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2013 AMC 12A problems

All 25 problems from the 2013 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 1Square ABCD has side length 10. Point E is on BC, and the area of △ ABE is 40. What is BE?Geometry
  2. 2Problem 2A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other…Number Theory
  3. 3Problem 3A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the…Algebra
  4. 4Problem 4What is the value of dfrac 2^2014 + 2^20122^2014 - 2^2012?Algebra
  5. 5Problem 5Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid…Algebra
  6. 6Problem 6In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of her three-point shots and…Algebra
  7. 7Problem 7The sequence S_1, S_2, S_3, …, S_10 has the property that every term beginning with the third is the sum of the previous two. That is, S_n = S_n-2 +…Algebra
  8. 8Problem 8Given that x and y are distinct nonzero real numbers such that x + 2/x = y + 2/y, what is xy?Algebra
  9. 9Problem 9In △ ABC, AB = AC = 28 and BC = 20. Points D, E, and F are on sides AB, BC, and AC, respectively, such that DE and EF are parallel to AC and AB…Geometry
  10. 10Problem 10Let S be the set of positive integers n for which 1/n has the repeating decimal representation 0.ab = 0.ababab…, with a and b different digits. What…Number Theory
  11. 11Problem 11Triangle ABC is equilateral with AB = 1. Points E and G are on AC and points D and F are on AB such that both DE and FG are parallel to BC…Geometry
  12. 12Problem 12The angles in a particular triangle are in arithmetic progression, and the side lengths are 4, 5, and x. The sum of the possible values of x equals a…Algebra
  13. 13Problem 13Let points A = (0, 0), B = (1, 2), C = (3, 3), and D = (4, 0). Quadrilateral ABCD is cut into equal area pieces by a line passing through A. This…Geometry
  14. 14Problem 14The sequence log _12 162, log _12 x, log _12 y, log _12 z, log _12 1250 is an arithmetic progression. What is x?Algebra
  15. 15Problem 15Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cottontail. These five rabbits are to be distributed to four different pet stores…Counting & Probability
  16. 16Problem 16A, B, and C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight…Algebra
  17. 17Problem 17A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The kth pirate to take a share takes k/12 of the…Number Theory
  18. 18Problem 18Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are…Geometry
  19. 19Problem 19In △ ABC, AB = 86, and AC = 97. A circle with center A and radius AB intersects BC at points B and X. Moreover BX and CX have integer lengths. What…Geometry
  20. 20Problem 20Let S be the set {1, 2, 3, …, 19}. For a, b ∈ S, define a succ b to mean that either 0 < a - b ≤ 9 or b - a > 9. How many ordered triples (x, y, z)…Number Theory
  21. 21Problem 21Consider A = tiny log (2013 + log (2012 + log (2011 + log (… + log (3 + log 2)…)))). Which of the following intervals contains A?Algebra
  22. 22Problem 22A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit…Number Theory
  23. 23Problem 23ABCD is a square of side length √(3) + 1. Point P is on AC such that AP = √(2). The square region bounded by ABCD is rotated 90° counterclockwise…Geometry
  24. 24Problem 24Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that…Geometry
  25. 25Problem 25Let f: mathbb C → mathbb C be defined by f(z) = z^2 + iz + 1. How many complex numbers z are there such that Im(z) > 0 and both the real and the…Algebra

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