2013 AMC 8 — Official Competition Problems (November 2019)
📅 2013 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
统计
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
💡 解题思路
hey guys it’s $13$ years after this problem was created. basically we just need to get to the closest multiple of $6$ , which is $24$ . meaning we need 23=\boxed{\textbf{(A)}\ 1} $more car. ~pereena
2
第 2 题
应用题
A sign at the fish market says, "50 \% off, today only: half-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars?
💡 解题思路
50% off the price of half a pound of fish is \$3, so 100%, the regular price, of a half pound of fish is \$6. If half a pound of fish costs \$6, then a whole pound of fish is $\boxed{\textbf{(D)}\ 12}
3
第 3 题
综合
What is the value of 4 · (-1+2-3+4-5+6-7+·s+1000) ?
💡 解题思路
We group the addends inside the parentheses two at a time: \begin{align*} -1 + 2 - 3 + 4 - 5 + 6 - 7 + \ldots + 1000 &= (-1 + 2) + (-3 + 4) + (-5 + 6) + \ldots + (-999 + 1000) \\ &= \underbrace{1+1+1+
4
第 4 题
应用题
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill?
💡 解题思路
Since Judi's 7 friends had to pay \$2.50 extra each to cover the total amount that Judi should have paid, we multiply $2.50\cdot7=17.50$ is the bill Judi would have paid if she had money. Hence, to ca
5
第 5 题
统计
Hammie is in 6^th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
💡 解题思路
Listing the elements from least to greatest, we have $(5, 5, 6, 8, 106)$ , we see that the median weight is 6 pounds. The average weight of the five kids is $\frac{5+5+6+8+106}{5} = \frac{130}{5} = 26
6
第 6 题
行程问题
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, 30 = 6×5 . What is the missing number in the top row? [图]
💡 解题思路
Let the value in the empty box in the middle row be $x$ , and the value in the empty box in the top row be $y$ . $y$ is the answer we're looking for.
7
第 7 题
计数
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
💡 解题思路
Clearly, for every $5$ seconds, $3$ cars pass. It's more convenient to have everything in seconds: $2$ minutes and $45$ seconds $=2\cdot60 + 45 = 165$ seconds. We then set up a ratio: \[\frac{3}{5}=\f
8
第 8 题
概率
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
💡 解题思路
There are $2^3 = 8$ ways to flip the coins, in order. There are two ways to get exactly two consecutive heads: HHT and THH. There is only one way to get three consecutive heads: HHH. Therefore, the pr
9
第 9 题
行程问题
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)?
💡 解题思路
This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that $2^{10}=1024$ .
10
第 10 题
数论
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
💡 解题思路
To find either the LCM or the GCF of two numbers, always prime factorize first.
11
第 11 题
计数
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
💡 解题思路
We use that fact that $d=rt$ . Let d= distance, r= rate or speed, and t=time. In this case, let $x$ represent the time.
12
第 12 题
分数与比例
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of \50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the \150 regular price did he save?
💡 解题思路
First, find the amount of money one will pay for three sandals without the discount. We have $\textdollar 50\times 3 \text{ sandals} = \textdollar 150$ .
13
第 13 题
方程
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one? https://www.youtube.com/watch?v=9FkjSCcdTqY ~David https://youtu.be/KBM2YN4kKGA ~savannahsolver
💡 解题思路
Let the two digits be $a$ and $b$ .
14
第 14 题
概率
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
💡 解题思路
The favorable responses are either they both show a green bean or they both show a red bean. The probability that both show a green bean is $\frac{1}{2}\cdot\frac{1}{4}=\frac{1}{8}$ . The probability
15
第 15 题
方程
If 3^p + 3^4 = 90 , 2^r + 44 = 76 , and 5^3 + 6^s = 1421 , what is the product of p , r , and s ? https://youtu.be/7an5wU9Q5hk?t=301 ~ pi_is_3.14 https://youtu.be/ew7QnjAAHcw ~savannahsolver
💡 解题思路
First, we're going to solve for $p$ . Start with $3^p+3^4=90$ . Then, change $3^4$ to $81$ . Subtract $81$ from both sides to get $3^p=9$ and see that $p$ is $2$ . Now, solve for $r$ . Since $2^r+44=7
16
第 16 题
分数与比例
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8^th -graders to 6^th -graders is 5:3 , and the the ratio of 8^th -graders to 7^th -graders is 8:5 . What is the smallest number of students that could be participating in the project?
💡 解题思路
We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:
17
第 17 题
规律与数列
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
💡 解题思路
The arithmetic mean of these numbers is $\frac{\frac{2013}{3}}{2}=\frac{671}{2}=335.5$ . Therefore the numbers are $333$ , $334$ , $335$ , $336$ , $337$ , $338$ , so the answer is $\boxed{\textbf{(B)}
18
第 18 题
综合
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? [图]
💡 解题思路
There are $10 \cdot 12 = 120$ cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 fe
19
第 19 题
综合
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?
💡 解题思路
If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did wor
20
第 20 题
几何·面积
A 1× 2 rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
💡 解题思路
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; usepackage("amsmath"); real labelscalefactor = 0.5; /* changes labe
21
第 21 题
方程
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take? https://youtu.be/9nveueuZiqs ~savannahsolver
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? [图] https://youtu.be/nNDdkv_zfOo ~savannahsolver
💡 解题思路
There are $61$ vertical columns with a length of $32$ toothpicks, and there are $33$ horizontal rows with a length of $60$ toothpicks, because $32$ and $60$ are the number of intervals. You can verify
23
第 23 题
几何·面积
Angle ABC of \triangle ABC is a right angle. The sides of \triangle ABC are the diameters of semicircles as shown. The area of the semicircle on \overline{AB} equals 8π , and the arc of the semicircle on \overline{AC} has length 8.5π . What is the radius of the semicircle on \overline{BC} ? [图] https://youtu.be/crR3uNwKjk0 ~savannahsolver
💡 解题思路
If the semicircle on $\overline{AB}$ were a full circle, the area would be $16\pi$ .
24
第 24 题
几何·面积
Squares ABCD , EFGH , and GHIJ are equal in area. Points C and D are the midpoints of sides IH and HE , respectively. What is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares? [图]
💡 解题思路
Let point X be the point that intersects line EI. We can split our original triangle into trapezoid ABCX and triangle XIJ. WLOG (Without Loss Of Generality), AB equals 1 unit. Then since, X is the mid
25
第 25 题
计数
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are R_1 = 100 inches, R_2 = 60 inches, and R_3 = 80 inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B? [图]
💡 解题思路
The total length of all of the arcs is $100\pi +80\pi +60\pi=240\pi$ . Since we want the path from the center, the actual distance will be subtracted by $2\pi$ because it's already half the circumfere