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2005 AMC 10A problems

All 25 problems from the 2005 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 1While eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill. What was the difference, in…Algebra
  2. 2Problem 2For each pair of real numbers a ≠ b, define the operation ★ as (a ★ b) = a+b/a-b. What is the value of ((1 ★ 2) ★ 3)?Algebra
  3. 3Problem 3The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x. What is the value of b?Algebra
  4. 4Problem 4A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle?Geometry
  5. 5Problem 5A 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…Algebra
  6. 6Problem 6The 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
  7. 7Problem 7Josh 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
  8. 8Problem 8In the figure, the length of side AB of square ABCD is √(50), E is between B and H, and BE = 1. What is the area of the inner square EFGH?Geometry
  9. 9Problem 9Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the…Counting & Probability
  10. 10Problem 10There 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
  11. 11Problem 11A 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
  12. 12Problem 12The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the…Geometry
  13. 13Problem 13How many positive integers n satisfy the following condition: (130n)^50 > n^100 > 2^200?Algebra
  14. 14Problem 14How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?Number Theory
  15. 15Problem 15How many positive cubes divide 3! · 5! · 7!?Number Theory
  16. 16Problem 16The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this…Number Theory
  17. 17Problem 17In 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 this order…Algebra
  18. 18Problem 18Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties…Counting & Probability
  19. 19Problem 19Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45°, as shown. Then it is centered and…Geometry
  20. 20Problem 20An equiangular octagon has four sides of length 1 and four sides of length √(2)/2, arranged so that no two consecutive sides have the same length…Geometry
  21. 21Problem 21For how many positive integers n does 1 + 2 + … + n evenly divide 6n?Number Theory
  22. 22Problem 22Let S be the set of the 2005 smallest positive multiples of 4, and let T be the set of the 2005 smallest positive multiples of 6. How many elements…Number Theory
  23. 23Problem 23Let 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
  24. 24Problem 24For each positive integer m > 1, let P(m) denote the greatest prime factor of m. For how many positive integers n is it true that both P(n) = √(n)…Number Theory
  25. 25Problem 25In △ ABC we have AB = 25, BC = 39, and AC = 42. Points D and E are on AB and AC respectively, with AD = 19 and AE = 14. What is the ratio of the area…Geometry

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