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2006 AMC 12B problems

All 25 problems from the 2006 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 1What is (-1)^1 + (-1)^2 + … + (-1)^2006?Algebra
  2. 2Problem 2For real numbers x and y, define x spadesuit y = (x + y)(x - y). What is 3 spadesuit (4 spadesuit 5)?Algebra
  3. 3Problem 3A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a…Algebra
  4. 4Problem 4Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a…Algebra
  5. 5Problem 5John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of…Algebra
  6. 6Problem 6Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon…Algebra
  7. 7Problem 7Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr…Counting & Probability
  8. 8Problem 8The lines x = tfrac 14 y + a, y = tfrac 14 x + b intersect at the point (1, 2). What is a + b?Algebra
  9. 9Problem 9How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?Number Theory
  10. 10Problem 10In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the…Algebra
  11. 11Problem 11Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2…Algebra
  12. 12Problem 12The parabola y = ax^2 + bx + c has vertex (p, p) and y-intercept (0, -p), where p ≠ 0. What is b?Algebra
  13. 13Problem 13Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24, and ∠ BAD = 60°. What is the area of rhombus BFDE?Geometry
  14. 14Problem 14Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4¢ per glob and J blobs of jam at 5¢ per blob. The…Number Theory
  15. 15Problem 15Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B are on the circle centered at O, and points…Geometry
  16. 16Problem 16Regular hexagon ABCDEF has vertices A and C at (0, 0) and (7, 1), respectively. What is its area?Geometry
  17. 17Problem 17For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6 on each die are in the ratio 1: 2: 3: 4: 5: 6. What is the…Counting & Probability
  18. 18Problem 18An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one…Geometry
  19. 19Problem 19Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which…Number Theory
  20. 20Problem 20Let x be chosen at random from the interval (0, 1). What is the probability that ⌊ log _10 4x ⌋ - ⌊ log _10 x ⌋ = 0? Here ⌊ x ⌋ denotes the greatest…Algebra
  21. 21Problem 21Rectangle ABCD has area 2006. An ellipse with area 2006π passes through A and C and has foci at B and D. What is the perimeter of the rectangle? (The…Algebra
  22. 22Problem 22Suppose a, b, and c are positive integers with a + b + c = 2006, and a! b! c! = m · 10^n, where m and n are integers and m is not divisible by 10…Number Theory
  23. 23Problem 23Isosceles △ ABC has a right angle at C. Point P is inside △ ABC, such that PA = 11, PB = 7, and PC = 6. Legs AC and BC have length s = √(a + b√(2))…Geometry
  24. 24Problem 24Let S be the set of all points (x, y) in the coordinate plane such that 0 ≤ x ≤ π/2 and 0 ≤ y ≤ π/2. What is the area of the subset of S for which…Geometry
  25. 25Problem 25A sequence a_1, a_2, … of non-negative integers is defined by the rule a_n+2 = |a_n+1 - a_n| for n ≥ 1. If a_1 = 999, a_2 < 999, and a_2006 = 1, how…Number Theory

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