Competition · AMC preparation · step 4 of 4

AMC 12 2008A: 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 2008A #1 grade 4+ rate-ratio
A bakery owner turns on his doughnut machine at 8:30 smallAM. At 11:10 smallAM the machine has completed one third of th…
AMC 12 2008A #2 grade 6+ arithmetic
What is the reciprocal of 1/2+2/3?
AMC 12 2008A #3 grade 6+ rate-ratio
Suppose that 2/3 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as 1/2 of 5 bananas?
AMC 12 2008A #4 grade 5+ algebra
Which of the following is equal to the product 8/4·12/8·16/12·…·(4n+4)/4n·…·2008/2004?
AMC 12 2008A #5 grade 7+ number-theory
Suppose that 2x/3-x/6 is an integer. Which of the following statements must be true about x?
AMC 12 2008A #6 grade 7+ algebra
Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed…
AMC 12 2008A #7 grade 6+ rate-ratio
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of…
AMC 12 2008A #8 grade 8+ geometry-3d
What is the volume of a cube whose surface area is twice that of a cube with volume 1?
AMC 12 2008A #9 grade 8+ rate-ratio
Older television screens have an aspect ratio of 4: 3. That is, the ratio of the width to the height is 4: 3. The aspect…
AMC 12 2008A #10 grade 6+ rate-ratio
Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and tak…
AMC 12 2008A #11 grade 6+ geometry-3d, logic
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the 1…
AMC 12 2008A #12 grade 9+ algebra
A function f has domain [0,2] and range [0,1]. (The notation [a,b] denotes {x:a ≤ x ≤ b }.) What are the domain and rang…
AMC 12 2008A #13 grade 8+ geometry-2d
Points A and B lie on a circle centered at O, and ∠ AOB = 60°. A second circle is internally tangent to the first and ta…
AMC 12 2008A #14 grade 9+ geometry-2d
What is the area of the region defined by the inequality |3x-18|+|2y+7|≤3?
AMC 12 2008A #15 grade 6+ number-theory
Let k=2008²+2²⁰⁰⁸. What is the units digit of k²+2^k?
AMC 12 2008A #16 grade 11+ algebra
The numbers log(a³b⁷), log(a⁵b¹²), and log(a⁸b¹⁵) are the first three terms of an arithmetic sequence, and the 12^th ter…
AMC 12 2008A #17 grade 7+ number-theory, counting
Let a₁,a₂,… be a sequence determined by the rule a_n=a_(n-1)/2 if a_(n-1) is even and a_n=3a_(n-1)+1 if a_(n-1) is odd.…
AMC 12 2008A #18 grade 10+ geometry-3d
Triangle ABC, with sides of length 5, 6, and 7, has one vertex on the positive x-axis, one on the positive y-axis, and o…
AMC 12 2008A #19 grade 8+ counting, algebra
In the expansion of (1 + x + x² + … + x²⁷)(1 + x + x² + … + x¹⁴)², what is the coefficient of x²⁸?
AMC 12 2008A #20 grade 10+ geometry-2d
Triangle ABC has AC=3, BC=4, and AB=5. Point D is on AB, and CD bisects the right angle. The inscribed circles of △ ADC…
AMC 12 2008A #21 grade 7+ counting
A permutation (a₁,a₂,a₃,a₄,a₅) of (1,2,3,4,5) is heavy-tailed if a₁ + a₂ < a₄ + a₅. What is the number of heavy-tailed p…
AMC 12 2008A #22 grade 8+ geometry-2d
A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x…
AMC 12 2008A #23 grade 12+ algebra, geometry-2d
The solutions of the equation z⁴+4z³i-6z²-4zi-i=0 are the vertices of a convex polygon in the complex plane. What is the…
AMC 12 2008A #24 grade 11+ geometry-2d
Triangle ABC has ∠ C = 60° and BC = 4. Point D is the midpoint of BC. What is the largest possible value of tan∠ BAD?
AMC 12 2008A #25 grade 11+ algebra
A sequence (a₁,b₁), (a₂,b₂), (a₃,b₃), … of points in the coordinate plane satisfies (a_(n + 1), b_(n + 1)) = (√3a_n - b_…

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