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2002 AMC 10A problems

All 25 problems from the 2002 AMC 10A, 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 1The ratio dfrac 10^2000+10^200210^2001+10^2001 is closest to which of the following numbers?Algebra
  2. 2Problem 2For the nonzero numbers a, b, and c, define (a,b,c)=a/b+b/c+c/a. Find (2,12,9).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 performed is…Algebra
  4. 4Problem 4For how many positive integers m does there exist at least one positive integer n such that m· n≤ m+n?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 6Cindy 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
  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 9Suppose A, B, and C are three numbers for which 1001C-2002A=4004 and 1001B+3003A=5005. The average of the three numbers A, B, and C isAlgebra
  10. 10Problem 10Compute the sum of all the roots of (2x+3)(x-4) +(2x+3)(x-6)=0.Algebra
  11. 11Problem 11Jamal 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
  12. 12Problem 12Mr. 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
  13. 13Problem 13The sides of a triangle have lengths of 15, 20, and 25. Find the length of the shortest altitude.Geometry
  14. 14Problem 14Both roots of the quadratic equation x^2-63x+k=0 are prime numbers. The number of possible values of k isNumber Theory
  15. 15Problem 15The digits 1, 2, 3, 4, 5, 6, 7, and 9 are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four…Number Theory
  16. 16Problem 16If a+1 =b+2 =c+3 =d+4 =a+b+c+d+5, then a+b+c+d isAlgebra
  17. 17Problem 17Sarah 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
  18. 18Problem 18A 3× 3× 3 cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.)…Algebra
  19. 19Problem 19Spot’s doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area…Geometry
  20. 20Problem 20Points A, B, C, D, E, and F lie, in that order, on AF, dividing it into five segments, each of length 1. Point G is not on line AF. Point H lies on…Geometry
  21. 21Problem 21The 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
  22. 22Problem 22A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and…Algebra
  23. 23Problem 23Points A, B, C, and D lie on a line, in that order, with AB=CD and BC=12. Point E is not on the line, and BE=CE=10. The perimeter of △ AED is twice…Geometry
  24. 24Problem 24Tina 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 probability…Counting & Probability
  25. 25Problem 25In trapezoid ABCD with bases AB and CD, we have AB=52, BC=12, CD=39, and DA=5. The area of ABCD isGeometry

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