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2005 AMC 12A problems

All 25 problems from the 2005 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 1Two is 10% of x and 20% of y. What is x - y?Algebra
  2. 2Problem 2The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution for x. What is the value of b?Algebra
  3. 3Problem 3A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle?Geometry
  4. 4Problem 4A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and…Counting & Probability
  5. 5Problem 5The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?Algebra
  6. 6Problem 6Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike’s house. A little later Mike started to ride his bicycle…Algebra
  7. 7Problem 7Square EFGH is inside square ABCD so that each side of EFGH can be extended to pass through a vertex of ABCD. Square ABCD has side length √(50), E is…Geometry
  8. 8Problem 8Let A, M, and C be digits with (100A + 10M + C) · (A + M + C) = 2005. What is A?Number Theory
  9. 9Problem 9There are two values of a for which the equation 4x^2 + ax + 8x + 9 = 0 has only one solution for x. What is the sum of those values of a?Algebra
  10. 10Problem 10A wooden cube n units on a side is painted red on all six faces and then cut into n^3 unit cubes. Exactly one-fourth of the total number of faces of…Geometry
  11. 11Problem 11How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?Number Theory
  12. 12Problem 12A line passes through A(1, 1) and B(100, 1000). How many other points with integer coordinates are on the line and strictly between A and B?Geometry
  13. 13Problem 13In the five-sided star shown, the letters A, B, C, D, and E are replaced by the numbers 3, 5, 6, 7, and 9, although not necessarily in that order…Algebra
  14. 14Problem 14On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability…Counting & Probability
  15. 15Problem 15Let AB be a diameter of a circle and C be a point on AB with 2 · AC = BC. Let D and E be points on the circle such that DC ⊥ AB and DE is a second…Geometry
  16. 16Problem 16Three circles of radius s are drawn in the first quadrant of the xy-plane. The first circle is tangent to both axes, the second is tangent to the…Geometry
  17. 17Problem 17A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner…Geometry
  18. 18Problem 18Call a number “prime-looking” if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There…Counting & Probability
  19. 19Problem 19A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile…Number Theory
  20. 20Problem 20For each x in [0, 1], define f(x) = 2x, 0 ≤ x ≤ 1/2, 2 - 2x, 1/2 < x ≤ 1. Let f^[2](x) = f(f(x)), and f^[n+1](x) = f^[n](f(x)) for each integer n ≥…Algebra
  21. 21Problem 21How many ordered triples of integers (a, b, c), with a ≥ 2, b ≥ 1, and c ≥ 0, satisfy both log _a b = c^2005 and a + b + c = 2005?Algebra
  22. 22Problem 22A rectangular box P is inscribed in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of its 12 edges is 112. What is r?Geometry
  23. 23Problem 23Two distinct numbers a and b are chosen randomly from the set {2, 2^2, 2^3, …, 2^25}. What is the probability that log _a b is an integer?Algebra
  24. 24Problem 24Let P(x) = (x - 1)(x - 2)(x - 3). For how many polynomials Q(x) does there exist a polynomial R(x) of degree 3 such that P(Q(x)) = P(x) · R(x)?Algebra
  25. 25Problem 25Let S be the set of all points with coordinates (x, y, z), where x, y, and z are each chosen from the set {0, 1, 2}. How many equilateral triangles…Geometry

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