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2006 AMC 12A problems

All 25 problems from the 2006 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 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 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 6The 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
  7. 7Problem 7Mary is 20% older than Sally, and Sally is 40% younger than Danielle. The sum of their ages is 23.2 years. How old will Mary be on her next birthday?Algebra
  8. 8Problem 8How many sets of two or more consecutive positive integers have a sum of 15?Algebra
  9. 9Problem 9Oscar buys 13 pencils and 3 erasers for $1.00. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total…Number Theory
  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 12A 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
  13. 13Problem 13The vertices of a 3–4–5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these…Geometry
  14. 14Problem 14Two 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
  15. 15Problem 15Suppose cos x = 0 and cos (x + z) = 1/2. What is the smallest possible positive value of z?Geometry
  16. 16Problem 16Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent intersects the circles at C and D, respectively. Lines AB…Geometry
  17. 17Problem 17Square ABCD has side length s, a circle centered at E has radius r, and r and s are both rational. The circle passes through D, and D lies on BE…Geometry
  18. 18Problem 18The function f has the property that for each real number x in its domain, 1/x is also in its domain and f(x) + f (1/x) = x. What is the largest set…Algebra
  19. 19Problem 19Circles with centers (2, 4) and (14, 9) have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in…Geometry
  20. 20Problem 20A 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
  21. 21Problem 21Let tiny S_1 = {(x, y) mid log _10(1 + x^2 + y^2) ≤ 1 + log _10(x + y)} and tiny S_2 = {(x, y) mid log _10(2 + x^2 + y^2) ≤ 2 + log _10(x + y)}. What…Algebra
  22. 22Problem 22A circle of radius r is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of the hexagon are…Geometry
  23. 23Problem 23Given a finite sequence S = (a_1, a_2, …, a_n) of n real numbers, let A(S) be the sequence small (a_1 + a_2/2, a_2 + a_3/2, …, frac a_n-1 + a_n2) of…Algebra
  24. 24Problem 24The expression (x + y + z)^2006 + (x - y - z)^2006 is simplified by expanding it and combining like terms. How many terms are in the simplified…Algebra
  25. 25Problem 25How many non-empty subsets S of {1, 2, 3, …, 15} have the following two properties? (1) No two consecutive integers belong to S. (2) If S contains k…Counting & Probability

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