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2002 AMC 10B problems

All 25 problems from the 2002 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 1What is the value of the ratio dfrac 2^2001 · 3^20036^2002?Algebra
  2. 2Problem 2For the nonzero numbers a, b, and c, define (a, b, c) = abc/a+b+c. What is (2, 4, 6)?Algebra
  3. 3Problem 3The arithmetic mean of the nine numbers in the set {9, 99, 999, 9999, …, 999999999} is a 9-digit number M, all of whose digits are distinct. Which…Number Theory
  4. 4Problem 4What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x = 4?Algebra
  5. 5Problem 5Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. What is the area of the shaded…Geometry
  6. 6Problem 6For how many positive integers n is n^2 - 3n + 2 a prime number?Algebra
  7. 7Problem 7Let n be a positive integer such that dfrac 12 + dfrac 13 + dfrac 17 + dfrac 1n is an integer. Which of the following statements is not true?Algebra
  8. 8Problem 8Suppose July of year N has five Mondays. Which of the following must occur five times in August of year N? (Note: both months have 31 days.)Algebra
  9. 9Problem 9Using the letters A, M, O, S, and U, we can form 120 five-letter “words.” If these “words” are arranged in alphabetical order, then the “word” USAMO…Counting & Probability
  10. 10Problem 10Suppose that a and b are nonzero real numbers, and that the equation x^2 + ax + b = 0 has solutions a and b. What is the pair (a, b)?Algebra
  11. 11Problem 11The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares?Algebra
  12. 12Problem 12For which of the following values of k does the equation x - 1/x - 2 = x - k/x - 6 have no solution for x?Algebra
  13. 13Problem 13What value of x makes 8xy - 12y + 2x - 3 = 0 true for all values of y?Algebra
  14. 14Problem 14The number 25^64 · 64^25 is the square of a positive integer N. In decimal representation, what is the sum of the digits of N?Algebra
  15. 15Problem 15The positive integers A, B, A - B, and A + B are all prime numbers. The sum of these four primes isNumber Theory
  16. 16Problem 16For how many integers n is n/20 - n the square of an integer?Number Theory
  17. 17Problem 17A regular octagon ABCDEFGH has sides of length two. What is the area of △ ADG?Geometry
  18. 18Problem 18Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?Counting & Probability
  19. 19Problem 19Suppose that {a_n} is an arithmetic sequence with a_1 + a_2 + … + a_100 = 100 and a_101 + a_102 + … + a_200 = 200. What is the value of a_2 - a_1?Algebra
  20. 20Problem 20Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. What is a^2 - b^2 + c^2?Algebra
  21. 21Problem 21Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower…Algebra
  22. 22Problem 22Let △ XOY be a right-angled triangle with m∠ XOY = 90°. Let M and N be the midpoints of legs OX and OY, respectively. Given that XN = 19 and YM = 22…Geometry
  23. 23Problem 23Let {a_k} be a sequence of integers such that a_1 = 1 and a_m+n = a_m + a_n + mn for all positive integers m and n. What is a_12?Algebra
  24. 24Problem 24Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feet and revolves at the constant rate of one…Geometry
  25. 25Problem 25When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is…Algebra

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