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2007 AMC 10A problems

All 25 problems from the 2007 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 1One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon…Algebra
  2. 2Problem 2Define a@b = ab - b^2 and a#b = a + b - ab^2. What is 6@2/6#2?Algebra
  3. 3Problem 3An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick…Geometry
  4. 4Problem 4The larger of two consecutive odd integers is three times the smaller. What is their sum?Algebra
  5. 5Problem 5A school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks…Algebra
  6. 6Problem 6At Euclid High School, the number of students taking the AMC 10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and is 85 in 2007…Algebra
  7. 7Problem 7Last year Mr. John Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state…Algebra
  8. 8Problem 8Triangles ABC and ADC are isosceles with AB = BC and AD = DC. Point D is inside △ ABC, ∠ ABC = 40°, and ∠ ADC = 140°. What is the degree measure of ∠…Geometry
  9. 9Problem 9Real numbers a and b satisfy the equations 3^a = 81^b+2 and 125^b = 5^a-3. What is ab?Algebra
  10. 10Problem 10The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20, the father is 48 years old…Algebra
  11. 11Problem 11The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this…Geometry
  12. 12Problem 12Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that…Counting & Probability
  13. 13Problem 13Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride…Algebra
  14. 14Problem 14A triangle with side lengths in the ratio 3: 4: 5 is inscribed in a circle of radius 3. What is the area of the triangle?Geometry
  15. 15Problem 15Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2, as shown. What is the area of the…Geometry
  16. 16Problem 16Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that ad…Counting & Probability
  17. 17Problem 17Suppose that m and n are positive integers such that 75m = n^3. What is the minimum possible value of m + n?Number Theory
  18. 18Problem 18Consider the 12-sided polygon ABCDEFGHIJKL, as shown. Each of its sides has length 4, and each two consecutive sides form a right angle. Suppose that…Geometry
  19. 19Problem 19A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What…Geometry
  20. 20Problem 20Suppose that the number a satisfies the equation 4 = a + a^-1. What is the value of a^4 + a^-4?Algebra
  21. 21Problem 21A sphere is inscribed in a cube that has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface…Geometry
  22. 22Problem 22A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens…Number Theory
  23. 23Problem 23How many ordered pairs (m, n) of positive integers, with m > n, have the property that their squares differ by 96?Number Theory
  24. 24Problem 24Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB, and OA = 2√(2). Segments OC and OD are tangent to the…Geometry
  25. 25Problem 25For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n)) = 2007?Number Theory

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