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2000 AMC 12 problems

All 25 problems from the 2000 AMC 12, 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 2What is 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 4The 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
  5. 5Problem 5If |x - 2| = p, where x < 2, then what is x - p?Algebra
  6. 6Problem 6Two 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
  7. 7Problem 7How many positive integers b have the property that log _b 729 is a positive integer?Number Theory
  8. 8Problem 8Figures 0, 1, 2, and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping…Number Theory
  9. 9Problem 9Mrs. 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
  10. 10Problem 10The point P = (1, 2, 3) is reflected in the xy-plane, then its image Q is rotated by 180° about the x-axis to produce R, and finally, R is translated…Geometry
  11. 11Problem 11Two non-zero real numbers, a and b, satisfy ab = a - b. Find a possible value of a/b + b/a - ab.Algebra
  12. 12Problem 12Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A · M · C + A · M + M · C + C · A?Algebra
  13. 13Problem 13One 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
  14. 14Problem 14When 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
  15. 15Problem 15Let 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
  16. 16Problem 16A checkerboard of 13 rows and 17 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered…Number Theory
  17. 17Problem 17A circle centered at O has radius 1 and contains the point A. Segment AB is tangent to the circle at A and ∠ AOB = θ. If point C lies on OA and BC…Geometry
  18. 18Problem 18In 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
  19. 19Problem 19In triangle ABC, AB = 13, BC = 14, and AC = 15. Let D denote the midpoint of BC and let E denote the intersection of BC with the bisector of angle…Geometry
  20. 20Problem 20If x, y, and z are positive numbers satisfying x + 1/y = 4, y + 1/z = 1, and z + 1/x = 7/3, then what is xyz?Algebra
  21. 21Problem 21Through 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
  22. 22Problem 22The graph below shows a portion of the curve defined by the quartic polynomial P(x) = x^4 + ax^3 + bx^2 + cx + d. Which of the following is the…Algebra
  23. 23Problem 23Professor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 through 46, inclusive. He chooses his numbers so…Algebra
  24. 24Problem 24If circular arcs AC and BC have centers at B and A, respectively, then there exists a circle tangent to both arc AC and arc BC, and to AB. If the…Geometry
  25. 25Problem 25Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there…Counting & Probability

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