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2015 AMC 12A problems

All 25 problems from the 2015 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 the value of (2^0-1+5^2+0)^-1× 5?Algebra
  2. 2Problem 2Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?Algebra
  3. 3Problem 3Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone’s test except Payton’s, the average grade for…Algebra
  4. 4Problem 4The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller number?Algebra
  5. 5Problem 5Amelia needs to estimate the quantity a/b-c, where a, b, and c are large positive integers. She rounds each of the integers so that the calculation…Algebra
  6. 6Problem 6Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will…Algebra
  7. 7Problem 7Two right circular cylinders have the same volume. The radius of the second cylinder is 10% more than the radius of the first. What is the…Geometry
  8. 8Problem 8The ratio of the length to the width of a rectangle is 4: 3. If the rectangle has diagonal of length d, then the area may be expressed as kd^2 for…Geometry
  9. 9Problem 9A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the…Counting & Probability
  10. 10Problem 10Integers x and y with x > y > 0 satisfy x+y+xy = 80. What is x?Algebra
  11. 11Problem 11On a sheet of paper, Isabella draws a circle of radius 2, a circle of radius 3, and all possible lines simultaneously tangent to both circles…Geometry
  12. 12Problem 12The parabolas y = ax^2 - 2 and y = 4 - bx^2 intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of…Geometry
  13. 13Problem 13A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team…Number Theory
  14. 14Problem 14What is the value of a for which 1/log _2 a + 1/log _3 a + 1/log _4 a = 1?Algebra
  15. 15Problem 15What is the minimum number of digits to the right of the decimal point needed to express the fraction dfrac 1234567892^26· 5^4 as a decimal?Algebra
  16. 16Problem 16Tetrahedron ABCD has AB = 5, AC = 3, BC = 4, BD = 4, AD = 3, and CD = 12/5√(2). What is the volume of the tetrahedron?Geometry
  17. 17Problem 17Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while…Counting & Probability
  18. 18Problem 18The zeros of the function f(x) = x^2 - ax + 2a are integers. What is the sum of the possible values of a?Algebra
  19. 19Problem 19For some positive integers p, there is a quadrilateral ABCD with positive integer side lengths, perimeter p, right angles at B and C, AB = 2, and CD…Geometry
  20. 20Problem 20Isosceles triangles T and T' are not congruent but have the same area and the same perimeter. The sides of T have lengths 5, 5, and 8, while those of…Algebra
  21. 21Problem 21A circle of radius r passes through both foci of, and exactly four points on, the ellipse with equation x^2 + 16y^2 = 16. The set of all possible…Geometry
  22. 22Problem 22For each positive integer n, let S(n) be the number of sequences of length n consisting solely of the letters A and B, with no more than three As in…Number Theory
  23. 23Problem 23Let S be a square of side length 1. Two points are chosen independently at random on the sides of S. The probability that the straight-line distance…Counting & Probability
  24. 24Problem 24Rational numbers a and b are chosen at random among all rational numbers in the interval [0, 2) that can be written as fractions n/d where n and d…Algebra
  25. 25Problem 25A collection of circles in the upper half-plane, all tangent to the x-axis, is constructed in layers as follows. Layer L_0 consists of two circles of…Algebra

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