📄 2011 AMC 8 真题

2011 AMC 8 — Official Competition Problems (November 2019)

📅 2011 年11月 📝 25题选择题 ⏱ 40分钟 🎯 满分25分 ✅ 含解题思路 👥 612 人已练习

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题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）

第 1 题

应用题

Margie bought 3 apples at a cost of 50 cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?

💡 解题思路

$50$ cents is equivalent to $\textdollar 0.50.$ Then the three apples cost $3 \times \textdollar 0.50 = \textdollar 1.50.$ The change Margie receives is $\textdollar 5.00 - \textdollar 1.50 = \boxed{\

第 2 题

几何·面积

Karl's rectangular vegetable garden is 20 feet by 45 feet, and Makenna's is 25 feet by 40 feet. Whose garden is larger in area? (A)\ Karl's garden is larger by 100 square feet. (B)\ Karl's garden is larger by 25 square feet. (C)\ The gardens are the same size. (D)\ Makenna's garden is larger by 25 square feet. (E)\ Makenna's garden is larger by 100 square feet.

💡 解题思路

The area of a rectangle is given by the formula length times width. Karl's garden is $20 \times 45 = 900$ square feet and Makenna's garden is $25 \times 40 = 1000$ square feet. Since $1000 > 900,$ Mak

第 3 题

几何·面积

Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? [图]

💡 解题思路

One way of approaching this is drawing the next circle of boxes around the current square. [asy] filldraw((-1,-1)--(6,-1)--(6,6)--(-1,6)--cycle,mediumgray,black); filldraw((0,0)--(5,0)--(5,5)--(0,5)--

第 4 题

统计

Here is a list of the numbers of fish that Tyler caught in nine outings last summer: \[2,0,1,3,0,3,3,1,2.\] Which statement about the mean, median, and mode is true?

💡 解题思路

First, put the numbers in increasing order.

第 5 题

行程问题

What time was it 2011 minutes after midnight on January 1, 2011? (A)\ January 1 at 9:31PM (B)\ January 1 at 11:51PM (C)\ January 2 at 3:11AM (D)\ January 2 at 9:31AM (E)\ January 2 at 6:01PM

💡 解题思路

There are $60$ minutes in an hour. $2011/60=33\text{r}31,$ or $33$ hours and $31$ minutes. There are $24$ hours in a day, so the time is $9$ hours and $31$ minutes after midnight on January 2, 2011. $

第 6 题

综合

In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?

💡 解题思路

By PIE , the number of adults who own both cars and motorcycles is $331+45-351=25.$ Out of the $331$ car owners, $25$ of them own motorcycles and $331-25=\boxed{\textbf{(D)}\ 306}$ of them don't.

第 7 题

综合

💡 解题思路

Assume that the area of each square is $1$ . Then, the area of the bolded region in the top left square is $\dfrac{1}{4}$ . The area of the top right bolded region is $\dfrac{1}{8}$ . The area of the

第 8 题

规律与数列

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?

💡 解题思路

By adding a number from Bag A and a number from Bag B together, the values we can get are $3, 5, 7, 5, 7, 9, 7, 9, 11.$ Therefore the number of different values is $\boxed{\textbf{(B)}\ 5}$ .

第 9 题

坐标几何

Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? [图]

💡 解题思路

We observe the graph and see that the shape of the graph does not matter. We only want the total time it took Carmen and the total distance she traveled. Based on the graph, Carmen traveled 35 miles f

第 10 题

行程问题

The taxi fare in Gotham City is 2.40 for the first \frac12 mile and additional mileage charged at the rate 0.20 for each additional 0.1 mile. You plan to give the driver a 2 tip. How many miles can you ride for 10?

💡 解题思路

Let $x$ be the number of miles you ride. The number of miles you ride after the first half mile is $x-0.5.$ We can write this equation:

第 11 题

坐标几何

The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha? [图]

💡 解题思路

Average the differences between each day. We get $10, -10,\text{ } 20,\text{ } 30,-20$ . We find the average of this list to get $\boxed{\textbf{(A)}\ 6}$ . ( In case you were wondering, the way to ca

第 12 题

几何·面积

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

💡 解题思路

If we designate a person to be on a certain side, then all placements of the other people can be considered unique. WLOG, assign Angie to be on the side. There are then $3!=6$ total seating arrangemen

第 13 题

几何·面积

Two congruent squares, ABCD and PQRS , have side length 15 . They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded? [图]

💡 解题思路

The length of BP is 5. the ratio of the areas is $\dfrac{5\cdot 15}{25\cdot 15}=\dfrac{5}{25}=20\%$ -Megacleverstarfish15

第 14 题

分数与比例

There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4 . There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4 : 5 . The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? https://youtu.be/rQUwNC0gqdg?t=697

💡 解题思路

At Colfax Middle School, there are $\frac49 \times 270 = 120$ girls. At Winthrop Middle School, there are $\frac59 \times 180 = 100$ girls. The ratio of girls to the total number of students is $\frac

第 15 题

数字运算

How many digits are in the product 4^5 · 5^{10} ? https://youtu.be/rQUwNC0gqdg?t=440

💡 解题思路

\[4^5 \cdot 5^{10} = 2^{10} \cdot 5^{10} = 10^{10}.\]

第 16 题

几何·面积

Let A be the area of the triangle with sides of length 25, 25 , and 30 . Let B be the area of the triangle with sides of length 25, 25, and 40 . What is the relationship between A and B ?

💡 解题思路

We can draw the altitude for the side with length 30. By HL Congruence, the two triangles formed are congruent. Thus the altitude splits the side with length 30 into two segments with length 15. By th

第 17 题

逻辑推理

Let w , x , y , and z be whole numbers. If 2^w · 3^x · 5^y · 7^z = 588 , then what does 2w + 3x + 5y + 7z equal?

💡 解题思路

The prime factorization of $588$ is $2^2\cdot3\cdot7^2.$ We can see $w=2, x=1,$ and $z=2.$ Because $5^0=1, y=0.$

第 18 题

概率

A fair 6 sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?

💡 解题思路

There are $6\cdot6=36$ ways to roll the two dice, and 6 of them result in two of the same number. Out of the remaining $36-6=30$ ways, the number of rolls where the first dice is greater than the seco

第 19 题

几何·面积

How many rectangles are in this figure? [图]

💡 解题思路

The figure can be divided into $7$ sections. The number of rectangles with just one section is $3.$ The number of rectangles with two sections is $5.$ There are none with only three sections. The numb

第 20 题

几何·面积

Quadrilateral ABCD is a trapezoid, AD = 15 , AB = 50 , BC = 20 , and the altitude is 12 . What is the area of the trapezoid? [图]

💡 解题思路

[asy] unitsize(1.5mm); defaultpen(linewidth(.9pt)+fontsize(10pt)); dotfactor=3; pair A,B,C,D,X,Y; A=(9,12); B=(59,12); C=(75,0); D=(0,0); X=(9,0); Y=(59,0); draw(A--B--C--D--cycle); draw(A--X); draw(B

第 21 题

数论

Students guess that Norb's age is 24, 28, 30, 32, 36, 38, 41, 44, 47 , and 49 . Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?

💡 解题思路

At least half the guesses are too low, so Norb's age must be greater than $36$ .

第 22 题

数字运算

What is the tens digit of 7^{2011} ?

💡 解题思路

Since we want the tens digit, we can find the last two digits of $7^{2011}$ . We can do this by using modular arithmetic. \[7^1\equiv 07 \pmod{100}.\] \[7^2\equiv 49 \pmod{100}.\] \[7^3\equiv 43 \pmod

第 23 题

数论

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit? This questions expects you to actually HAVE a 5 in it.

💡 解题思路

We can separate this into two cases. If an integer is a multiple of $5,$ the last digit must be either $0$ or $5.$

第 24 题

数论

In how many ways can 10001 be written as the sum of two primes?

💡 解题思路

For the sum of two numbers to be odd, one must be odd and the other must be even, because all odd numbers are of the form $2n+1$ where n is an integer, and all even numbers are of the form $2m$ where

第 25 题

几何·面积

A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares? [图]

💡 解题思路

The area of the smaller square is one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagona

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