2012 AMC 8 — Official Competition Problems (November 2019)
📅 2012 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
综合
Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic?
💡 解题思路
Since Rachelle uses $3$ pounds of meat to make $8$ hamburgers, she uses $\frac{3}{8}$ pounds of meat to make one hamburger. She'll need 24 times that amount of meat for 24 hamburgers, or $\frac{3}{8}
2
第 2 题
计数
In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?
💡 解题思路
There are $24\text{ hours}\div8\text{ hours} = 3$ births and one death everyday in East Westmore. Therefore, the population increases by $3$ - $1$ = $2$ people everyday. Thus, there are $2 \times 365
3
第 3 题
时间问题
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was 6:57\textsc{am} , and the sunset as 8:15\textsc{pm} . The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
💡 解题思路
The problem wants us to find the time of sunset and gives us the length of daylight and time of sunrise. So all we have to do is add the length of daylight to the time of sunrise to obtain the answer.
4
第 4 题
分数与比例
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
💡 解题思路
Peter ate $1 + \frac{1}{2} = \frac{3}{2}$ slices. The pizza has $12$ slices total. Taking the ratio of the amount of slices Peter ate to the amount of slices in the pizza, we find that Peter ate $\dfr
5
第 5 题
行程问题
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , X in centimeters? [图]
💡 解题思路
$1 + 1 + 1 + 2 + X = 1 + 2 + 1 + 6\\ 5 + X = 10\\ X = 5$
6
第 6 题
几何·面积
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches?
💡 解题思路
In order to find the area of the frame, we need to subtract the area of the photograph from the area of the photograph and the frame together. The area of the photograph is $8 \times 10 = 80$ square i
7
第 7 题
统计
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
💡 解题思路
Isabella wants an average grade of $95$ on her 4 tests; this also means that she wants the sum of her test scores to be at least $95 \times 4 = 380$ (if she goes over this number, she'll be over her g
8
第 8 题
分数与比例
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
💡 解题思路
Let the original price of an item be $x$ .
9
第 9 题
计数
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?
💡 解题思路
Let the number of two-legged birds be $x$ and the number of four-legged mammals be $y$ . We can now use systems of equations to solve this problem.
10
第 10 题
数字运算
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?
💡 解题思路
For this problem, all we need to do is find the amount of valid 4-digit numbers that can be made from the digits of $2012$ , since all of the valid 4-digit number will always be greater than $1000$ .
11
第 11 题
统计
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and x are all equal. What is the value of x ?
💡 解题思路
We can eliminate answer choices ${\textbf{(A)}\ 5}$ and ${\textbf{(C)}\ 7}$ , because of the above statement. Now we need to test the remaining answer choices.
12
第 12 题
方程
What is the units digit of 13^{2012} ? https://youtu.be/7xlBdxcsP3I?si=sFJGKXnBUUN7Su-C https://youtu.be/6RGNZj0tt2w ~David https://youtu.be/7an5wU9Q5hk?t=1186 ~ pi_is_3.14 https://youtu.be/6c_s967T7cA ~savannahsolver
💡 解题思路
The problem wants us to find the units digit of $13^{2012}$ , therefore, we can eliminate the tens digit of $13$ , because the tens digit will not affect the final result. So our new expression is $3^
13
第 13 题
应用题
Jamar bought some pencils costing more than a penny each at the school bookstore and paid \textdollar 1.43 . Sharona bought some of the same pencils and paid \textdollar 1.87 . How many more pencils did Sharona buy than Jamar?
💡 解题思路
We assume that the price of the pencils remains constant. Convert $\textdollar 1.43$ and $\textdollar 1.87$ to cents. Since the price of the pencils is more than one penny, we can find the price of on
14
第 14 题
综合
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
💡 解题思路
This problem is very similar to a handshake problem. We use the formula $\frac{n(n-1)}{2}$ to usually find the number of games played (or handshakes). Now we have to use the formula in reverse.
15
第 15 题
数论
The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers? https://youtu.be/rQUwNC0gqdg?t=172 https://www.youtube.com/watch?v=Vfsb4nwvopU ~David https://youtu.be/hOnw5UtBSqI ~savannahsolver
💡 解题思路
To find the answer to this problem, we need to find the least common multiple of $3$ , $4$ , $5$ , $6$ and add $2$ to the result. To calculate the least common multiple, we need to find the prime fact
16
第 16 题
规律与数列
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
💡 解题思路
In order to maximize the sum of the numbers, the numbers must have their digits ordered in decreasing value. There are only two numbers from the answer choices with this property: $76531$ and $87431$
17
第 17 题
几何·面积
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? (A)\hspace{.05in}3 (B)\hspace{.05in}4 (C)\hspace{.05in}5 (D)\hspace{.05in}6 (E)\hspace{.05in}7
💡 解题思路
The first answer choice ${\textbf{(A)}\ 3}$ , can be eliminated since there must be $10$ squares with integer side lengths. We then test the next smallest side length, which is $4$ . The square with a
18
第 18 题
几何·面积
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
💡 解题思路
The problem states that the answer cannot be a perfect square or have prime factors less than $50$ . Therefore, the answer will be the product of at least two different primes greater than $50$ . The
19
第 19 题
综合
In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?
💡 解题思路
$6$ are blue and green - $b+g=6$
20
第 20 题
综合
What is the correct ordering of the three numbers \frac{5}{19} , \frac{7}{21} , and \frac{9}{23} , in increasing order?
💡 解题思路
The value of $\frac{7}{21}$ is $\frac{1}{3}$ . Now we give all the fractions a common denominator.
21
第 21 题
综合
💡 解题思路
If Marla evenly distributes her $300$ square feet of paint between the 6 faces, each face will get $300\div6 = 50$ square feet of paint. The surface area of one of the faces of the cube is $10^2 = 100
22
第 22 题
统计
Let R be a set of nine distinct integers. Six of the elements are 2 , 3 , 4 , 6 , 9 , and 14 . What is the number of possible values of the median of R ?
💡 解题思路
Let the values of the missing integers be $x, y, z$ . We will find the bound of the possible medians.
23
第 23 题
几何·面积
An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon?
💡 解题思路
Let the perimeter of the equilateral triangle be $3s$ . The side length of the equilateral triangle would then be $s$ and the sidelength of the hexagon would be $\frac{s}{2}$ .
24
第 24 题
几何·面积
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle? [图]
A square with area 4 is inscribed in a square with area 5 , with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length a , and the other of length b . What is the value of ab ? [图]
💡 解题思路
The total area of the four congruent triangles formed by the squares is $5-4 = 1$ . Therefore, the area of one of these triangles is $\frac{1}{4}$ . The height of one of these triangles is $a$ and the