美国数学竞赛AMC题目及答案

2023年12月10日发(作者：自制九年级数学试卷)

.

is the value of

?

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?

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?

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?

1 / 17 .

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?

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?

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. Let the two digits be and .

The correct score was

which factors into

choice that is a multiple of 9 is

. Clara misinterpreted it as . The difference between the two is

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

.

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

15. If ,

2 / 17 .

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?

--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of

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?

3 / 17 .

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

equals

are the diameters of semicircles as

has length . shown. The area of the semicircle on , and the arc of the semicircle on

What is the radius of the semicircle on ?

24. Squares

and

, , and are equal in area. Points and are the midpoints of sides

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

areas of the three squares?

4 / 17 .

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?

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

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

dollars.

Therefore, the answer is

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

5 / 17 .

median here is obviously less than the mean, so option (A) and (B) are out.

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

The average weight of the five kids is

Therefore, the average weight is bigger, by

.

.

pounds, making the answer

on 1: Working Backwards

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 see that , making .

It follows that , so .

Solution 2: Jumping Back to the Start

6 / 17 .

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.)

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:

Trey saw , then he saw .

seconds. 2 minutes and 45 seconds can also be expressed as

Trey\'s rate of seeing cars,

preserve the same rate):

, can be multiplied by on the top and bottom (and

. It follows that the most likely number of cars is .

Solution 2

minutes and seconds is equal to .

7 / 17 .

Since Trey probably counts around cars every

Trey most likely counts. Since

, there are

seconds, there are

, the closest answer choice is

groups of cars that

.

ways to flip the coins, in order.

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 .

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 .

and not , the solution to the problem is However, because the first term is

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

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

It is a well known fact that

.

Dividing by yields .

. So we have,

Therefore, .

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

11. We use that fact that

represent the time.

8 / 17 .

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 + + = .

per day. Set up We now find the amount of time Grandfather would have taken if he walked at

the equation, .

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

this to minutes, we multiply by .

- = . To convert

Thus, the solution to this problem is

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

.

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

Finding the percentage yields .

To find the percent saved, we have

13. Let the two digits be and .

The correct score was

which factors into

choice that is a multiple of 9 is

. Clara misinterpreted it as . The difference between the two is

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

.

14. The probability that both show a green bean is . The probability that both show a red bean

is . Therefore the probability is

15.

9 / 17 .

Therefore, .

Therefore, .

To most people, it would not be immediately evident that

the desired number:

, so

Therefore the answer is

.

.

, so we can multiply 6\'s until we get

16. Solution 1: Algebra

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:

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

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

. Since the ratio is in lowest

.

Solution 2: Fakesolving

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

10 / 17 .

17. Solution 1

The mean of these numbers is

, so the answer is

. Therefore the numbers are

Solution 2

Let the number be . Then our desired number is .

Our integers are , so we have that

.

Solution 3

Let the first term be . Our integers are

. We have,

18. Solution 1

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 2

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

height will be feet. So the volume of the interior box is

The volume of the original box is

the fort is .

.

box. Each

interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the

. Therefore, the number of blocks contained in

19. 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 .

11 / 17 .

20.

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.

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. There are

length of

vertical columns with a length of toothpicks, and there are horizontal rows with a

grid of toothpicks. toothpicks. An effective way to verify this is to try a small case, i.e. a

. Thus, our answer is

12 / 17 .

23. Solution 1

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 2

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 .

24.

First let

extension of

is

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

. Call this point . The area of triangle then is

until it hits the

The area of rectangle

. Thus, our desired area is . Now, the ratio of the shaded area to the

combined area of the three squares is .

13 / 17 .

Solution 2

Let the side length of each square be .

Let the intersection of

Since

congruent. We also have

So we have by

and

,

be .

. Since and are vertical angles, they are

by definition.

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

The area of the squares is

is

.

.

Therefore, .

14 / 17 .

Solution 3

Let the intersection of

Now we have

and be

.

.

.

.

.

into the position of

.

and

Because both triangles has a side on congruent squares therefore

Because

Also both

and

and

are vertical angles

are right angles so

Therefore by AAS(Angle, Angle, Side)

Then translating/rotating the shaded

So the shaded area now completely covers the square

Set the area of a square as

Therefore, .

25. Solution 1

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.

15 / 17 .

So, the departure from the length of the track

means that the answer is .

Solution 2

The total length of all of the arcs is . Since we want the path from the center,

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

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.

古希腊哲学大师亚里士多德说： 人有两种，一种即“吃饭是为了活着”,一种是“活着是为了吃饭”.一个人之所以伟大，首先是因为他有超于常人的心。“志当存高远”,“风物长宜放眼量”,这些古语皆鼓舞人们要树立雄心壮志，要有远大的理想。

有一位心理学家到一个建筑工地，分别问三个正在砌砖的工人：“你在干什么？”

第一个工人懒洋洋地说：“我在砌砖。” 第二个工人缺乏热情地说：“我在砌一堵墙。” 第三个工人满怀憧憬地说：“我在建一座高楼！”

16 / 17 .

听完回答，心理学家判定： 第一个人心中只有砖，他一辈子能把砖砌好就不错了；第二个人眼中只有墙，好好干或许能当一位技术员；而第三个人心中已经立起了一座殿堂，因为他心态乐观，胸怀远大的志向！

井底之蛙，只能看到巴掌大的天空；摸到大象腿的盲人，只能认为大象长得像柱子；登上五岳的人，才能感觉“一览众山小”;看到大海的人，就会顿感心胸开阔舒畅；

心中没有希望的人，是世界上最贫穷的人；心中没有梦想的人，是普天下最平庸的人；目光短浅的人，是最没有希望的人。

清代“红顶商人”胡雪岩说：“做生意顶要紧的是眼光，看得到一省，就能做一省的生意；看得到天下，就能做天下的生意；看得到外国，就能做外国的生意。”可见，一个人的心胸和眼光，决定了他志向的短浅或高远；一个人的希望和梦想，决定了他的人生暗淡或辉煌。

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