2010 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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1
第 1 题
综合
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest?
💡 解题思路
Given that these are the only math teachers at Euclid Middle School and we are told how many from each class are taking the AMC 8, we simply add the three numbers to find the total. $11+8+9=\boxed{\te
2
第 2 题
行程问题
If a @ b = \frac{a× b}{a+b} for a,b positive integers, then what is 5 @10 ?
💡 解题思路
Substitute $a=5$ and $b=10$ into the expression for $a @ b$ to get:
3
第 3 题
坐标几何
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? [图]
💡 解题思路
The highest price was in Month 1, which was \$17. The lowest price was in Month 3, which was \$10. 17 is $\frac{17}{10}\cdot100=170\%$ of 10, and is $170-100=70\%$ more than 10. Therefore, the answer
4
第 4 题
统计
What is the sum of the mean, median, and mode of the numbers 2,3,0,3,1,4,0,3 ?
💡 解题思路
Putting the numbers in numerical order we get the list $0,0,1,2,3,3,3,4.$ The mode is $3.$ The median is $\frac{2+3}{2}=2.5.$ The average is $\frac{0+0+1+2+3+3+3+4}{8}=\frac{16}{8}=2.$ The sum of all
5
第 5 题
行程问题
Alice needs to replace a light bulb located 10 centimeters below the ceiling in her kitchen. The ceiling is 2.4 meters above the floor. Alice is 1.5 meters tall and can reach 46 centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
💡 解题思路
Convert everything to the same unit. Since the answer is in centimeters, change meters to centimeters by moving the decimal place two places to the right.
6
第 6 题
几何·面积
Which of the following figures has the greatest number of lines of symmetry? (A)\ equilateral triangle(B)\ non-square rhombus(C)\ non-square rectangle(D)\ isosceles trapezoid(E)\ square
💡 解题思路
An equilateral triangle has $3$ lines of symmetry. A non-square rhombus has $2$ lines of symmetry. A non-square rectangle has $2$ lines of symmetry. An isosceles trapezoid has $1$ line of symmetry. A
7
第 7 题
概率
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
💡 解题思路
To make any units digit, you can use 4 pennies and 1 nickel. Using 3 quarters, you can make 75, so you will need 2 more dimes. Therefore, you will need $\boxed{4 + 1 + 2 + 3 = B(10)}$ coins.
8
第 8 题
行程问题
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction 1/2 mile in front of her. After she passes him, she can see him in her rear mirror until he is 1/2 mile behind her. Emily rides at a constant rate of 12 miles per hour, and Emerson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Emerson?
💡 解题思路
Because they are both moving in the same direction, Emily is riding relative to Emerson $12-8=4$ mph. Now we can look at it as if Emerson is not moving at all [on his skateboard] and Emily is riding a
9
第 9 题
分数与比例
Ryan got 80\% of the problems correct on a 25 -problem test, 90\% on a 40 -problem test, and 70\% on a 10 -problem test. What percent of all the problems did Ryan answer correctly?
💡 解题思路
Ryan answered $(0.8)(25)=20$ problems correct on the first test, $(0.9)(40)=36$ on the second, and $(0.7)(10)=7$ on the third. This amounts to a total of $20+36+7=63$ problems correct. The total numbe
10
第 10 题
几何·面积
Six pepperoni circles will exactly fit across the diameter of a 12 -inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?
💡 解题思路
The pepperoni circles' diameter is $2$ , since $\dfrac{12}{6} = 2$ . From that we see that the area of the $24$ circles of pepperoni is $\left ( \frac{2}{2} \right )^2 (24\pi) = 24\pi$ . The large piz
11
第 11 题
分数与比例
The top of one tree is 16 feet higher than the top of another tree. The heights of the two trees are in the ratio 3:4 . In feet, how tall is the taller tree?
💡 解题思路
Let the height of the taller tree be $h$ and let the height of the smaller tree be $h-16$ . Since the ratio of the smaller tree to the larger tree is $\frac{3}{4}$ , we have $\frac{h-16}{h}=\frac{3}{4
12
第 12 题
综合
Of the 500 balls in a large bag, 80\% are red and the rest are blue. How many of the red balls must be removed so that 75\% of the remaining balls are red?
💡 解题思路
Since 80 percent of the 500 balls are red, there are 400 red balls. Therefore, there must be 100 blue balls. For the 100 blue balls to be 25% or $\dfrac{1}{4}$ of the bag, there must be 400 balls in t
13
第 13 题
几何·面积
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 30\% of the perimeter. What is the length of the longest side?
💡 解题思路
Let $n$ , $n+1$ , and $n+2$ be the lengths of the sides of the triangle. Then the perimeter of the triangle is $n + (n+1) + (n+2) = 3n+3$ . Using the fact that the length of the smallest side is $30\%
14
第 14 题
数论
What is the sum of the prime factors of 2010 ?
💡 解题思路
First, we must find the prime factorization of $2010$ . $2010=2\cdot 3 \cdot 5 \cdot 67$ . We add the factors up to get $\boxed{\textbf{(C)}\ 77}$
15
第 15 题
综合
A jar contains 5 different colors of gumdrops. 30\% are blue, 20\% are brown, 15\% are red, 10\% are yellow, and other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
💡 解题思路
We do $100-30-20-15-10$ to find the percent of gumdrops that are green. We find that $25\%$ of the gumdrops are green. That means there are $120$ gumdrops. If we replace half of the blue gumdrops with
16
第 16 题
几何·面积
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
💡 解题思路
Let the side length of the square be $s$ , and let the radius of the circle be $r$ . Thus we have $s^2=r^2\pi$ . Dividing each side by $r^2$ , we get $\frac{s^2}{r^2}=\pi$ . Since $\left(\frac{s}{r}\r
17
第 17 题
几何·面积
The diagram shows an octagon consisting of 10 unit squares. The portion below \overline{PQ} is a unit square and a triangle with base 5 . If \overline{PQ} bisects the area of the octagon, what is the ratio \dfrac{XQ}{QY} ? [图]
💡 解题思路
We see that half the area of the octagon is $5$ . We see that the triangle area is $5-1 = 4$ . That means that $\frac{5h}{2} = 4 \rightarrow h=\frac{8}{5}$ . \[\text{QY}=\frac{8}{5} - 1 = \frac{3}{5}\
18
第 18 题
几何·面积
A decorative window is made up of a rectangle with semicircles at either end. The ratio of AD to AB is 3:2 . And AB is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles? [图]
💡 解题思路
We can set a proportion:
19
第 19 题
几何·面积
The two circles pictured have the same center C . Chord \overline{AD} is tangent to the inner circle at B , AC is 10 , and chord \overline{AD} has length 16 . What is the area between the two circles? [图]
💡 解题思路
Since $\triangle ACD$ is isosceles, $CB$ bisects $AD$ . Thus $AB=BD=8$ . From the Pythagorean Theorem, $CB=6$ . Thus the area between the two circles is $100\pi - 36\pi=64\pi$ $\boxed{\textbf{(C)}\ 64
20
第 20 题
综合
In a room, 2/5 of the people are wearing gloves, and 3/4 of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and gloves?
💡 解题思路
Let $x$ be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of 4 and
21
第 21 题
逻辑推理
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful . On the first day, Hui read 1/5 of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?
💡 解题思路
Let $x$ be the number of pages in the book. After the first day, Hui had $\frac{4x}{5}-12$ pages left to read. After the second, she had $\left(\frac{3}{4}\right)\left(\frac{4x}{5}-12\right)-15 = \fra
22
第 22 题
数字运算
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
💡 解题思路
Let the hundreds, tens, and units digits of the original three-digit number be $a$ , $b$ , and $c$ , respectively. We are given that $a=c+2$ . The original three-digit number is equal to $100a+10b+c =
23
第 23 题
几何·面积
Semicircles POQ and ROS pass through the center O . What is the ratio of the combined areas of the two semicircles to the area of circle O ? [图]
💡 解题思路
By the Pythagorean Theorem, the radius of the larger circle turns out to be $\sqrt{1^2 + 1^2} = \sqrt{2}$ . Therefore, the area of the larger circle is $(\sqrt{2})^2\pi = 2\pi$ . Using the coordinate
24
第 24 题
综合
What is the correct ordering of the three numbers, 10^8 , 5^{12} , and 2^{24} ?
💡 解题思路
Since all of the exponents are multiples of four, we can simplify the problem by taking the fourth root of each number. Evaluating we get $10^2=100$ , $5^3=125$ , and $2^6=64$ . Since $64<100<125$ , i
25
第 25 题
计数
Everyday at school, Jo climbs a flight of 6 stairs. Jo can take the stairs 1 , 2 , or 3 at a time. For example, Jo could climb 3 , then 1 , then 2 . In how many ways can Jo climb the stairs?
💡 解题思路
A dynamics programming approach is quick and easy. The number of ways to climb one stair is $1$ . There are $2$ ways to climb two stairs: $1$ , $1$ or $2$ . For 3 stairs, there are $4$ ways: ( $1$ , $