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2008 AMC 10B problems

All 25 problems from the 2008 AMC 10B, 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 basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total…Counting & Probability
  2. 2Problem 2A 4 × 4 block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth…Algebra
  3. 3Problem 3Assume that x is a positive real number. Which is equivalent to √(x√(x))?Algebra
  4. 4Problem 4A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000, and that the total of all…Algebra
  5. 5Problem 5For real numbers a and b, define a $ b=(a-b)^2. What is (x-y)^2 $ (y-x)^2?Algebra
  6. 6Problem 6Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what…Algebra
  7. 7Problem 7An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles…Geometry
  8. 8Problem 8A class collects $50 to buy flowers for a classmate who is in the hospital. Roses cost $3 each, and carnations cost $2 each. No other flowers are to…Number Theory
  9. 9Problem 9A quadratic equation ax^2-2ax+b=0 has two real solutions. What is the average of the solutions?Algebra
  10. 10Problem 10Points A and B are on a circle of radius 5 and AB=6. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC?Geometry
  11. 11Problem 11Suppose that (u_n) is a sequence of real numbers satisfying u_n+2=2u_n+1+u_n, and that u_3=9 and u_6=128. What is u_5?Algebra
  12. 12Problem 12Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to…Algebra
  13. 13Problem 13For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence?Algebra
  14. 14Problem 14Triangle OAB has O=(0,0), B=(5,0), and A in the first quadrant. In addition, ∠ ABO=90° and ∠ AOB=30°. Suppose that OA is rotated 90° counterclockwise…Geometry
  15. 15Problem 15How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1, where b< 100?Geometry
  16. 16Problem 16Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls…Counting & Probability
  17. 17Problem 17A poll shows that 70% of all voters approve of the mayor’s work. On three separate occasions a pollster selects a voter at random. What is the…Counting & Probability
  18. 18Problem 18Bricklayer Brenda would take 9 hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together…Algebra
  19. 19Problem 19A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume…Geometry
  20. 20Problem 20The faces of a cubical die are marked with the numbers 1, 2, 2, 3, 3, and 4. The faces of a second cubical die are marked with the numbers 1, 3, 4…Counting & Probability
  21. 21Problem 21Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10. Five married couples are to sit in the chairs with men…Counting & Probability
  22. 22Problem 22Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are…Counting & Probability
  23. 23Problem 23A rectangular floor measures a feet by b feet, where a and b are positive integers with b> a. An artist paints a rectangle on the floor with the…Algebra
  24. 24Problem 24Quadrilateral ABCD has AB=BC=CD, ∠ ABC=70°, and ∠ BCD=170°. What is the degree measure of ∠ BAD?Geometry
  25. 25Problem 25Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200 feet along the path. A garbage truck…Algebra

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