2018 AMC 8 — Official Competition Problems (November 2019)
📅 2018 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）
1
第 1 题
分数与比例
An amusement park has a collection of scale models, with a ratio of 1: 20 , of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?
💡 解题思路
You can see that since the ratio of real building's heights to the model building's height is $1:20$ . We also know that the U.S Capitol is $289$ feet in real life, so to find the height of the model,
2
第 2 题
综合
What is the value of the product \[(1+\frac{1}{1})·(1+\frac{1}{2})·(1+\frac{1}{3})·(1+\frac{1}{4})·(1+\frac{1}{5})·(1+\frac{1}{6})?\]
💡 解题思路
By adding up the numbers in each of the $6$ parentheses, we get:
3
第 3 题
几何·面积
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
💡 解题思路
The five numbers which cause people to leave the circle are $7, 14, 17, 21,$ and $27.$
4
第 4 题
几何·面积
The twelve-sided figure shown has been drawn on 1 cm× 1 cm graph paper. What is the area of the figure in cm^2 ? [图]
💡 解题思路
We count $3 \cdot 3=9$ unit squares in the middle, and $8$ small triangles, which gives 4 rectangles each with an area of $1$ . Thus, the answer is $9+4=\boxed{\textbf{(C) } 13}$ .
5
第 5 题
综合
What is the value of 1+3+5+·s+2017+2019-2-4-6-·s-2016-2018 ?
💡 解题思路
Rearranging the terms, we get $(1-2)+(3-4)+(5-6)+...(2017-2018)+2019$ , and our answer is $-1009+2019=\boxed{\textbf{(E) }1010}$ .
6
第 6 题
行程问题
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
💡 解题思路
Since Anh spends half an hour to drive 10 miles on the coastal road, his speed is [mathjax]r=\dfrac dt=\dfrac{10}{0.5}=20[/mathjax] mph. His speed on the highway then is [mathjax]60[/mathjax] mph. He
7
第 7 题
数论
The 5 -digit number \underline{2}\underline{0}\underline{1}\underline{8}\underline{U} is divisible by 9 . What is the remainder when this number is divided by 8 ?
💡 解题思路
We use the property that the digits of a number must sum to a multiple of $9$ if it are divisible by $9$ . This means $2+0+1+8+U$ must be divisible by $9$ . The only possible value for $U$ then must b
8
第 8 题
坐标几何
John Pork asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students. [图] What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in John Pork's class? (Real problem uses Mr. Garcia, not John Pork)
💡 解题思路
The mean, or average number of days is the total number of days divided by the total number of students. The total number of days is $1\cdot 1+2\cdot 3+3\cdot 2+4\cdot 6+5\cdot 8+6\cdot 3+7\cdot 2=109
9
第 9 题
几何·面积
Tyler is tiling the floor of his 12-foot by 16-foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?
💡 解题思路
He will place $(12\cdot2)+((16-2)\cdot2)=52$ tiles around the border. For the inner part of the room, we have $10\cdot14=140$ square feet. Each tile takes up $2^{2}$ , or $4$ square feet, so he will u
10
第 10 题
统计
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
💡 解题思路
The sum of the reciprocals is $\frac{1}{1} + \frac{1}{2} + \frac{1}{4}= \frac{7}{4}$ . Their average is $\frac{7}{12}$ . Taking the reciprocal of this gives $\boxed{\textbf{(C) }\frac{12}{7}}$ .
11
第 11 题
概率
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. \begin{eqnarray*} X& X &X ; X& X &X \end{eqnarray*} If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?
💡 解题思路
There are a total of $6 !$ ways to arrange the kids.
12
第 12 题
行程问题
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
💡 解题思路
We see that every $35$ minutes the clock passes, the watch passes $30$ minutes. That means that the clock is $\frac{7}{6}$ as fast the watch, so we can set up proportions. $\dfrac{\text{car clock}}{\t
13
第 13 题
统计
Lalia took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?
💡 解题思路
Say Laila gets a value of $x$ on her first 4 tests, and a value of $y$ on her last test. Thus, $4x+y=82 \cdot 5=410$ .
14
第 14 题
规律与数列
Let N be the greatest five-digit number whose digits have a product of 120 . What is the sum of the digits of N ?
💡 解题思路
If we start off with the first digit, we know that it can't be $9$ since $9$ is not a factor of $120$ . We go down to the digit $8$ , which does work since it is a factor of $120$ . Now, we have to kn
15
第 15 题
几何·面积
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units? [图]
💡 解题思路
Let the radius of the large circle be $R$ . Then, the radius of the smaller circles are $\frac R2$ . The areas of the circles are directly proportional to the square of the radii, so the ratio of the
16
第 16 题
统计
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
💡 解题思路
To solve this, treat the two Arabic books as one unit and the four Spanish books as another unit. Along with the three German books, you now have five units to arrange.
17
第 17 题
行程问题
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers 2 \tfrac{1}{2} feet with each step. How many steps will Bella take by the time she meets Ella?
💡 解题思路
Every 10 feet Bella goes, Ella goes 50 feet, which means a total of 60 feet. They need to travel that 60 feet $10560\div60=176$ times to travel the entire 2 miles. Since Bella goes 10 feet 176 times,
18
第 18 题
数论
How many positive factors does 23,232 have?
💡 解题思路
We can first find the prime factorization of $23,232$ , which is $2^6\cdot3^1\cdot11^2$ . Now, we add one to our powers and multiply. Therefore, the answer is $(6+1)\cdot(1+1)\cdot(2+1)=7\cdot2\cdot3=
19
第 19 题
计数
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? [图]
💡 解题思路
You could just make out all of the patterns that make the top positive. In this case, you would have the following patterns:
20
第 20 题
几何·面积
In \triangle ABC, a point E is on \overline{AB} with AE=1 and EB=2. Point D is on \overline{AC} so that \overline{DE} \parallel \overline{BC} and point F is on \overline{BC} so that \overline{EF} \parallel \overline{AC}. What is the ratio of the area of CDEF to the area of \triangle ABC? [图]
💡 解题思路
By similar triangles, we have $[ADE] = \frac{1}{9}[ABC]$ . Similarly, we see that [mathjax][BEF] = \dfrac{4}{9}[ABC][/mathjax]. Using this information, we get \[[ACFE] = \frac{5}{9}[ABC].\] Then, sinc
21
第 21 题
数论
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
💡 解题思路
Looking at the values, we notice that $11-7=4$ , $9-5=4$ and $6-2=4$ . This means we are looking for a value that is four less than a multiple of $11$ , $9$ , and $6$ . The least common multiple of th
22
第 22 题
几何·面积
Point E is the midpoint of side \overline{CD} in square ABCD, and \overline{BE} meets diagonal \overline{AC} at F. The area of quadrilateral AFED is 45. What is the area of ABCD? [图]
💡 解题思路
We can use analytic geometry for this problem.
23
第 23 题
几何·面积
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? [图]
💡 解题思路
Firstly, we calculate the amount of all possible triangles. It is $C_8^3=56$ . Then, we consider how many triangles have at least a edge which is also the edge of octagon.
24
第 24 题
几何·面积
In the cube ABCDEFGH with opposite vertices C and E,J and I are the midpoints of segments \overline{FB} and \overline{HD}, respectively. Let R be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is R^2? [图]
💡 解题思路
Note that $EJCI$ is a rhombus by symmetry. Let the side length of the cube be $s$ . By the Pythagorean theorem, $EC= s\sqrt 3$ and $JI= s\sqrt 2$ . Since the area of a rhombus is half the product of i
25
第 25 题
立体几何
How many perfect cubes lie between 2^8+1 and 2^{18}+1 , inclusive?
💡 解题思路
First, $2^8+1=257$ . Then, $2^{18}+1=262145$ . Now, we can see how many perfect cubes are between these two parameters. By guessing and checking, we find that it starts from $7$ and ends with $64$ . N