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2013 AMC 10A problems

All 25 problems from the 2013 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 1A taxi ride costs $1.50 plus $0.25 per mile traveled. How much does a 5-mile taxi ride cost?Algebra
  2. 2Problem 2Alice is making a batch of cookies and needs 21/2 cups of sugar. Unfortunately, her measuring cup holds only 1/4 cup of sugar. How many times must…Algebra
  3. 3Problem 3Square ABCD has side length 10. Point E is on BC, and the area of △ ABE is 40. What is BE?Geometry
  4. 4Problem 4A 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
  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 6Joey and his five brothers are ages 3, 5, 7, 9, 11, and 13. One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers…
  7. 7Problem 7A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This…Counting & Probability
  8. 8Problem 8What is the value of dfrac 2^2014+2^20122^2014-2^2012?Algebra
  9. 9Problem 9In 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
  10. 10Problem 10A 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
  11. 11Problem 11A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10…Algebra
  12. 12Problem 12In △ 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
  13. 13Problem 13How many three-digit numbers are not divisible by 5, have digits that sum to less than 20, and have the first digit equal to the third digit?Number Theory
  14. 14Problem 14A solid cube of side length 1 is removed from each corner of a solid cube of side length 3. How many edges does the remaining solid have?Geometry
  15. 15Problem 15Two sides of a triangle have lengths 10 and 15. The length of the altitude to the third side is the average of the lengths of the altitudes to the…Geometry
  16. 16Problem 16A triangle with vertices (6, 5), (8, -3), and (9, 1) is reflected about the line x = 8 to create a second triangle. What is the area of the union of…Geometry
  17. 17Problem 17Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day…Counting & Probability
  18. 18Problem 18Let 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 line intersects…Geometry
  19. 19Problem 19In base 10, the number 2013 ends in the digit 3. In base 9, on the other hand, the same number is written as (2676)_9 and ends in the digit 6. For…Number Theory
  20. 20Problem 20A unit square is rotated 45° about its center. What is the area of the region swept out by the interior of the square?Geometry
  21. 21Problem 21A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^th pirate to take a share takes k/12 of the…Number Theory
  22. 22Problem 22Six 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
  23. 23Problem 23In △ 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 is…Geometry
  24. 24Problem 24Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require…Counting & Probability
  25. 25Problem 25All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more…Counting & Probability

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