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2002 AMC 12A problems

All 25 problems from the 2002 AMC 12A, 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 1Compute the sum of all the roots of (2x+3)(x-4) +(2x+3)(x-6)=0.Algebra
  2. 2Problem 2Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the…Algebra
  3. 3Problem 3According to the standard convention for exponentiation, 2^2^2^2 = 2^(2^(2^2)) = 2^16 = 65,536. If the order in which the exponentiations are…Algebra
  4. 4Problem 4Find the degree measure of an angle whose complement is 25% of its supplement.Algebra
  5. 5Problem 5Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those…Geometry
  6. 6Problem 6For how many positive integers m does there exist at least one positive integer n such that m · n ≤ m + n?Algebra
  7. 7Problem 7If an arc of 45° on circle A has the same length as an arc of 30° on circle B, then the ratio of the area of circle A to the area of circle B isGeometry
  8. 8Problem 8Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let B be the total area of the blue triangles, W…Geometry
  9. 9Problem 9Jamal wants to store 30 computer files on floppy disks, each of which has a capacity of 1.44 megabytes (mb). Three of his files require 0.8 mb of…Algebra
  10. 10Problem 10Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the…Algebra
  11. 11Problem 11Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three…Algebra
  12. 12Problem 12Both roots of the quadratic equation x^2 - 63x + k = 0 are prime numbers. The number of possible values of k isNumber Theory
  13. 13Problem 13Two different positive numbers a and b each differ from their reciprocals by 1. What is a + b?Algebra
  14. 14Problem 14For all positive integers n, let f(n) = log _2002 n^2. Let N = f(11) + f(13) + f(14). Which of the following relations is true?Algebra
  15. 15Problem 15The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this…Algebra
  16. 16Problem 16Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, …, 10}. The…Counting & Probability
  17. 17Problem 17Several sets of prime numbers, such as {7, 83, 421, 659}, use each of the nine nonzero digits exactly once. What is the smallest possible sum such a…Number Theory
  18. 18Problem 18Let C_1 and C_2 be circles defined by (x - 10)^2 + y^2 = 36 and (x + 15)^2 + y^2 = 81, respectively. What is the length of the shortest line segment…Geometry
  19. 19Problem 19The graph of the function f is shown below. How many solutions does the equation f(f(x)) = 6 have?Algebra
  20. 20Problem 20Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.ab is expressed as a fraction in lowest terms. How many…Number Theory
  21. 21Problem 21Consider the sequence of numbers 4, 7, 1, 8, 9, 7, 6, … For n > 2, the nth term of the sequence is the units digit of the sum of the two previous…Algebra
  22. 22Problem 22Triangle ABC is a right triangle with ∠ ACB as its right angle, m∠ ABC = 60°, and AB = 10. Let P be randomly chosen inside △ ABC, and extend BP to…Geometry
  23. 23Problem 23In triangle ABC, side AC and the perpendicular bisector of BC meet in point D, and BD bisects ∠ ABC. If AD = 9 and DC = 7, what is the area of…Geometry
  24. 24Problem 24Find the number of ordered pairs of real numbers (a, b) such that (a + bi)^2002 = a - bi.Algebra
  25. 25Problem 25The nonzero coefficients of a polynomial P with real coefficients are all replaced by their mean to form a polynomial Q. Which of the following could…Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.