AMC 12A&B 2008 Problems & Answers

历年美国数学竞赛(The American Mathematics Competitions)AMC12 考题和答案

10 February 2008

S

1A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job.

At what time will the doughnut machine complete the job?

1:50 PM 3:00 PM 3:30 PM 4:30 PM

5:50 PM 2What is the reciprocal of

3Suppose that of

4Which of the following is equal to the product

bananas are worth as much as oranges. How many oranges are worth as much is of bananas?

?

S

S

S

5Suppose that

S

is an integer. Which of the following statements must be true about ?

6Heather compares the price of a new computer at two different stores. Store A offers off the sticker price followed by a

rebate, and store B offers off the same sticker price with no rebate. Heather saves by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?

7While Steve and LeRoy are fishing mile from shore, their boat springs a leak, and water comes in at a constant rate of

gallons per minute. The boat will sink if it takes in more than gallons of water. Steve starts rowing toward the shore at a constant rate of miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?

8What is the volume of a cube whose surface area is twice that of a cube with volume ?

S

S

S

S . That is, the ratio of the width to the height is . The aspect ratio of 9Older television screens have an aspect ratio of

many movies is not , so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of and is shown on an older television screen with a -inch diagonal. What is the height, in inches, of each darkened strip?

10Doug can paint a room in hours. Dave can paint the same room in hours. Doug and Dave paint the room together and take a S

one-hour break for lunch. Let be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by ?

历年美国数学竞赛(The American Mathematics Competitions)AMC12 考题和答案

11Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the

visible numbers have the greatest possible sum. What is that sum?

S

and range 12A function has domain

respectively, of the function defined by

. A second circle is internally tangent to the first and tangent to 13Points and lie on a circle centered at , and

both and . What is the ratio of the area of the smaller circle to that of the larger circle?

14What is the area of the region defined by the inequality

15Let

16The numbers

sequence is

be a sequence of integers determined by the rule if is even and 17Let

odd. For how many positive integers is it true that is less than each of , , and ?

, with sides of length , , and , has one vertex on the positive -axis, one on the positive -axis, and one on 18Triangle

the positive -axis. Let be the origin. What is the volume of tetrahedron ?

19In the expansion of

S S

if

is

S

,

. What is ?

, and

are the first three terms of an arithmetic sequence, and the

term of the S

. What is the units digit of

?

S

?

S S

. (The notation

?

denotes

.) What are the domain and range,

S

what is the coefficient of

20Triangle has

and

?

, and . Point is on and , respectively. What is

, and ?

bisects the right angle. The inscribed circles of S

,

have radii

21A permutation

permutations?

S 22A round table has radius . Six rectangular place mats are placed on the table. Each place mat has width and length as

shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is ?

of

is heavy-tailed if

. What is the number of heavy-tailed

S

历年美国数学竞赛(The American Mathematics Competitions)AMC12 考题和答案

USA AMC 12/AHSME 2008 Art of Problem SolvingPage 3of 5

23The solutions of the equation

are the vertices of a convex polygon in the complex plane. What is S

the area of the polygon?

24Triangle

has

and

. Point

is the midpoint of

. What is the largest possible value of

?

25A sequence

,

,

,

of points in the coordinate plane satisfies

for

.

Suppose that . What is

?

B

27 February 2008

1A basketball player made baskets during a game. Each basket was worth either or points. How many different numbers

could represent the total points scored by the player?

2A

block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?

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