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2005 AMC 12B problems

All 25 problems from the 2005 AMC 12B, 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 scout troop buys 1000 candy bars at a price of five for $2. They sell all the candy bars at a price of two for $1. What was their profit, in…Algebra
  2. 2Problem 2A positive number x has the property that x% of x is 4. What is x?Algebra
  3. 3Problem 3Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third…Algebra
  4. 4Problem 4At the beginning of the school year, Lisa’s goal was to earn an A on at least 80% of her 50 quizzes for the year. She earned an A on 22 of the first…Algebra
  5. 5Problem 5An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of…Geometry
  6. 6Problem 6In △ ABC, we have AC = BC = 7 and AB = 2. Suppose that D is a point on line AB such that B lies between A and D and CD = 8. What is BD?Geometry
  7. 7Problem 7What is the area enclosed by the graph of |3x| + |4y| = 12?Algebra
  8. 8Problem 8For how many values of a is it true that the line y = x + a passes through the vertex of the parabola y = x^2 + a^2?Algebra
  9. 9Problem 9On a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points. What…Algebra
  10. 10Problem 10The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2005th term of the…Algebra
  11. 11Problem 11An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability…Counting & Probability
  12. 12Problem 12The quadratic equation x^2 + mx + n = 0 has roots that are twice those of x^2 + px + m = 0, and none of m, n, and p is zero. What is the value of n/p?Algebra
  13. 13Problem 13Suppose that 4^x_1 = 5, 5^x_2 = 6, 6^x_3 = 7, …, 127^x_124 = 128. What is x_1 x_2 … x_124?Algebra
  14. 14Problem 14A circle having center (0, k), with k > 6, is tangent to the lines y = x, y = -x and y = 6. What is the radius of this circle?Geometry
  15. 15Problem 15The sum of four two-digit numbers is 221. None of the eight digits is 0 and no two of them are the same. Which of the following is not included among…Number Theory
  16. 16Problem 16Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the…Geometry
  17. 17Problem 17How many distinct four-tuples (a, b, c, d) of rational numbers are there with alog _10 2 + blog _10 3 + clog _10 5 + dlog _10 7 = 2005?Algebra
  18. 18Problem 18Let A(2, 2) and B(7, 7) be points in the plane. Define R as the region in the first quadrant consisting of those points C such that △ ABC is an acute…Geometry
  19. 19Problem 19Let x and y be two-digit integers such that y is obtained by reversing the digits of x. The integers x and y satisfy x^2 - y^2 = m^2 for some…Number Theory
  20. 20Problem 20Let a, b, c, d, e, f, g and h be distinct elements in the set {-7, -5, -3, -2, 2, 4, 6, 13}. What is the minimum possible value of (a + b + c + d)^2…Algebra
  21. 21Problem 21A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n?Number Theory
  22. 22Problem 22A sequence of complex numbers z_0, z_1, z_2, … is defined by the rule z_n+1 = i z_n/z_n, where z_n is the complex conjugate of z_n and i^2 = -1…Algebra
  23. 23Problem 23Let S be the set of ordered triples (x, y, z) of real numbers for which log _10(x + y) = z and log _10(x^2 + y^2) = z + 1. There are real numbers a…Algebra
  24. 24Problem 24All three vertices of an equilateral triangle are on the parabola y = x^2, and one of its sides has a slope of 2. The x-coordinates of the three…Geometry
  25. 25Problem 25Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently…Counting & Probability

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