Competition · AMC preparation · step 4 of 4

AMC 10 2006A: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2006A #1 grade 3+ arithmetic
Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiche…
AMC 10 2006A #2 grade 6+ algebra
Define xotimes y=x³-y. What is hotimes (hotimes h)?
AMC 10 2006A #3 grade 6+ rate-ratio
The ratio of Mary's age to Alice's age is 3:5. Alice is 30 years old. How old is Mary?
AMC 10 2006A #4 grade 3+ arithmetic
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display…
AMC 10 2006A #5 grade 5+ arithmetic
Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half t…
AMC 10 2006A #6 grade 8+ arithmetic
What non-zero real value for x satisfies (7x)¹⁴=(14x)⁷?
AMC 10 2006A #7 grade 6+ geometry-2d
The 8×18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be reposit…
AMC 10 2006A #8 grade 8+ algebra
A parabola with equation y=x²+bx+c passes through the points (2,3) and (4,3). What is c?
AMC 10 2006A #9 grade 6+ counting
How many sets of two or more consecutive positive integers have a sum of 15?
AMC 10 2006A #10 grade 8+ algebra, counting
For how many real values of x is √(120-√x) an integer?
AMC 10 2006A #11 grade 8+ geometry-2d
Which of the following describes the graph of the equation (x+y)²=x²+y²?
AMC 10 2006A #12 grade 7+ geometry-2d
Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawin…
AMC 10 2006A #13 grade 7+ probability
A player pays textdollar 5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the nu…
AMC 10 2006A #14 grade 4+ geometry-2d, pattern
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outs…
AMC 10 2006A #15 grade 7+ geometry-2d
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with…
AMC 10 2006A #16 grade 8+ geometry-2d
A 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…
AMC 10 2006A #17 grade 8+ geometry-2d
In rectangle ADEH, points B and C trisect AD, and points G and F trisect HE. In addition, AH=AC=2, and AD=3. What is the…
AMC 10 2006A #18 grade 7+ arithmetic
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily d…
AMC 10 2006A #19 grade 8+ arithmetic
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progressio…
AMC 10 2006A #20 grade 7+ probability
Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair…
AMC 10 2006A #21 grade 4+ arithmetic
How many four-digit positive integers have at least one digit that is a 2 or a 3?
AMC 10 2006A #22 grade 7+ number-theory
Two farmers agree that pigs are worth 300 dollars and that goats are worth 210 dollars. When one farmer owes the other m…
AMC 10 2006A #23 grade 8+ geometry-2d
Circles with centers A and B have radius 3 and 8, respectively. A common internal tangent intersects the circles at C an…
AMC 10 2006A #24 grade 8+ geometry-3d
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
AMC 10 2006A #25 grade 7+ probability, geometry-3d
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vert…

A parent dashboard for the family lives at sensimlab.com.