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2000 AMC 10 problems

All 25 problems from the 2000 AMC 10, 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 1In the year 2001, the United States will host the International Mathematical Olympiad. Let I, M, and O be distinct positive integers such that the…Algebra
  2. 2Problem 2Which of the following is equal to 2000 · 2000^2000?Algebra
  3. 3Problem 3Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many…Algebra
  4. 4Problem 4Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her…Algebra
  5. 5Problem 5Points M and N are the midpoints of sides PA and PB of △ PAB. As P moves along a line that is parallel to side AB, how many of the four quantities…Geometry
  6. 6Problem 6The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, … starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the…Algebra
  7. 7Problem 7In rectangle ABCD, AD = 1, P is on AB, and DB and DP trisect ∠ ADC. What is the perimeter of △ BDP?Geometry
  8. 8Problem 8At Olympic High School, tfrac 25 of the freshmen and tfrac 45 of the sophomores took the AMC 10. Given that the number of freshmen and sophomore…Algebra
  9. 9Problem 9If |x - 2| = p, where x < 2, then x - p =Algebra
  10. 10Problem 10The sides of a triangle with positive area have lengths 4, 6, and x. The sides of a second triangle with positive area have lengths 4, 6, and y. What…Algebra
  11. 11Problem 11Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be…Number Theory
  12. 12Problem 12Figures 0, 1, 2, and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping…Algebra
  13. 13Problem 13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the…Counting & Probability
  14. 14Problem 14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the…Number Theory
  15. 15Problem 15Two non-zero real numbers, a and b, satisfy ab = a - b. Find a possible value of a/b + b/a - ab.Algebra
  16. 16Problem 16The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E. Find the length of segment AE.Geometry
  17. 17Problem 17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies…Algebra
  18. 18Problem 18Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km…Geometry
  19. 19Problem 19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a…Geometry
  20. 20Problem 20Let A, M, and C be nonnegative integers such that A + M + C = 10. What is the maximum value of A · M · C + A · M + M · C + C · A?Algebra
  21. 21Problem 21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? I. All alligators are creepy…Counting & Probability
  22. 22Problem 22One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but…Algebra
  23. 23Problem 23When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression…Algebra
  24. 24Problem 24Let f be a function for which f(x/3) = x^2 + x + 1. Find the sum of all values of z for which f(3z) = 7.Algebra
  25. 25Problem 25In year N, the 300th day of the year is a Tuesday. In year N + 1, the 200th day is also a Tuesday. On what day of the week did the 100th day of year…Algebra

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