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2004 AMC 10B problems

All 25 problems from the 2004 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 1Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?Counting & Probability
  2. 2Problem 2How many two-digit positive integers have at least one 7 as a digit?Number Theory
  3. 3Problem 3At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48…Algebra
  4. 4Problem 4A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P?Number Theory
  5. 5Problem 5In the expression c · a^b - d, the values of a, b, c, and d are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible…Algebra
  6. 6Problem 6Which of the following numbers is a perfect square?Number Theory
  7. 7Problem 7On a trip from the United States to Canada, Isabella took d U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for…Algebra
  8. 8Problem 8Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the…Geometry
  9. 9Problem 9A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by…Geometry
  10. 10Problem 10A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains…Algebra
  11. 11Problem 11Two eight-sided dice each have faces numbered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What…Counting & Probability
  12. 12Problem 12An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c, with b > c. Let OX be a radius of…Geometry
  13. 13Problem 13In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these…Algebra
  14. 14Problem 14A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only 1/3 of the marbles…Algebra
  15. 15Problem 15Patty has 20 coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have 70 cents more. How much are…Algebra
  16. 16Problem 16Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?Geometry
  17. 17Problem 17The two digits in Jack’s age are the same as the digits in Bill’s age, but in reverse order. In five years Jack will be twice as old as Bill will be…Number Theory
  18. 18Problem 18In right triangle △ ACE, we have AC = 12, CE = 16, and EA = 20. Points B, D, and F are located on AC, CE, and EA, respectively, so that AB = 3, CD =…Geometry
  19. 19Problem 19In the sequence 2001, 2002, 2003, …, each term after the third is found by subtracting the previous term from the sum of the two terms that precede…Algebra
  20. 20Problem 20In △ ABC points D and E lie on BC and AC, respectively. If AD and BE intersect at T so that AT/DT = 3 and BT/ET = 4, what is CD/BD?Geometry
  21. 21Problem 21Let 1, 4, … and 9, 16, … be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers…Algebra
  22. 22Problem 22A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?Geometry
  23. 23Problem 23Each face of a cube is painted either red or blue, each with probability tfrac 12. The color of each face is determined independently. What is the…Geometry
  24. 24Problem 24In △ ABC we have AB = 7, AC = 8, and BC = 9. Point D is on the circumscribed circle of the triangle so that AD bisects ∠ BAC. What is the value of…Geometry
  25. 25Problem 25A circle of radius 1 is internally tangent to two circles of radius 2 at points A and B, where AB is a diameter of the smaller circle. What is the…Geometry

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