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2012 AMC 10A problems

All 25 problems from the 2012 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 1Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5…Algebra
  2. 2Problem 2A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?Geometry
  3. 3Problem 3A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?Algebra
  4. 4Problem 4Let ∠ ABC = 24° and ∠ ABD = 20°. What is the smallest possible degree measure for ∠ CBD?Algebra
  5. 5Problem 5Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied…Algebra
  6. 6Problem 6The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of…Algebra
  7. 7Problem 7In a bag of marbles, 3/5 of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays…Algebra
  8. 8Problem 8The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?Algebra
  9. 9Problem 9A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers…Counting & Probability
  10. 10Problem 10Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic…Algebra
  11. 11Problem 11Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles…Geometry
  12. 12Problem 12A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th…Algebra
  13. 13Problem 13An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the…Algebra
  14. 14Problem 14Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every corner and alternate red and…Number Theory
  15. 15Problem 15Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of △ ABC?Geometry
  16. 16Problem 16Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining…Algebra
  17. 17Problem 17Let a and b be relatively prime integers with a > b > 0 and a^3-b^3/(a-b)^3 = 73/3. What is a-b?Algebra
  18. 18Problem 18The closed curve in the figure is made up of 9 congruent circular arcs each of length 2π/3, where each of the centers of the corresponding circles is…Geometry
  19. 19Problem 19Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same…Algebra
  20. 20Problem 20A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen…Counting & Probability
  21. 21Problem 21Let points A=(0,0,0), B=(1,0,0), C=(0,2,0), and D=(0,0,3). Points E, F, G, and H are midpoints of line segments BD, AB, AC, and DC respectively. What…Geometry
  22. 22Problem 22The sum of the first m positive odd integers is 212 more than the sum of the first n positive even integers. What is the sum of all possible values…Algebra
  23. 23Problem 23Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of…Counting & Probability
  24. 24Problem 24Let a, b, and c be positive integers with a≥ b≥ c such that a^2-b^2-c^2+ab=2011 and a^2+3b^2+3c^2-3ab-2ac-2bc=-1997. What is a?Algebra
  25. 25Problem 25Real numbers x, y, and z are chosen independently and at random from the interval [0,n] for some positive integer n. The probability that no two of…Geometry

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