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2003 AMC 10B problems

All 25 problems from the 2003 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 1Which of the following is the same as 2-4+6-8+10-12+14/3-6+9-12+15-18+21?Algebra
  2. 2Problem 2Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill…Algebra
  3. 3Problem 3The sum of 5 consecutive even integers is 4 less than the sum of the first 8 consecutive odd counting numbers. What is the smallest of the even…Algebra
  4. 4Problem 4Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular…Algebra
  5. 5Problem 5Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make…Algebra
  6. 6Problem 6Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a…Algebra
  7. 7Problem 7The symbolism ⌊ x ⌋ denotes the largest integer not exceeding x. For example, ⌊ 3 ⌋ = 3, and ⌊ 9/2 ⌋ = 4. Compute ⌊ √(1) ⌋ + ⌊ √(2) ⌋ + ⌊ √(3) ⌋ + ……Algebra
  8. 8Problem 8The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?Algebra
  9. 9Problem 9Find the value of x that satisfies the equation 25^-2 = dfrac 5^48/x5^26/x · 25^17/x.Algebra
  10. 10Problem 10Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new…Algebra
  11. 11Problem 11A line with slope 3 intersects a line with slope 5 at the point (10, 15). What is the distance between the x-intercepts of these two lines?Geometry
  12. 12Problem 12Al, Betty, and Clare split $1000 among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a…Algebra
  13. 13Problem 13Let clubsuit (x) denote the sum of the digits of the positive integer x. For example, clubsuit (8)=8 and clubsuit (123)=1+2+3=6. For how many…Number Theory
  14. 14Problem 14Given that 3^8 · 5^2 = a^b, where both a and b are positive integers, find the smallest possible value for a+b.Number Theory
  15. 15Problem 15There are 100 players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In…Counting & Probability
  16. 16Problem 16A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a…Algebra
  17. 17Problem 17An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream…Geometry
  18. 18Problem 18What is the largest integer that is a divisor of (n+1)(n+3)(n+5) · (n+7)(n+9) for all positive even integers n?Number Theory
  19. 19Problem 19Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four…Geometry
  20. 20Problem 20In rectangle ABCD, AB=5 and BC=3. Points F and G are on CD so that DF=1 and GC=2. Lines AF and BG intersect at E. Find the area of △ AEB.Geometry
  21. 21Problem 21A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color…Counting & Probability
  22. 22Problem 22A clock chimes once at 30 minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon…Algebra
  23. 23Problem 23A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?Geometry
  24. 24Problem 24The first four terms in an arithmetic sequence are x+y, x-y, xy, and x/y, in that order. What is the fifth term?Algebra
  25. 25Problem 25How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits?Number Theory

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