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2017 AMC 12A problems

All 25 problems from the 2017 AMC 12A, 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 1Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the…Algebra
  2. 2Problem 2The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?Algebra
  3. 3Problem 3Ms. 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…
  4. 4Problem 4Jerry 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
  5. 5Problem 5At 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
  6. 6Problem 6Joy 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
  7. 7Problem 7Define a function on the positive integers recursively by f(1)=2, f(n)=f(n-1)+1 if n is even, and f(n)=f(n-2)+2 if n is odd and greater than 1. What…Algebra
  8. 8Problem 8The 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
  9. 9Problem 9Let 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
  10. 10Problem 10Chloé chooses a real number uniformly at random from the interval [0,2017]. Independently, Laurent chooses a real number uniformly at random from the…Counting & Probability
  11. 11Problem 11Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 2017. She then discovers that she forgot to…Algebra
  12. 12Problem 12There are 10 horses, named Horse 1, Horse 2, …, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular…Number Theory
  13. 13Problem 13Driving at a constant speed, Sharon usually takes 180 minutes to drive from her house to her mother’s house. One day Sharon begins the drive at her…Algebra
  14. 14Problem 14Alice 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
  15. 15Problem 15Let f(x)=sin x+2cos x+3tan x, using radian measure for the variable x. In what interval does the smallest positive value of x for which f(x)=0 lie?Geometry
  16. 16Problem 16In the figure below, semicircles with centers at A and B and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with…Geometry
  17. 17Problem 17There are 24 different complex numbers z such that z^24=1. For how many of these is z^6 a real number?Algebra
  18. 18Problem 18Let 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
  19. 19Problem 19A 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
  20. 20Problem 20How many ordered pairs (a,b) such that a is a positive real number and b is an integer between 2 and 200, inclusive, satisfy the equation (log _b…Algebra
  21. 21Problem 21A set S is constructed as follows. To begin, S={0,10}. Repeatedly, as long as possible, if x is an integer root of some polynomial a_nx^n+a_n-1x^n-1…Algebra
  22. 22Problem 22A square is drawn in the Cartesian coordinate plane with vertices at (2,2), (-2,2), (-2,-2), and (2,-2). A particle starts at (0,0). Every second it…Counting & Probability
  23. 23Problem 23For 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
  24. 24Problem 24Quadrilateral ABCD is inscribed in circle O and has sides AB=3, BC=2, CD=6, and DA=8. Let X and Y be points on BD such that DX/BD=1/4 and…Geometry
  25. 25Problem 25The vertices V of a centrally symmetric hexagon in the complex plane are given by V={ sqrt 2 i, -sqrt 2 i, 1/sqrt 8(1+i), 1/sqrt 8(-1+i), 1/sqrt…Algebra

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