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2017 AMC 10A problems

All 25 problems from the 2017 AMC 10A, 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 1What is the value of the following expression? 2(2(2(2(2(2+1)+1) +1)+1)+1)+1Algebra
  2. 2Problem 2Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-popsicle boxes for $3. What…Algebra
  3. 3Problem 3Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown…Geometry
  4. 4Problem 4Mia is “helping” her mom pick up 30 toys that are strewn on the floor. Mia’s mom manages to put 3 toys into the toy box every 30 seconds, but each…Algebra
  5. 5Problem 5The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?Algebra
  6. 6Problem 6Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of…
  7. 7Problem 7Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach…Algebra
  8. 8Problem 8At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people…Counting & Probability
  9. 9Problem 9Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph, and uphill at 5 kph. Penny rides on a flat road at 30 kph, downhill…Algebra
  10. 10Problem 10Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3 cm, 7 cm, and 15 cm on a table…Geometry
  11. 11Problem 11The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume 216π. What is the length AB?Geometry
  12. 12Problem 12Let S be the set of points (x,y) in the coordinate plane such that two of the three quantities 3, x+2, and y-4 are equal and the third of the three…Geometry
  13. 13Problem 13Define a sequence recursively by F_0=0, ~F_1=1, and F_n= the remainder when F_n-1+F_n-2 is divided by 3, for all n≥ 2. Thus the sequence starts 0, 1…Algebra
  14. 14Problem 14Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger’s allowance was A dollars. The cost of his movie ticket…Algebra
  15. 15Problem 15Chloé chooses a real number uniformly at random from the interval [0, 2017]. Independently, Laurent chooses a real number uniformly at random from…Counting & Probability
  16. 16Problem 16There are 10 horses, named Horse 1, Horse 2, and so on through Horse 10. They get their names from how many minutes it takes them to run one lap…Number Theory
  17. 17Problem 17Distinct points P, Q, R, S lie on the circle x^2+y^2=25 and have integer coordinates. The distances PQ and RS are irrational numbers. What is the…Geometry
  18. 18Problem 18Amelia has a coin that lands heads with probability 1/3, and Blaine has a coin that lands on heads with probability 2/5. Amelia and Blaine…Counting & Probability
  19. 19Problem 19Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of…Counting & Probability
  20. 20Problem 20Let S(n) equal the sum of the digits of positive integer n. For example, S(1507) = 13. For a particular positive integer n, S(n) = 1274. Which of the…Number Theory
  21. 21Problem 21A square with side length x is inscribed in a right triangle with sides of length 3, 4, and 5 so that one vertex of the square coincides with the…Geometry
  22. 22Problem 22Sides AB and AC of equilateral triangle ABC are tangent to a circle at points B and C respectively. What fraction of the area of △ ABC lies outside…Geometry
  23. 23Problem 23How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5…Counting & Probability
  24. 24Problem 24For certain real numbers a, b, and c, the polynomial g(x) = x^3 + ax^2 + x + 10 has three distinct roots, and each root of g(x) is also a root of the…Algebra
  25. 25Problem 25How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For…Number Theory

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