2024年4月11日发(作者：丹东模考数学试卷)

2003 AMC 10B

1、Which of the following is the same as

Solution

2、Al gets the disease algebritis and must take one green pill and one

pink pill each day for two weeks. A green pill costs more than a pink

pill, and Al’s pills cost a total of for the two weeks. How much does

one green pill cost?

Solution

3、The sum of 5 consecutive even integers is less than the sum of the

first consecutive odd counting numbers. What is the smallest of the

even integers?

Solution

4、Rose fills each of the rectangular regions of her rectangular flower

bed with a different type of flower. The lengths, in feet, of the

rectangular regions in her flower bed are as shown in the figure. She

plants one flower per square foot in each region. Asters cost 1 each,

begonias 1.50 each, cannas 2 each, dahlias 2.50 each, and Easter

lilies 3 each. What is the least possible cost, in dollars, for her garden?

Solution

5、Moe uses a mower to cut his rectangular -foot by -foot lawn.

The swath he cuts is inches wide, but he overlaps each cut by

inches to make sure that no grass is missed. He walks at the rate of

- 1 -

feet per hour while pushing the mower. Which of the following is

closest to the number of hours it will take Moe to mow his lawn?

Solution

.

6、Many television screens are rectangles that are measured by the

length of their diagonals. The ratio of the horizontal length to the

height in a standard television screen is . The horizontal length of

a “-inch” television screen is closest, in inches, to which of the

following?

Solution

7、The symbolism

example.

denotes the largest integer not exceeding . For

. Compute

, and

Solution

.

8、The second and fourth terms of a geometric sequence are and .

Which of the following is a possible first term?

Solution

9、Find the value of that satisfies the equation

Solution

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10、Nebraska, the home of the AMC, changed its license plate scheme.

Each old license plate consisted of a letter followed by four digits. Each

new license plate consists of three letters followed by three digits. By

how many times is the number of possible license plates increased?

Solution

11、A line with slope intersects a line with slope at the point .

What is the distance between the -intercepts of these two lines?

Solution

12、Al, Betty, and Clare split among them to be invested in

different ways. Each begins with a different amount. At the end of one

year they have a total of . Betty and Clare have both doubled

their money, whereas Al has managed to lose . What was Al’s

original portion?

Solution

.

13、Let

example,

values of is

Solution

denote the sum of the digits of the positive integer . For

and

?

. For how many two-digit

14、Given that , where both and are positive integers,

find the smallest possible value for .

Solution

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15、There are players in a singles tennis tournament. The

tournament is single elimination, meaning that a player who loses a

match is eliminated. In the first round, the strongest players are

given a bye, and the remaining players are paired off to play. After

each round, the remaining players play in the next round. The match

continues until only one player remains unbeaten. The total number of

matches played is

Solution

16、A restaurant offers three desserts, and exactly twice as many

appetizers as main courses. A dinner consists of an appetizer, a main

course, and a dessert. What is the least number of main courses that

the restaurant should offer so that a customer could have a different

dinner each night in the year ?

Solution

.

17、An ice cream cone consists of a sphere of vanilla ice cream and a

right circular cone that has the same diameter as the sphere. If the ice

cream melts, it will exactly fill the cone. Assume that the melted ice

cream occupies of the volume of the frozen ice cream. What is the

ratio of the cone’s height to its radius?

Solution

18、What is the largest integer that is a divisor of

for all positive even integers ?

Solution

19、Three semicircles of radius are constructed on diameter of a

semicircle of radius . The centers of the small semicircles divide

into four line segments of equal length, as shown. What is the area of

the shaded region that lies within the large semicircle but outside the

smaller semicircles?

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Solution

20、In rectangle ,

so that and

the area of .

and

. Lines

. Points and are on

and intersect at . Find

Solution

21、A bag contains two red beads and two green beads. You reach into

the bag and pull out a bead, replacing it with a red bead regardless of

the color you pulled out. What is the probability that all beads in the

bag are red after three such replacements?

Solution

22、A clock chimes once at minutes past each hour and chimes on

the hour according to the hour. For example, at 1 PM there is one

chime and at noon and midnight there are twelve chimes. Starting at

11:15 AM on February , , on what date will the chime

occur?

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Solution

23、A regular octagon

What is the area of the rectangle

has an area of one square unit.

?

Solution

24、The first four terms in an arithmetic sequence are

and , in that order. What is the fifth term?

Solution

, , ,

25、How many distinct four-digit numbers are divisible by and have

as their last two digits?

Solution

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