9A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the…Counting & Probability
  10. 10Problem 10The lines with equations ax-2y=c and 2x+by=-c are perpendicular and intersect at (1, -5). What is c?Geometry
  11. 11Problem 11At Typico High School, 60% of the students like dancing, and the rest dislike it. Of those who like dancing, 80% say that they like it, and the rest…Algebra
  12. 12Problem 12Elmer’s new car gets 50% better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is…Algebra
  13. 13Problem 13There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of…Algebra
  14. 14Problem 14An integer N is selected at random in the range 1≤ N ≤ 2020. What is the probability that the remainder when N^16 is divided by 5 is 1?Number Theory
  15. 15Problem 15Rectangle ABCD has AB=3 and BC=4. Point E is the foot of the perpendicular from B to diagonal AC. What is the area of △ ADE?Geometry
  16. 16Problem 16How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0?Number Theory
  17. 17Problem 17Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a…Counting & Probability
  18. 18Problem 18In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be…Counting & Probability
  19. 19Problem 19Let ABC be an equilateral triangle. Extend side AB beyond B to a point B' so that BB'=3 · AB. Similarly, extend side BC beyond C to a point C' so…Geometry
  20. 20Problem 20The number 21! =51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that…Number Theory
  21. 21Problem 21In △ ABC, AB=6, AC=8, BC=10, and D is the midpoint of BC. What is the sum of the radii of the circles inscribed in △ ADB and △ ADC?Geometry
  22. 22Problem 22The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD=3. Point E is chosen so that ED=5 and line ED is…Geometry
  23. 23Problem 23Let N=123456789101112dots 4344 be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the…Number Theory
  24. 24Problem 24The vertices of an equilateral triangle lie on the hyperbola xy=1, and a vertex of this hyperbola is the centroid of the triangle. What is the square…Geometry
  25. 25Problem 25Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that…Algebra

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