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2018 AMC 12A problems

All 25 problems from the 2018 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 1A large urn contains 100 balls, of which 36% are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red…Algebra
  2. 2Problem 2While exploring a cave, Carl comes across a collection of 5-pound rocks worth $14 each, 4-pound rocks worth $11 each, and 1-pound rocks worth $2…Algebra
  3. 3Problem 3How many ways can a student schedule 3 mathematics courses—algebra, geometry, and number theory—in a 6-period day if no two mathematics courses can…Counting & Probability
  4. 4Problem 4Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, “We are at least 6 miles away,” Bob…Algebra
  5. 5Problem 5What is the sum of all possible values of k for which the polynomials x^2 - 3x + 2 and x^2 - 5x + k have a root in common?Algebra
  6. 6Problem 6For positive integers m and n such that m + 10 < n + 1, both the mean and the median of the set {m, m + 4, m + 10, n + 1, n + 2, 2n} are equal to n…Algebra
  7. 7Problem 7For how many (not necessarily positive) integer values of n is the value of 4000 · (2/5)^n an integer?Algebra
  8. 8Problem 8All of the triangles in the diagram below are similar to isosceles triangle ABC, in which AB = AC. Each of the 7 smallest triangles has area 1, and △…Geometry
  9. 9Problem 9Which of the following describes the largest subset of values of y within the closed interval [0, π] for which sin (x + y) ≤ sin (x) + sin (y) for…Algebra
  10. 10Problem 10How many ordered pairs of real numbers (x, y) satisfy the following system of equations? x + 3y = 3 big | |x| - |y| big | = 1Algebra
  11. 11Problem 11A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point A falls on point B. What is the length in inches of the…Geometry
  12. 12Problem 12Let S be a set of 6 integers taken from {1, 2, …, 12} with the property that if a and b are elements of S with a < b, then b is not a multiple of a…Algebra
  13. 13Problem 13How many nonnegative integers can be written in the form a_7 · 3^7 + a_6 · 3^6 + a_5 · 3^5 + a_4 · 3^4 + a_3 · 3^3 + a_2 · 3^2 + a_1 · 3^1 + a_0 ·…Number Theory
  14. 14Problem 14The solution to the equation log _3x 4 = log _2x 8, where x is a positive real number other than tfrac 13 or tfrac 12, can be written as p/q, where p…Algebra
  15. 15Problem 15A scanning code consists of a 7 × 7 grid of squares, with some of its squares colored black and the rest colored white. There must be at least one…Counting & Probability
  16. 16Problem 16Which of the following describes the set of values of a for which the curves x^2 + y^2 = a^2 and y = x^2 - a in the real xy-plane intersect at…Geometry
  17. 17Problem 17Farmer Pythagoras has a field in the shape of a right triangle. The right triangle’s legs have lengths of 3 and 4 units. In the corner where those…Geometry
  18. 18Problem 18Triangle ABC with AB = 50 and AC = 10 has area 120. Let D be the midpoint of AB, and let E be the midpoint of AC. The angle bisector of ∠ BAC…Geometry
  19. 19Problem 19Let A be the set of positive integers that have no prime factors other than 2, 3, or 5. The infinite sum 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 +…Algebra
  20. 20Problem 20Triangle ABC is an isosceles right triangle with AB = AC = 3. Let M be the midpoint of hypotenuse BC. Points I and E lie on sides AC and AB…Geometry
  21. 21Problem 21Which of the following polynomials has the greatest real root?Algebra
  22. 22Problem 22The solutions to the equations z^2 = 4 + 4√(15) i and z^2 = 2 + 2√(3) i, where i = √(-1), form the vertices of a parallelogram in the complex plane…Algebra
  23. 23Problem 23In △ PAT, ∠ P = 36°, ∠ A = 56°, and PA = 10. Points U and G lie on sides TP and TA, respectively, so that PU = AG = 1. Let M and N be the midpoints…Geometry
  24. 24Problem 24Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is…Algebra
  25. 25Problem 25For a positive integer n and nonzero digits a, b, and c, let A_n be the n-digit integer each of whose digits is equal to a; let B_n be the n-digit…Algebra

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