2008 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）
1
第 1 题
应用题
Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?
💡 解题思路
If Susan spent 12 dollars, then twice that much on rides, then she spent $12+12 \times 2=36$ dollars in total. We subtract $36$ from $50$ to get $\boxed{\textbf{(B)}\ 14}$
2
第 2 题
数字运算
The ten-letter code BEST OF LUCK represents the ten digits 0-9 , in order. What 4-digit number is represented by the code word CLUE ?
💡 解题思路
We can derive that $C=8$ , $L=6$ , $U=7$ , and $E=1$ . Therefore, the answer is $\boxed{\textbf{(A)}\ 8671}$ ~edited by OwenTang
3
第 3 题
综合
If February is a month that contains Friday the 13^{th} , what day of the week is February 1?
💡 解题思路
We can go backwards by days, but we can also backwards by weeks. If we go backwards by weeks, we see that February 6 is a Friday. If we now go backwards by days, February 1 is a $\boxed{\textbf{(A)}\
4
第 4 题
几何·面积
In the figure, the outer equilateral triangle has area 16 , the inner equilateral triangle has area 1 , and the three trapezoids are congruent. What is the area of one of the trapezoids? [图] https://www.youtube.com/watch?v=43JZqmiUeLw ~David
💡 解题思路
The area outside the small triangle but inside the large triangle is $16-1=15$ . This is equally distributed between the three trapezoids. Each trapezoid has an area of $15/3 = \boxed{\textbf{(C)}\ 5}
5
第 5 题
统计
Barney Schwinn notices that the odometer on his bicycle reads 1441 , a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661 . What was his average speed in miles per hour?
💡 解题思路
Barney travels $1661-1441=220$ miles in $4+6=10$ hours for an average of $220/10=\boxed{\textbf{(E)}\ 22}$ miles per hour.
6
第 6 题
几何·面积
In the figure, what is the ratio of the area of the gray squares to the area of the white squares? [图]
💡 解题思路
Dividing the gray square into four smaller squares, there are $6$ gray tiles and $10$ white tiles, giving a ratio of $\boxed{\textbf{(D)}\ 3:5}$ .
7
第 7 题
综合
If \frac{3}{5}=\frac{M}{45}=\frac{60}{N} , what is M+N ?
💡 解题思路
Separate into two equations $\frac35 = \frac{M}{45}$ and $\frac35 = \frac{60}{N}$ and solve for the unknowns. $M=27$ and $N=100$ , therefore $M+N=\boxed{\textbf{(E)}\ 127}$ .
8
第 8 题
统计
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? [图]
💡 解题思路
There are a total of $100+60+40+120=320$ dollars of sales spread through $4$ months, for an average of $320/4 = \boxed{\textbf{(D)}\ 80}$ .
9
第 9 题
应用题
In 2005 Tycoon Tammy invested 100 dollars for two years. During the first year her investment suffered a 15\% loss, but during the second year the remaining investment showed a 20\% gain. Over the two-year period, what was the change in Tammy's investment?
💡 解题思路
After the $15 \%$ loss, Tammy has $100 \cdot 0.85 = 85$ dollars. After the $20 \%$ gain, she has $85 \cdot 1.2 = 102$ dollars. This is an increase in $2$ dollars from her original $100$ dollars, a $\b
10
第 10 题
统计
The average age of the 6 people in Room A is 40 . The average age of the 4 people in Room B is 25 . If the two groups are combined, what is the average age of all the people?
💡 解题思路
The total of all their ages over the number of people is
11
第 11 题
综合
Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
💡 解题思路
The union of two sets is equal to the sum of each set minus their intersection. The number of students that have both a dog and a cat is $20+26-39 = \boxed{\textbf{(A)}\ 7}$ .
12
第 12 题
综合
A ball is dropped from a height of 3 meters. On its first bounce it rises to a height of 2 meters. It keeps falling and bouncing to \frac{2}{3} of the height it reached in the previous bounce. On which bounce will it not rise to a height of 0.5 meters?
💡 解题思路
Each bounce is $2/3$ times the height of the previous bounce. The first bounce reaches $2$ meters, the second $4/3$ , the third $8/9$ , the fourth $16/27$ , and the fifth $32/81$ . Half of $81$ is $40
13
第 13 题
行程问题
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122 , 125 and 127 pounds. What is the combined weight in pounds of the three boxes?
💡 解题思路
Each box is weighed twice during this, so the combined weight of the three boxes is half the weight of these separate measures:
14
第 14 题
综合
Solution
💡 解题思路
There are $2$ ways to place the remaining $\text{As}$ , $2$ ways to place the remaining $\text{Bs}$ , and $1$ way to place the remaining $\text{Cs}$ for a total of $(2)(2)(1) = \boxed{\textbf{(C)}\ 4}
15
第 15 题
统计
In Theresa's first 8 basketball games, she scored 7, 4, 3, 6, 8, 3, 1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
💡 解题思路
The total number of points from the first $8$ games is $7+4+3+6+8+3+1+5=37$ . We have to make this a multiple of $9$ by scoring less than $10$ points. The smallest multiple of $9$ that is greater than
16
第 16 题
几何·面积
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [图]
💡 解题思路
The volume is of seven unit cubes which is $7$ . The surface area is the same for each of the protruding cubes which is $5\cdot 6=30$ . The ratio of the volume to the surface area is $\boxed{\textbf{(
17
第 17 题
几何·面积
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
💡 解题思路
A rectangle's area is maximized when its length and width are equivalent, or the two side lengths are closest together in this case with integer lengths. This occurs with the sides $12 \times 13 = 156
18
第 18 题
几何·面积
Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K . How many meters does the aardvark run? [图]
💡 解题思路
We will deal with this part by part: Part 1: 1/4 circumference of big circle= $\frac{2\pi r}{4}=\frac{\pi r}{2}=\frac{20\pi}{2}=10\pi$ Part 2: Big radius minus small radius= $20-10=10$ Part 3: 1/4 cir
19
第 19 题
综合
💡 解题思路
The two points are one unit apart at $8$ places around the edge of the square. There are $8 \choose 2$ $= 28$ ways to choose two points. The probability is
20
第 20 题
综合
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and \frac{3}{4} of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
💡 解题思路
Let $b$ be the number of boys and $g$ be the number of girls.
21
第 21 题
行程问题
Jerry cuts a wedge from a 6 -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? [图]
💡 解题思路
The slice is cutting the cylinder into two equal wedges with equal volume. The cylinder's volume is $\pi r^2 h = \pi (4^2)(6) = 96\pi$ . The volume of the wedge is half this which is $48\pi \approx \b
22
第 22 题
数字运算
For how many positive integer values of n are both \frac{n}{3} and 3n three-digit whole numbers?
💡 解题思路
Instead of finding $n$ , we find $x=\frac{n}{3}$ . We want $x$ and $9x$ to be three-digit whole numbers. The smallest three-digit whole number is $100$ , so that is our minimum value for $x$ , since i
23
第 23 题
几何·面积
In square ABCE , AF=2FE and CD=2DE . What is the ratio of the area of \triangle BFD to the area of square ABCE ? [图]
💡 解题思路
The area of $\triangle BFD$ is the area of square $ABCE$ subtracted by the the area of the three triangles around it. Arbitrarily assign the side length of the square to be $6$ .
24
第 24 题
几何·面积
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? https://www.youtube.com/watch?v=ZqPFm9cU0MY ~David
💡 解题思路
The numbers can at most multiply to be $60$ . The squares less than $60$ are $1,4,9,16,25,36,$ and $49$ . The possible pairs are $(1,1),(1,4),(2,2),(4,1),(3,3),(9,1),(4,4),(8,2),(5,5),(6,6),$ and $(9,
25
第 25 题
几何·面积
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black? [图]
💡 解题思路
Let the smallest circle be 1, the second smallest circle be 2, the third smallest circle be 3, etc. \[\begin{array}{c|cc} \text{circle \#} & \text{radius} & \text{area} \\ \hline 1 & 2 & 4\pi \\ 2 & 4