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2010 AMC 12A problems

All 25 problems from the 2010 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 1What is (20-(2010-201)) +(2010-(201-20))?Algebra
  2. 2Problem 2A ferry boat shuttles tourists to an island every hour starting at 10 am until its last trip, which starts at 3 pm. One day the boat captain notes…Algebra
  3. 3Problem 3Rectangle ABCD, pictured below, shares 50% of its area with square EFGH. Square EFGH shares 20% of its area with rectangle ABCD. What is AB/AD?Algebra
  4. 4Problem 4If xlt 0, then which of the following must be positive?Algebra
  5. 5Problem 5Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores…Algebra
  6. 6Problem 6A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x + 32 are three-digit and four-digit…Number Theory
  7. 7Problem 7Logan is constructing a scaled model of his town. The city’s water tower stands 40 meters high, and the top portion is a sphere that holds 100,000…Geometry
  8. 8Problem 8Triangle ABC has AB=2 · AC. Let D and E be on AB and BC, respectively, such that ∠ BAE = ∠ ACD. Let F be the intersection of segments AE and CD, and…Geometry
  9. 9Problem 9A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the…Geometry
  10. 10Problem 10The first four terms of an arithmetic sequence are p, 9, 3p-q, and 3p+q. What is the 2010th term of this sequence?Algebra
  11. 11Problem 11The solution of the equation 7^x+7=8^x can be expressed in the form x=log _b 7^7. What is b?Algebra
  12. 12Problem 12In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always…Counting & Probability
  13. 13Problem 13For how many integer values of k do the graphs of x^2+y^2=k^2 and xy=k not intersect?Geometry
  14. 14Problem 14Nondegenerate △ ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC = 8. What is the smallest possible value of the perimeter?Geometry
  15. 15Problem 15A coin is altered so that the probability that it lands on heads is less than dfrac 12, and when the coin is flipped four times, the probability of…Algebra
  16. 16Problem 16Bernardo randomly picks 3 distinct numbers from the set {1,2,3,4,5,6,7,8,9} and arranges them in descending order to form a 3-digit number. Silvia…Counting & Probability
  17. 17Problem 17Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of △ ACE is 70% of the area of the hexagon. What is the sum of all…Geometry
  18. 18Problem 18A 16-step path is to go from (-4,-4) to (4,4) with each step increasing either the x-coordinate or the y-coordinate by 1. How many such paths stay…Counting & Probability
  19. 19Problem 19Each of 2010 boxes in a line contains a single red marble, and for 1 ≤ k ≤ 2010, the box in the kth position also contains k white marbles. Isabella…Algebra
  20. 20Problem 20Arithmetic sequences (a_n) and (b_n) have integer terms with a_1=b_1=1< a_2≤ b_2 and a_nb_n=2010 for some n. What is the largest possible value of n?Number Theory
  21. 21Problem 21The graph of y=x^6-10x^5 +29x^4-4x^3+ax^2 lies above the line y=bx+c except at three values of x, where the graph and the line intersect. What is the…Algebra
  22. 22Problem 22What is the minimum value of f(x) = |x-1|+|2x-1| +|3x-1|+…+|119x-1|?Algebra
  23. 23Problem 23The number obtained from the last two nonzero digits of 90! is equal to n. What is n?Number Theory
  24. 24Problem 24Let f(x)=log _10big (sin (π x) ·sin (2π x)·sin (3π x) ·sin (4π x)·sin (5π x) ·sin (6π x)·sin (7π x) ·sin (8π x)big ). The intersection of the domain…Geometry
  25. 25Problem 25Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic…Counting & Probability

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