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2009 AMC 10A problems

All 25 problems from the 2009 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 1One can holds 12 ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?Number Theory
  2. 2Problem 2Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be…Algebra
  3. 3Problem 3Which of the following is equal to 1 + cfrac 11 + cfrac 11 + 1?Algebra
  4. 4Problem 4Eric plans to compete in a triathlon. He can average 2 miles per hour in the tfrac 14-mile swim and 6 miles per hour in the 3-mile run. His goal is…Algebra
  5. 5Problem 5What is the sum of the digits of the square of 111,111,111?Number Theory
  6. 6Problem 6A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the…Geometry
  7. 7Problem 7A carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole milk. What is the percentage of…Algebra
  8. 8Problem 8Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50%…Algebra
  9. 9Problem 9Positive integers a, b, and 2009, with a < b < 2009, form a geometric sequence with an integer ratio. What is a?Algebra
  10. 10Problem 10Triangle ABC has a right angle at B. Point D is the foot of the altitude from B, AD = 3, and DC = 4. What is the area of △ ABC?Geometry
  11. 11Problem 11One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5…Algebra
  12. 12Problem 12In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an integer. What is BD?Algebra
  13. 13Problem 13Suppose that P = 2^m and Q = 3^n. Which of the following is equal to 12^mn for every pair of integers (m, n)?Algebra
  14. 14Problem 14Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of…Geometry
  15. 15Problem 15The figures F_1, F_2, F_3, and F_4 shown are the first in a sequence of figures. For n ≥ 3, F_n is constructed from F_n-1 by surrounding it with a…Algebra
  16. 16Problem 16Let a, b, c, and d be real numbers with |a - b| = 2, |b - c| = 3, and |c - d| = 4. What is the sum of all possible values of |a - d|?Algebra
  17. 17Problem 17Rectangle ABCD has AB = 4 and BC = 3. Segment EF is constructed through B so that EF ⊥ DB, and A and C lie on DE and DF, respectively. What is EF?Geometry
  18. 18Problem 18At Jefferson Summer Camp, 60% of the children play soccer, 30% of the children swim, and 40% of the soccer players swim. To the nearest whole…Counting & Probability
  19. 19Problem 19Circle A has radius 100. Circle B has an integer radius r < 100 and remains internally tangent to circle A as it rolls once around the circumference…Number Theory
  20. 20Problem 20Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between…Algebra
  21. 21Problem 21Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure…Geometry
  22. 22Problem 22Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time…Counting & Probability
  23. 23Problem 23Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and △ AED and △ BEC have equal areas. What is AE?Geometry
  24. 24Problem 24Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points…Counting & Probability
  25. 25Problem 25For k > 0, let I_k = 10ldots 064, where there are k zeros between the 1 and the 6. Let N(k) be the number of factors of 2 in the prime factorization…Algebra

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