AMC 美国数学竞赛试题+详解 英文版

1.

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?

2.

A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per

package." What is the regular price for a full pound of fish, in dollars?

What is the value of

?

3.

4.

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? 5.

Hammie is in the

grade and weighs 106 pounds. His quadruplet sisters are tiny babies

and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?

The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?

7.

Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?

8.

A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?

9.

The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?

10.

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?

11.

Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?

12.

At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?

13.

When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?

14.

Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?

15.

If , , and , what is the product of , , and ?

16.

A number of students from Fibonacci Middle School are taking part in a community service

project. The ratio of -graders to -graders is , and the the ratio of -graders to

-graders is . What is the smallest number of students that could be participating in the project?

17.

The sum of six consecutive positive integers is 2013. What is the largest of these six integers?

18.

Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

19.

Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?

20.

A rectangle is inscribed in a semicircle with longer side on the diameter. What is the

area of the semicircle?

21.

Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

22.

Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?

23.

Angle of is a right angle. The sides of are the diameters of semicircles

as shown. The area of the semicircle on equals , and the arc of the semicircle on

has length . What is the radius of the semicircle on ?

24.

Squares , , and are equal in area. Points and are the midpoints

of sides and , respectively. What is the ratio of the area of the shaded pentagon

to the sum of the areas of the three squares?

25.

A ball with diameter 4 inches starts at point A to roll along the track shown. The track is

comprised of 3 semicircular arcs whose radii are inches, inches, and

inches, respectively. The ball always remains in contact with the track and does not

slip. What is the distance the center of the ball travels over the course from A to B?

2013 AMC8 Problems/Solutions

1. Problem

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?

Solution:

In order to have her model cars in perfect, complete rows of 6, Danica must have a number of

cars that is a multiple of 6. The smallest multiple of 6 which is larger than 23 is 24, so she'll need to buy more model car.

2.

A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars?

Problem

Solution: The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.

What is the value of

?

3. Problem

Notice that we can pair up every two numbers to make a sum of 1:

Solution

Therefore, the answer is .

4. Problem

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill.

What was the total bill?

Each of her seven friends paid

to cover Judi's portion. Therefore, Judi's portion must

be

. Since Judi was supposed to pay

of the total bill, the total bill must be

.

Solution

5.

Hammie is in the

grade and weighs 106 pounds. His quadruplet sisters are tiny babies

and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five

children or the median weight, and by how many pounds?

Problem

Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds. Solution

The average weight of the five kids is .

Therefore, the average weight is bigger, by

pounds, making the answer

.

6. The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example,

. What is the missing number in the top row?

Problem

Solution

Let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.

Solution 1: Working Backwards

We see that

, making

.

It follows that

, so

.

Another way to do this problem is to realize what makes up the bottommost number. This

method doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)

Solution 2: Jumping Back to the Start

Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.

We can write some equations:

Now we can substitute into the first equation using the two others:

7. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass,

Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear

the crossing at a constant speed. Which of the following was the most likely number of cars in

the train?

Problem

If Trey saw

, then he saw

.

Solution 1

2 minutes and 45 seconds can also be expressed as

seconds.

Trey's rate of seeing cars,

, can be multiplied by

on the top and

bottom (and preserve the same rate):

. It follows that the most likely number of cars is

.

2 minutes and 45 seconds is equal to

.

Solution 2

Since Trey probably counts around 6 cars every 10 seconds, there are groups of 6

cars that Trey most likely counts. Since

, the closest answer choice is

.

8. A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?

Problem

First, there are

ways to flip the coins, in order.

Solution The ways to get two consecutive heads are HHT and THH. The way to get three consecutive heads is HHH.

Therefore, the probability of flipping at least two consecutive heads is .

9. The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on

which jump will he first be able to jump more than 1 kilometer?

Problem

This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that

. Solution

However, because the first term is

and not

, the solution to the problem is

10. What is the ratio of the least common multiple of 180 and 594 to the greatest common factor

of 180 and 594?

Problem

To find either the LCM or the GCF of two numbers, always prime factorize first. Solution 1

The prime factorization of . The prime factorization of .

Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.

For the GCF of 180 and 594, use the least power of all of the numbers that are in both

factorizations and multiply. = 18. Thus the answer = =

.

We start off with a similar approach as the original solution. From the prime factorizations, the GCF is 18.

Similar Solution

It is a well known fact that

. So we have,

.

Dividing by 18 yields .

Therefore, .

11. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?

Problem

We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let

represent the time.

Solution

On Monday, he was at a rate of . So,

.

For Wednesday, he walked at a rate of . Therefore,

.

On Friday, he walked at a rate of

. So,

. Adding up the hours yields

+

+

=

.

We now find the amount of time Grandfather would have taken if he walked at

per

day. Set up the equation,

.

To find the amount of time saved, subtract the two amounts: -

=

.

To convert this to minutes, we multiply by 60.

Thus, the solution to this problem is

12. At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?

Problem

First, find the amount of money one will pay for three sandals without the discount. We have

.

Solution

Then, find the amount of money using the discount: .

Finding the percentage yields .

To find the percent saved, we have

13. Problem

When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?

Let the two digits be and

. Solution

The correct score was . Clara misinterpreted it as

. The difference between the

two is

which factors into

. Therefore, since the difference is a multiple of 9,

the only answer choice that is a multiple of 9 is

.

14.

Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?

Problem

The probability that both show a green bean is

. The probability that both show a

red bean is . Therefore the probability is

Solution

15. If ,

, and , what is the product of

, , and ?

Problem

Solution

Therefore,

.

Therefore,

.

To most people, it would not be immediately evident that , so we can multiply 6's

until we get the desired number:

, so

.

Therefore the answer is

16. A number of students from Fibonacci Middle School are taking part in a community service

project. The ratio of

-graders to

-graders is

, and the the ratio of

-graders to

-graders is . What is the smallest number of students that could be participating in

the project?

Problem

Solution

We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:

Solution 1: Algebra

Therefore, the ratio of 8th graders to 7th graders to 6th graders is

. Since the ratio

is in lowest terms, the smallest number of students participating in the project is

.

The number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and

7th graders. The numbers of

students is

Solution 2: Fakesolving

17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?

Problem

The mean of these numbers is

. Therefore the numbers are

, so the answer is

Solution 1

Let the

number be . Then our desired number is

.

Solution 2

Our integers are , so we have that

.

Let the first term be

. Our integers are

. We have,

Solution 3

18.

Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

Problem

There are

cubes on the base of the box. Then, for each of the 4 layers above

the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are

4 feet left), there are

cubes. Hence, the answer is

.

Solution 1 We can just calculate the volume of the prism that was cut out of the original

box. Each interior side of the fort will be 2 feet shorter than each side of the outside. Since the

floor is 1 foot, the height will be 4 feet. So the volume of the interior box is

.

Solution 2

The volume of the original box is . Therefore, the number of blocks

contained in the fort is

19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?

Problem

If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if

Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get

the highest in the class. Therefore, Hannah did better than Bridget, so our order is

Solution

20. A

rectangle is inscribed in a semicircle with longer side on the diameter. What is the

area of the semicircle?

Problem

Solution

A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem,

. The area is

21. Problem

Solution

The number of ways to get from Samantha's house to City Park is

, and the number of

ways to get from City Park to school is

. Since there's one way to go through City

Park (just walking straight through), the number of different ways to go from Samantha's house to City Park to school

22.

Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?

Problem

There are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rows

with a length of 60 toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks. Thus, our answer is

Solution

23.

Angle

of is a right angle. The sides of

are the diameters of semicircles as shown. The area of the semicircle on equals

, and the arc of the semicircle on

has length . What is the radius of the semicircle on

?

Problem

If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is

.

Solution 1

We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is , and the middle one is , so the radius is .

Solution 2

24. Squares

, , and

are equal in area. Points

and

are the midpoints

of sides

and

, respectively. What is the ratio of the area of the shaded pentagon

to the sum of the areas of the three squares?

Problem

Solution 1

First let

(where

is the side length of the squares) for simplicity. We can extend

until it hits the extension of

. Call this point

. The area of triangle

then is

The area of rectangle

is

. Thus, our desired area is

. Now, the ratio of the shaded area to the combined area of the three squares is

.

Solution 2

Let the side length of each square be 1.

Let the intersection of

and

be .

Since

, . Since

and are vertical angles, they

are congruent. We also have

by definition.

So we have

by congruence. Therefore,

.

Since and

are midpoints of sides,

. This combined with

yields

.

The area of trapezoid

is

.

The area of triangle

is

.

So the area of the pentagon is .

The area of the 3 squares is . Therefore, .

Solution 3

Let the intersection of and

be .

Now we have

and

.

Because both triangles has a side on congruent squares therefore

.

Because

and are vertical angles

. Also both

and

are right angles so

.

Therefore by AAS (Angle, Angle, Side) .

Then translating/rotating the shaded

into the position of

So the shaded area now completely covers the square

Set the area of a square as

Therefore, .

25.

A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are

inches,

inches, and

inches, respectively. The ball always remains in contact with the track and does not

slip. What is the distance the center of the ball travels over the course from A to B?

Problem

The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses

inches, and it gains

inches on B.

So, the departure

from the length of the track means that the answer is

.

Solution 1

The total length of all of the arcs is

. Since we want the path from

the center, the actual distance will be shorter. Therefore, the only answer choice less than

is

. This solution may be invalid because the actual distance can be longer if

the path the center travels is on the outside of the curve, as it is in the middle bump. Solution 2