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2006 AMC 10A problems

All 25 problems from the 2006 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 1Sandwiches at Joe’s Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?Algebra
  2. 2Problem 2Define x otimes y = x^3 - y. What is h otimes (h otimes h)?Algebra
  3. 3Problem 3The ratio of Mary’s age to Alice’s age is 3: 5. Alice is 30 years old. How many years old is Mary?Algebra
  4. 4Problem 4A digital watch displays hours and minutes with am and pm. What is the largest possible sum of the digits in the display?Algebra
  5. 5Problem 5Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half of the pizza. The cost of a…Algebra
  6. 6Problem 6What non-zero real value for x satisfies (7x)^14 = (14x)^7?Algebra
  7. 7Problem 7The 8 × 18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to…Geometry
  8. 8Problem 8A parabola with equation y = x^2 + bx + c passes through the points (2, 3) and (4, 3). What is c?Algebra
  9. 9Problem 9How many sets of two or more consecutive positive integers have a sum of 15?Algebra
  10. 10Problem 10For how many real values of x is √(120 - √(x)) an integer?Algebra
  11. 11Problem 11Which of the following describes the graph of the equation (x + y)^2 = x^2 + y^2?Algebra
  12. 12Problem 12Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these…Geometry
  13. 13Problem 13A player pays $5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is…Counting & Probability
  14. 14Problem 14A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the…Algebra
  15. 15Problem 15Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters…Algebra
  16. 16Problem 16A circle of radius 1 is tangent to a circle of radius 2. The sides of △ ABC are tangent to the circles as shown, and the sides AB and AC are…Geometry
  17. 17Problem 17In rectangle ADEH, points B and C trisect AD, and points G and F trisect HE. In addition, AH = AC = 2. What is the area of quadrilateral WXYZ shown…Geometry
  18. 18Problem 18A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters…Counting & Probability
  19. 19Problem 19How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?Algebra
  20. 20Problem 20Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a…Number Theory
  21. 21Problem 21How many four-digit positive integers have at least one digit that is a 2 or a 3?Number Theory
  22. 22Problem 22Two farmers agree that pigs are worth $300 and that goats are worth $210. When one farmer owes the other money, he pays the debt in pigs or goats…Number Theory
  23. 23Problem 23Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent touches the circles at C and D, as shown. Lines AB and CD…Geometry
  24. 24Problem 24Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?Geometry
  25. 25Problem 25A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to…Counting & Probability

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