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2011 AMC 10A problems

All 25 problems from the 2011 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 cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text…Algebra
  2. 2Problem 2A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the…Number Theory
  3. 3Problem 3Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is the value of the following expression? {{1 1 0}…Algebra
  4. 4Problem 4Let X and Y be the following sums of arithmetic sequences: X = 10+12+14+…+100, Y = 12+14+16+…+102. What is the value of Y - X?Algebra
  5. 5Problem 5At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively…Algebra
  6. 6Problem 6Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪ B, the union of A and B?Algebra
  7. 7Problem 7Which of the following equations does not have a solution?Algebra
  8. 8Problem 8Last summer 30 % of the birds living on Town Lake were geese, 25 % were swans, 10 % were herons, and 35 % were ducks. What percent of the birds that…Algebra
  9. 9Problem 9A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the…Geometry
  10. 10Problem 10A majority of the 30 students in Ms. Demeanor’s class bought pencils at the school bookstore. Each of these students bought the same number of…Number Theory
  11. 11Problem 11Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7· EB. What is the ratio of the area of EFGH to the area of ABCD?Geometry
  12. 12Problem 12The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with…Algebra
  13. 13Problem 13How many even integers are there between 200 and 700 whose digits are all different and come from the set {1,2,5,7,8,9}?Number Theory
  14. 14Problem 14A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that…Algebra
  15. 15Problem 15Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on…Algebra
  16. 16Problem 16Which of the following is equal to √(9-6√(2))+√(9+6√(2))?Algebra
  17. 17Problem 17In the eight term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A+H?Algebra
  18. 18Problem 18Circles A, B, and C each have radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What…Geometry
  19. 19Problem 19In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect…Algebra
  20. 20Problem 20Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a…Geometry
  21. 21Problem 21Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the…Counting & Probability
  22. 22Problem 22Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different…Counting & Probability
  23. 23Problem 23Seven students count from 1 to 1000 as follows: • Alice says all the numbers, except she skips the middle number in each consecutive group of three…Algebra
  24. 24Problem 24Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the…Geometry
  25. 25Problem 25Let R be a square region and n ≥ 4 an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that…Geometry

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