2017 AMC 8 — Official Competition Problems (November 2019)
📅 2017 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
Which of the following values is largest?
💡 解题思路
We will compute each expression.
2
第 2 题
逻辑推理
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together? [图]
💡 解题思路
Let $x$ be the total amount of votes casted. From the chart, Brenda received $30\%$ of the votes and had $36$ votes. We can express this relationship as $\frac{30}{100}x=36$ . Solving for $x$ , we get
3
第 3 题
综合
What is the value of the expression √(16\sqrt{8\sqrt{4)}} ?
When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following?
💡 解题思路
We can approximate $7,928,564$ to $8,000,000$ and $0.000315$ to $0.0003.$ Multiplying the two yields $2400.$ Thus, it shows our answer is $\boxed{\textbf{(D)}\ 2400}.$
5
第 5 题
综合
What is the value of the expression \frac{1 · 2 · 3 · 4 · 5 · 6 · 7 · 8}{1+2+3+4+5+6+7+8} ?
💡 解题思路
Directly calculating:
6
第 6 题
几何·面积
If the degree measures of the angles of a triangle are in the ratio 3:3:4 , what is the degree measure of the largest angle of the triangle?
💡 解题思路
The sum of the ratios is $10$ . Since the sum of the angles of a triangle is $180^{\circ}$ , the ratio can be scaled up to $54:54:72$ $(3\cdot 18:3\cdot 18:4\cdot 18).$ The numbers in the ratio $54:54
7
第 7 题
数论
Let Z be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of Z ?
💡 解题思路
To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits. If th
8
第 8 题
数论
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true." (1) It is prime. (2) It is even. (3) It is divisible by 7. (4) One of its digits is 9. This information allows Malcolm to determine Isabella's house number. What is its units digit?
💡 解题思路
Notice that (1) cannot be true. Otherwise, the number would have to be prime and be either even or divisible by 7. This only happens if the number is 2 or 7, neither of which are two-digit numbers, so
9
第 9 题
综合
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?
💡 解题思路
The $6$ green marbles and yellow marbles form $1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}$ of the total marbles. Now, suppose the total number of marbles is $x$ . We know the number of yellow marble
10
第 10 题
概率
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
💡 解题思路
There are $\binom{5}{3}$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$ , $2$ , or $3$ , for a total $\binom{3}{2}$ gro
11
第 11 题
几何·面积
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
💡 解题思路
Since the number of tiles lying on both diagonals is $37$ , counting one tile twice, there are $37=2x-1\implies x=19$ tiles on each side, where x is the number of tiles on the side length of the squar
12
第 12 题
数论
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
💡 解题思路
Since the remainder is the same for all numbers, then we will only need to find the lowest common multiple of the three given numbers, and add the given remainder.The $\operatorname{LCM}(4,5,6)$ is $6
13
第 13 题
综合
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
💡 解题思路
Given $n$ games, there must be a total of $n$ wins and $n$ losses. Hence, $4 + 3 + K = 2 + 3 + 3$ where $K$ is Kyler's wins. $K = 1$ , so our final answer is $\boxed{\textbf{(B)}\ 1}.$ ~CHECKMATE2021
14
第 14 题
分数与比例
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80\% of the problems she solved alone, but overall 88\% of her answers were correct. Zoe had correct answers to 90\% of the problems she solved alone. What was Zoe's overall percentage of correct answers?
💡 解题思路
Let the number of questions that they solved alone be $x$ . Let the percentage of problems they correctly solve together be $a$ %. As given, \[\frac{80x}{100} + \frac{ax}{100} = \frac{2 \cdot 88x}{100
15
第 15 题
统计
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture. [图]
💡 解题思路
Notice that the upper-most section contains a 3 by 3 square that looks like:
16
第 16 题
几何·面积
In the figure below, choose point D on \overline{BC} so that \triangle ACD and \triangle ABD have equal perimeters. What is the area of \triangle ABD ? [图]
💡 解题思路
We know that the perimeters of the two small triangles are $3+CD+AD$ and $4+BD+AD$ . Setting both equal and using $BD+CD = 5$ , we have $BD = 2$ and $CD = 3$ . Now, we simply have to find the area of
17
第 17 题
概率
Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have?
💡 解题思路
We can represent the amount of gold with $g$ and the amount of chests with $c$ . We can use the problem to make the following equations: \[9c-18 = g\] \[6c+3 = g\]
18
第 18 题
几何·面积
In the non-convex quadrilateral ABCD shown below, \angle BCD is a right angle, AB=12 , BC=4 , CD=3 , and AD=13 . What is the area of quadrilateral ABCD ? [图]
💡 解题思路
We first connect point $B$ with point $D$ .
19
第 19 题
数论
For any positive integer M , the notation M! denotes the product of the integers 1 through M . What is the largest integer n for which 5^n is a factor of the sum 98!+99!+100! ?
💡 解题思路
Factoring out $98!+99!+100!$ , we have $98! (1+99+99\cdot100)$ , which is $98! (10000)$ . Next, $98!$ has $\left\lfloor\frac{98}{5}\right\rfloor + \left\lfloor\frac{98}{25}\right\rfloor = 19 + 3 = 22$
20
第 20 题
概率
An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
💡 解题思路
There are $5$ options for the last digit $(1,3,5,7,9)$ as the integer must be odd. The first digit now has $8$ options left (it can't be $0$ or the same as the last digit). The second digit also has $
21
第 21 题
综合
Suppose a , b , and c are nonzero real numbers, and a+b+c=0 . What are the possible value(s) for \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|} ?
💡 解题思路
There are $2$ cases to consider:
22
第 22 题
几何·面积
In the right triangle ABC , AC=12 , BC=5 , and angle C is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? [图]
💡 解题思路
We can draw another radius from the center to the point of tangency. This angle, $\angle{ODB}$ , is $90^\circ$ . Label the center $O$ , the point of tangency $D$ , and the radius $r$ . [asy] draw((0,0
23
第 23 题
行程问题
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
💡 解题思路
It is well known that $\text{Distance}=\text{Speed} \cdot \text{Time}$ . In the question, we want distance. From the question, we have that the time is $60$ minutes or $1$ hour. By the equation derive
24
第 24 题
综合
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
💡 解题思路
We use Principle of Inclusion-Exclusion. There are $365$ days in the year, and we subtract the days that she gets at least $1$ phone call, which is \[\left \lfloor \frac{365}{3} \right \rfloor + \left
25
第 25 题
几何·面积
In the figure shown, \overline{US} and \overline{UT} are line segments each of length 2, and m\angle TUS = 60^\circ . Arcs \overarc{TR} and \overarc{SR} are each one-sixth of a circle with radius 2. What is the area of the region shown? [图]