2015 AMC 8 — Official Competition Problems (November 2019)
📅 2015 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
几何·面积
Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is 12 feet long and 9 feet wide? (There are 3 feet in a yard.)
💡 解题思路
We need $12\cdot9$ square feet of carpet to cover the floor. Since there are $9$ square feet in a square yard, we divide this by $9$ to get $\bold{\boxed{\textbf{(A) }12}}$ square yards.
2
第 2 题
几何·面积
Point O is the center of the regular octagon ABCDEFGH , and X is the midpoint of the side \overline{AB}. What fraction of the area of the octagon is shaded? [图]
💡 解题思路
Since octagon $ABCDEFGH$ is a regular octagon, it is split into $8$ equal parts, such as triangles $\bigtriangleup ABO, \bigtriangleup BCO, \bigtriangleup CDO$ , etc. These parts, since they are all e
3
第 3 题
行程问题
Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of 10 miles per hour. Jack walks to the pool at a constant speed of 4 miles per hour. How many minutes before Jack does Jill arrive?
💡 解题思路
Using $d=rt$ , we can set up an equation for when Jill arrives at the swimming pool:
4
第 4 题
坐标几何
The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
💡 解题思路
There are $2! = 2$ ways to order the boys on the ends, and there are $3!=6$ ways to order the girls in the middle. We get the answer to be $2 \cdot 6 = \boxed{\textbf{(E) }12}$ .
5
第 5 题
综合
Billy's basketball team scored the following points over the course of the first 11 games of the season. If his team scores 40 in the 12^{th} game, which of the following statistics will show an increase? \[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\]
💡 解题思路
When they score a $40$ on the next game, the range increases from $73-42=31$ to $73-40=33$ . This means the $\boxed{\textbf{(A) } \text{range}}$ increased.
6
第 6 题
几何·面积
In \bigtriangleup ABC , AB=BC=29 , and AC=42 . What is the area of \bigtriangleup ABC ?
💡 解题思路
We know the semi-perimeter of $\triangle ABC$ is $\frac{29+29+42}{2}=50$ . Next, we use Heron's Formula to find that the area of the triangle is just $\sqrt{50(50-29)^2(50-42)}=\sqrt{50 \cdot 21^2 \cd
7
第 7 题
概率
Each of two boxes contains three chips numbered 1 , 2 , 3 . A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
💡 解题思路
(This solution is similar to Solution 2.) Let's make this a problem with boxes.
8
第 8 题
几何·面积
What is the smallest whole number larger than the perimeter of any triangle with a side of length 5 and a side of length 19 ?
💡 解题思路
We know from the Triangle Inequality that the last side, $s$ , fulfills $s<5+19=24$ . Adding $5+19$ to both sides of the inequality, we get $s+5+19<48$ , and because $s+5+19$ is the perimeter of our t
9
第 9 题
工程问题
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?
💡 解题思路
First, we recognize that the number of widgets Janabel sells each day forms an arithmetic sequence. On day 1, she sells 1 widget, on day 2, she sells 3 widgets, on day 3, she sells 5 widgets, and so o
10
第 10 题
数字运算
How many integers between 1000 and 9999 have four distinct digits?
💡 解题思路
There are $9$ choices for the first number, since it cannot be $0$ , there are only $9$ choices left for the second number since it must differ from the first, $8$ choices for the third number, since
11
第 11 题
概率
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
💡 解题思路
There is one favorable case, which is the license plate says "AMC8." We must now find how many total cases there are. There are $5$ choices for the first letter (since it must be a vowel), $21$ choice
12
第 12 题
几何·角度
How many pairs of parallel edges, such as \overline{AB} and \overline{GH} or \overline{EH} and \overline{FG} , does a cube have? [图]
💡 解题思路
We first count the number of pairs of parallel lines that are in the same direction as $\overline{AB}$ . The pairs of parallel lines are $\overline{AB}\text{ and }\overline{EF}$ , $\overline{CD}\text{
13
第 13 题
统计
How many subsets of two elements can be removed from the set \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} so that the mean (average) of the remaining numbers is 6?
💡 解题思路
Since there will be $9$ elements after removal, and their mean is $6$ , we know their sum is $54$ . We also know that the sum of the set pre-removal is $66$ . Thus, the sum of the $2$ elements removed
14
第 14 题
规律与数列
Which of the following integers cannot be written as the sum of four consecutive odd integers?
💡 解题思路
Let our $4$ numbers be $n, n+2, n+4, n+6$ , where $n$ is odd. Then, our sum is $4n+12$ . The only answer choice that cannot be written as $4n+12$ , where $n$ is odd, is $\boxed{\textbf{(D)}\text{ 100}
15
第 15 题
综合
At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?
💡 解题思路
We are given:
16
第 16 题
分数与比例
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If \frac{1}{3} of all the ninth graders are paired with \frac{2}{5} of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
💡 解题思路
Let the number of sixth graders be $s$ , and the number of ninth graders be $n$ . Thus, $\frac{n}{3}=\frac{2s}{5}$ , which simplifies to $n=\frac{6s}{5}$ . Since we are trying to find the value of $\f
17
第 17 题
行程问题
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?
💡 解题思路
For starters, we identify d as distance and v as velocity (speed)
18
第 18 题
几何·面积
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2,5,8,11,14 is an arithmetic sequence with five terms, in which the first term is 2 and the constant added is 3 . Each row and each column in this 5×5 array is an arithmetic sequence with five terms. The square in the center is labelled X as shown. What is the value of X ? [图]
💡 解题思路
We begin filling in the table. The top row has a first term $1$ and a fifth term $25$ , so we have the common difference is $\frac{25-1}4=6$ . This means we can fill in the first row of the table: [as
19
第 19 题
几何·面积
A triangle with vertices as A=(1,3) , B=(5,1) , and C=(4,4) is plotted on a 6×5 grid. What fraction of the grid is covered by the triangle? [图]
💡 解题思路
The area of $\triangle ABC$ is equal to half the product of its base and height. By the Pythagorean Theorem, we find its height is $\sqrt{1^2+2^2}=\sqrt{5}$ , and its base is $\sqrt{2^2+4^2}=\sqrt{20}
20
第 20 题
应用题
Ralph went to the store and bought 12 pairs of socks for a total of \24 . Some of the socks he bought cost \1 a pair, some of the socks he bought cost \3 a pair, and some of the socks he bought cost \4 a pair. If he bought at least one pair of each type, how many pairs of \1$ socks did Ralph buy?
💡 解题思路
So, let there be $x$ pairs of $\$1$ socks, $y$ pairs of $\$3$ socks, and $z$ pairs of $\$4$ socks.
21
第 21 题
几何·面积
In the given figure, hexagon ABCDEF is equiangular, ABJI and FEHG are squares with areas 18 and 32 respectively, \triangle JBK is equilateral and FE=BC . What is the area of \triangle KBC ? [图]
💡 解题思路
Clearly, since $\overline{FE}$ is a side of a square with area $32$ , $\overline{FE} = \sqrt{32} = 4 \sqrt{2}$ . Now, since $\overline{FE} = \overline{BC}$ , we have $\overline{BC} = 4 \sqrt{2}$ .
22
第 22 题
综合
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
💡 解题思路
Since we know the number must be a multiple of $15$ , we can eliminate $A$ . We also know that after $12$ days, the students can't find any more arrangements, meaning the number has $12$ factors. Now,
23
第 23 题
规律与数列
Tom has twelve slips of paper which he wants to put into five cups labeled A , B , C , D , E . He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from A to E . The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5 . If a slip with 2 goes into cup E and a slip with 3 goes into cup B , then the slip with 3.5 must go into what cup?
💡 解题思路
The numbers have a sum of $6+5+12+4+8=35$ , which averages to $7$ , which means $A, B, C, D,$ and $E$ will have the values $5, 6, 7, 8$ and $9$ , respectively. Now, it's the process of elimination: Cu
24
第 24 题
综合
A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with N>2M and M>4 . Each team plays a 76 game schedule. How many games does a team play within its own division?
💡 解题思路
On one team they play $3N$ games in their division and $4M$ games in the other. This gives $3N+4M=76$ .
25
第 25 题
几何·面积
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space? [图]
💡 解题思路
We can draw a diagram as shown: [asy] size(75); draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw(