Competition · AMC preparation · step 4 of 4

AMC 12 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 12 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 12 2006A #2 grade 6+ algebra
Define xotimes y=x³-y. What is hotimes (hotimes h)?
AMC 12 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 12 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 12 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 12 2006A #6 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 12 2006A #7 grade 7+ algebra
Mary is 20% older than Sally, and Sally is 40% younger than Danielle. The sum of their ages is 23.2 years. How old will…
AMC 12 2006A #8 grade 6+ counting
How many sets of two or more consecutive positive integers have a sum of 15?
AMC 12 2006A #9 grade 7+ number-theory
Oscar buys 13 pencils and 3 erasers for $1.00. A pencil costs more than an eraser, and both items cost a whole number of…
AMC 12 2006A #10 grade 8+ algebra, counting
For how many real values of x is √(120-√x) an integer?
AMC 12 2006A #11 grade 8+ geometry-2d
Which of the following describes the graph of the equation (x+y)²=x²+y²?
AMC 12 2006A #12 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 12 2006A #13 grade 8+ geometry-2d
The vertices of a 3-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is t…
AMC 12 2006A #14 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 12 2006A #15 grade 11+ algebra
Suppose cos x=0 and cos (x+z)=1/2. What is the smallest possible positive value of z?
AMC 12 2006A #16 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 12 2006A #17 grade 10+ geometry-2d
Square ABCD has side length s, a circle centered at E has radius r, and r and s are both rational. The circle passes thr…
AMC 12 2006A #18 grade 9+ algebra
The 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 W…
AMC 12 2006A #19 grade 10+ geometry-2d
Circles with centers (2,4) and (14,9) have radii 4 and 9, respectively. The equation of a common external tangent to the…
AMC 12 2006A #20 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…
AMC 12 2006A #21 grade 11+ algebra, geometry-2d
Let S₁={(x,y)|log₁₀(1+x²+y²)≤ 1+log₁₀(x+y)} and S₂={(x,y)|log₁₀(2+x²+y²)≤ 2+log₁₀(x+y)}. What is the ratio of the area o…
AMC 12 2006A #22 grade 11+ geometry-2d, probability
A circle of radius r is concentric with and outside a regular hexagon of side length 2. The probability that three entir…
AMC 12 2006A #23 grade 9+ algebra, pattern
Given a finite sequence S=(a₁,a₂,… ,a_n) of n real numbers, let A(S) be the sequence ((a₁+a₂)/2,(a₂+a₃)/2,… ,(a_(n-1)+a_…
AMC 12 2006A #24 grade 8+ counting, algebra
The expression (x+y+z)²⁰⁰⁶+(x-y-z)²⁰⁰⁶ is simplified by expanding it and combining like terms. How many terms are in the…
AMC 12 2006A #25 grade 11+ counting
How many non- empty subsets S of {1,2,3,… ,15} have the following two properties? (1) No two consecutive integers belong…

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