Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 2008A #2 grade 6+ geometry-2d
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is 2:1. The ratio…
AMC 10 2008A #3 grade 4+ number-theory
For the positive integer n, let ⟨ n⟩ denote the sum of all the positive divisors of n with the exception of n itself. Fo…
AMC 10 2008A #4 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 10 2008A #5 grade 5+ algebra
Which of the following is equal to the product 8/4·12/8·16/12·…·(4n+4)/4n·…·2008/2004?
AMC 10 2008A #6 grade 7+ rate-ratio
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The…
AMC 10 2008A #7 grade 8+ algebra
The fraction ((3²⁰⁰⁸)²-(3²⁰⁰⁶)²)/((3²⁰⁰⁷)²-(3²⁰⁰⁵)²) simplifies to which of the following?
AMC 10 2008A #8 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 10 2008A #9 grade 7+ number-theory
Suppose that 2x/3-x/6 is an integer. Which of the following statements must be true about x?
AMC 10 2008A #10 grade 6+ geometry-2d
Each of the sides of a square S₁ with area 16 is bisected, and a smaller square S₂ is constructed using the bisection po…
AMC 10 2008A #11 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 10 2008A #12 grade 7+ algebra
In a collection of red, blue, and green marbles, there are 25% more red marbles than blue marbles, and there are 60% mor…
AMC 10 2008A #13 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 10 2008A #14 grade 8+ geometry-2d
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 10 2008A #15 grade 7+ rate-ratio
Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longe…
AMC 10 2008A #16 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 10 2008A #17 grade 7+ geometry-2d
An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the tri…
AMC 10 2008A #18 grade 8+ geometry-2d
A right triangle has perimeter 32 and area 20. What is the length of its hypotenuse?
AMC 10 2008A #19 grade 8+ geometry-2d
Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90° clockwise about R, then rotated 90…
AMC 10 2008A #20 grade 8+ geometry-2d
Trapezoid ABCD has bases AB and CD and diagonals intersecting at K. Suppose that AB = 9, DC = 12, and the area of △ AKD…
AMC 10 2008A #21 grade 8+ geometry-3d
A cube with side length 1 is sliced by a plane that passes through two diagonally opposite vertices A and C and the midp…
AMC 10 2008A #22 grade 7+ probability
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To gene…
AMC 10 2008A #23 grade 7+ counting
Two subsets of the set S=lbrace a,b,c,d,erbrace are to be chosen so that their union is S and their intersection contain…
AMC 10 2008A #24 grade 6+ number-theory
Let k=2008²+2²⁰⁰⁸. What is the units digit of k²+2^k?
AMC 10 2008A #25 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…

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