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2003 AMC 12A problems

All 25 problems from the 2003 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 1What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers?Algebra
  2. 2Problem 2Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member…Algebra
  3. 3Problem 3A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the…Algebra
  4. 4Problem 4It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same…Algebra
  5. 5Problem 5The sum of the two 5-digit numbers AMC10 and AMC12 is 123422. What is A + M + C?Number Theory
  6. 6Problem 6Define xheartsuit y to be |x - y| for all real numbers x and y. Which of the following statements is not true?Algebra
  7. 7Problem 7How many non-congruent triangles with perimeter 7 have integer side lengths?Geometry
  8. 8Problem 8What is the probability that a randomly drawn positive factor of 60 is less than 7?Number Theory
  9. 9Problem 9A set S of points in the xy-plane is symmetric about the origin, both coordinate axes, and the line y = x. If (2, 3) is in S, what is the smallest…Geometry
  10. 10Problem 10Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of 3: 2: 1, respectively…Algebra
  11. 11Problem 11A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the square and B be the area of the…Geometry
  12. 12Problem 12Sally has five red cards numbered 1 through 5 and four blue cards numbered 3 through 6. She stacks the cards so that the colors alternate and so that…Number Theory
  13. 13Problem 13The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to…Geometry
  14. 14Problem 14Points K, L, M, and N lie in the plane of the square ABCD so that AKB, BLC, CMD, and DNA are equilateral triangles. If ABCD has an area of 16, find…Geometry
  15. 15Problem 15A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the…Geometry
  16. 16Problem 16A point P is chosen at random in the interior of equilateral triangle ABC. What is the probability that △ ABP has a greater area than each of △ ACP…Geometry
  17. 17Problem 17Square ABCD has sides of length 4, and M is the midpoint of CD. A circle with radius 2 and center M intersects a circle with radius 4 and center A at…Geometry
  18. 18Problem 18Let n be a 5-digit number, and let q and r be the quotient and remainder, respectively, when n is divided by 100. For how many values of n is q + r…Number Theory
  19. 19Problem 19A parabola with equation y = ax^2 + bx + c is reflected about the x-axis. The parabola and its reflection are translated horizontally five units in…Geometry
  20. 20Problem 20How many 15-letter arrangements of 5 A’s, 5 B’s, and 5 C’s have no A’s in the first 5 letters, no B’s in the next 5 letters, and no C’s in the last 5…Counting & Probability
  21. 21Problem 21The graph of the polynomial P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e has five distinct x-intercepts, one of which is at (0, 0). Which of the…Algebra
  22. 22Problem 22Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts at (0, 0) and each of its…Counting & Probability
  23. 23Problem 23How many perfect squares are divisors of the product 1! · 2! · 3! … 9!?Number Theory
  24. 24Problem 24If a ≥ b > 1, what is the largest possible value of log _a(a/b) + log _b(b/a)?Algebra
  25. 25Problem 25Let f(x) = √(ax^2 + bx). For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the…Algebra

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