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2003 AMC 10A problems

All 25 problems from the 2003 AMC 10A, 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…Geometry
  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 5Let d and e denote the solutions of 2x^2 + 3x - 5 = 0. What is the value of (d - 1)(e - 1)?Algebra
  6. 6Problem 6Define x heartsuit 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 9Simplify √(x√(x√(x√(x)))).Algebra
  10. 10Problem 10The 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
  11. 11Problem 11The sum of the two 5-digit numbers AMC10 and AMC12 is 123422. What is A + M + C?Number Theory
  12. 12Problem 12A point (x, y) is randomly picked from inside the rectangle with vertices (0, 0), (4, 0), (4, 1), and (0, 1). What is the probability that x < y?Geometry
  13. 13Problem 13The sum of three numbers is 20. The first is 4 times the sum of the other two. The second is seven times the third. What is the product of all three?Algebra
  14. 14Problem 14Let n be the largest integer that is the product of exactly 3 distinct prime numbers, d, e, and 10d + e, where d and e are single digits. What is the…Number Theory
  15. 15Problem 15What is the probability that an integer in the set {1, 2, 3, …, 100} is divisible by 2 and not divisible by 3?Counting & Probability
  16. 16Problem 16What is the units digit of 13^2003?Number Theory
  17. 17Problem 17The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is…Geometry
  18. 18Problem 18What is the sum of the reciprocals of the roots of the equation 2003/2004x + 1 + 1/x = 0?Algebra
  19. 19Problem 19A 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
  20. 20Problem 20A base-10 three-digit number n is selected at random. Which of the following is closest to the probability that the base-9 representation and the…Counting & Probability
  21. 21Problem 21Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these…Counting & Probability
  22. 22Problem 22In rectangle ABCD, we have AB = 8, BC = 9, H is on BC with BH = 6, E is on AD with DE = 4, line EC intersects line AH at G, and F is on line AD with…Algebra
  23. 23Problem 23A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 3…Algebra
  24. 24Problem 24Sally 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
  25. 25Problem 25Let 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

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