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2001 AMC 12 problems

All 25 problems from the 2001 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 1The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two…Algebra
  2. 2Problem 2Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a…Number Theory
  3. 3Problem 3The state income tax where Kristin lives is levied at the rate of p% of the first $28000 of annual income plus (p + 2)% of any amount above $28000…Algebra
  4. 4Problem 4The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their…Algebra
  5. 5Problem 5What is the product of all positive odd integers less than 10,000?Algebra
  6. 6Problem 6A telephone number has the form ABC - DEF - GHIJ, where each letter represents a different digit. The digits in each part of the number are in…Number Theory
  7. 7Problem 7A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price…Number Theory
  8. 8Problem 8Which of the cones below can be formed from a 252° sector of a circle of radius 10 by aligning the two straight sides?Geometry
  9. 9Problem 9Let f be a function satisfying f(xy) = f(x)/y for all positive real numbers x and y. If f(500) = 3, what is the value of f(600)?Algebra
  10. 10Problem 10The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toGeometry
  11. 11Problem 11A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are…Counting & Probability
  12. 12Problem 12How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?Counting & Probability
  13. 13Problem 13The parabola with equation y = ax^2 + bx + c and vertex (h, k) is reflected about the line y = k. This results in the parabola with equation y = dx^2…Geometry
  14. 14Problem 14Given the nine-sided regular polygon A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9, how many distinct equilateral triangles in the plane of the polygon have at…Counting & Probability
  15. 15Problem 15An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the…Geometry
  16. 16Problem 16A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that…Counting & Probability
  17. 17Problem 17A point P is selected at random from the interior of the pentagon with vertices A = (0, 2), B = (4, 0), C = (2π + 1, 0), D = (2π + 1, 4), and E = (0…Geometry
  18. 18Problem 18A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the…Geometry
  19. 19Problem 19The polynomial P(x) = x^3 + ax^2 + bx + c has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are…Algebra
  20. 20Problem 20Points A = (3, 9), B = (1, 1), C = (5, 3), and D = (a, b) lie in the first quadrant and are the vertices of quadrilateral ABCD. The quadrilateral…Geometry
  21. 21Problem 21Four positive integers a, b, c, and d have a product of 8! and satisfy ab + a + b = 524,bc + b + c = 146,cd + c + d = 104. What is a - d?Algebra
  22. 22Problem 22In rectangle ABCD, points F and G lie on AB so that AF = FG = GB and E is the midpoint of DC. Also, AC intersects EF at H and EG at J. The area of…Geometry
  23. 23Problem 23A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the…Algebra
  24. 24Problem 24In triangle ABC, ∠ ABC = 45°. Point D is on BC so that 2 · BD = CD and ∠ DAB = 15°. Find ∠ ACB.Geometry
  25. 25Problem 25Consider sequences of positive real numbers of the form x, 2000, y, …, in which every term after the first is 1 less than the product of its two…Algebra

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