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2005 AMC 10B problems

All 25 problems from the 2005 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 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 3A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is…Algebra
  4. 4Problem 4For real numbers a and b, define a ◆ b = √(a^2 + b^2). What is the value of (5 ◆ 12) ◆ ((-12) ◆ (-5))?Algebra
  5. 5Problem 5Brianna 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
  6. 6Problem 6At 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
  7. 7Problem 7A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of…Geometry
  8. 8Problem 8An 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
  9. 9Problem 9One fair die has faces 1, 1, 2, 2, 3, 3 and another has faces 4, 4, 5, 5, 6, 6. The dice are rolled and the numbers on the top faces are added. What…Counting & Probability
  10. 10Problem 10In △ 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
  11. 11Problem 11The 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
  12. 12Problem 12Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?Counting & Probability
  13. 13Problem 13How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?Counting & Probability
  14. 14Problem 14Equilateral △ ABC has side length 2, M is the midpoint of AC, and C is the midpoint of BD. What is the area of △ CDM?Geometry
  15. 15Problem 15An 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
  16. 16Problem 16The 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
  17. 17Problem 17Suppose that 4^a = 5, 5^b = 6, 6^c = 7, and 7^d = 8. What is a · b · c · d?Algebra
  18. 18Problem 18All of David’s telephone numbers have the form 555–abc–defg, where a, b, c, d, e, f, and g are distinct digits and in increasing order, and none is…Counting & Probability
  19. 19Problem 19On 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
  20. 20Problem 20What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?Number Theory
  21. 21Problem 21Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are…Counting & Probability
  22. 22Problem 22For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + … + n?Number Theory
  23. 23Problem 23In trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC, and F as the midpoint of DA. The area of ABEF is twice the area of FECD. What…Geometry
  24. 24Problem 24Let 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
  25. 25Problem 25A subset B of the set of integers from 1 to 100, inclusive, has the property that no two elements of B sum to 125. What is the maximum possible…Counting & Probability

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