2019 AMC 8 — Official Competition Problems (November 2019)
📅 2019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
计数
Ike and Mike go into a sandwich shop with a total of \30.00 to spend. Sandwiches cost \4.50 each and soft drinks cost \1.00$ each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
💡 解题思路
We know that the sandwiches cost $4.50$ dollars. Guessing will bring us to multiplying $4.50$ by 6, which gives us $27.00$ . Since they can spend $30.00$ they have $3$ dollars left. Since soft drinks
2
第 2 题
几何·面积
Three identical rectangles are put together to form rectangle ABCD , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle ABCD ? [图]
💡 解题思路
We can see that there are $2$ rectangles lying on top of the other and that is the same as the length of one rectangle. Now we know that the shorter side is $5$ , so the bigger side is $10$ , if we do
3
第 3 题
分数与比例
Which of the following is the correct order of the fractions \frac{15}{11},\frac{19}{15}, and \frac{17}{13}, from least to greatest?
💡 解题思路
We take a common denominator: \[\frac{15}{11},\frac{19}{15}, \frac{17}{13} = \frac{15\cdot 15 \cdot 13}{11\cdot 15 \cdot 13},\frac{19 \cdot 11 \cdot 13}{15\cdot 11 \cdot 13}, \frac{17 \cdot 11 \cdot 1
4
第 4 题
几何·面积
Quadrilateral ABCD is a rhombus with perimeter 52 meters. The length of diagonal \overline{AC} is 24 meters. What is the area in square meters of rhombus ABCD ? [图]
A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance d traveled by the two animals over time t from start to finish?
💡 解题思路
First, the tortoise walks at a constant rate, ruling out $(D)$ . Second, when the hare is resting, the distance will stay the same, ruling out $(E)$ and $(C)$ . Third, the tortoise wins the race, mean
6
第 6 题
几何·面积
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point P is in the center of the square. Given that point Q is randomly chosen among the other 80 points, what is the probability that the line PQ is a line of symmetry for the square? [图]
Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76 , 94 , and 87 . In order to average 81 for all five tests, what is the lowest score she could earn on one of the other two tests?
💡 解题思路
We should notice that we can turn the information we are given into a linear equation and just solve for our set variables. I'll use the variables $x$ and $y$ for the scores on the last two tests. \[\
8
第 8 题
分数与比例
Gilda has a bag of marbles. She gives 20\% of them to her friend Pedro. Then Gilda gives 10\% of what is left to another friend, Ebony. Finally, Gilda gives 25\% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
💡 解题思路
After Gilda gives $20$ % of the marbles to Pedro, she has $80$ % of the marbles left. If she then gives $10$ % of what's left to Ebony, she has $(0.8*0.9)$ = $72$ % of what she had at the beginning. F
9
第 9 题
分数与比例
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
💡 解题思路
Using the formula for the volume of a cylinder $\pi r^2 h$ , we get Alex, $108\pi$ , and Felicia, $216\pi$ . We can quickly notice that $\pi$ cancels out on both sides and that Alex's volume is $1/2$
10
第 10 题
统计
The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually 21 participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made? (A) The mean increases by 1 and the median does not change. (B) The mean increases by 1 and the median increases by 1 . (C) The mean increases by 1 and the median increases by 5 . (D) The mean increases by 5 and the median increases by 1 . (E) The mean increases by 5 and the median increases by 5 .
💡 解题思路
On Monday, $20$ people come. On Tuesday, $26$ people come. On Wednesday, $16$ people come. On Thursday, $22$ people come. Finally, on Friday, $16$ people come. $20+26+16+22+16=100$ , so the mean is $2
11
第 11 题
综合
The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
💡 解题思路
Let $x$ be the number of students taking both a math and a foreign language class.
12
第 12 题
立体几何
The faces of a cube are painted in six different colors: red (R) , white (W) , green (G) , brown (B) , aqua (A) , and purple (P) . Three views of the cube are shown below. What is the color of the face opposite the aqua face? [图]
💡 解题思路
$B$ is on the top, and $R$ is on the side, and $G$ is on the right side. That means that (image $2$ ) $W$ is on the left side. From the third image, you know that $P$ must be on the bottom since $G$ i
13
第 13 题
规律与数列
A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let N be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of N ?
💡 解题思路
Note that the only positive 2-digit palindromes are multiples of 11, namely $11, 22, \ldots, 99$ . Since $N$ is the sum of 2-digit palindromes, $N$ is necessarily a multiple of 11. The smallest 3-digi
14
第 14 题
几何·面积
Isabella has 6 coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
💡 解题思路
Let $\text{Day }1$ to $\text{Day\\ }2$ denote a day where one coupon is redeemed and the day when the second coupon is redeemed.
15
第 15 题
概率
On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is \frac{2}{5} . If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
💡 解题思路
The number of people wearing caps and sunglasses is $\frac{2}{5}\cdot35=14$ . So then, 14 people out of the 50 people wearing sunglasses also have caps.
16
第 16 题
统计
Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
💡 解题思路
The only option that is easily divisible by $55$ is $110$ , which gives 2 hours of travel. And, the formula is $\frac{15}{30} + \frac{110}{55} = \frac{5}{2}$ .
17
第 17 题
综合
What is the value of the product \[(\frac{1·3}{2·2})(\frac{2·4}{3·3})(\frac{3·5}{4·4})·s(\frac{97·99}{98·98})(\frac{98·100}{99·99})?\]
💡 解题思路
We rewrite: \[\frac{1}{2}\cdot\left(\frac{3\cdot2}{2\cdot3}\right)\left(\frac{4\cdot3}{3\cdot4}\right)\cdots\left(\frac{99\cdot98}{98\cdot99}\right)\cdot\frac{100}{99}\]
18
第 18 题
概率
The faces of each of two fair dice are numbered 1 , 2 , 3 , 5 , 7 , and 8 . When the two dice are tossed, what is the probability that their sum will be an even number?
💡 解题思路
We have $2$ dice with $2$ evens and $4$ odds on each die. For the sum to be even, the 2 rolls must be $2$ odds or $2$ evens.
19
第 19 题
综合
In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
💡 解题思路
Each team wins once and loses once to get the highest points. For a win, we have $3$ points, so a team gets $3\times2=6$ points if they each win a game and lose a game. Against the other 3 teams, ther
20
第 20 题
方程
How many different real numbers x satisfy the equation \[(x^{2}-5)^{2}=16?\]
💡 解题思路
We have that $(x^2-5)^2 = 16$ if and only if $x^2-5 = \pm 4$ . If $x^2-5 = 4$ , then $x^2 = 9 \implies x = \pm 3$ , giving 2 solutions. If $x^2-5 = -4$ , then $x^2 = 1 \implies x = \pm 1$ , giving 2 m
21
第 21 题
几何·面积
What is the area of the triangle formed by the lines y=5 , y=1+x , and y=1-x ?
💡 解题思路
First, we need to find the coordinates where the graphs intersect.
22
第 22 题
分数与比例
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was 84\% of the original price, by what percent was the price increased and decreased ?
💡 解题思路
Suppose the fraction of discount is $x$ . That means $(1-x)(1+x)=0.84$ ; so, $1-x^{2}=0.84$ , and $(x^{2})=0.16$ , procuring $x=0.4$ . Therefore, the price was increased and decreased by $40$ %, or $\
23
第 23 题
规律与数列
After Euclid High School's last basketball game, it was determined that \frac{1}{4} of the team's points were scored by Alexa and \frac{2}{7} were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?
💡 解题思路
Given the information above, we start with the equation $\frac{t}{4}+\frac{2t}{7} + 15 + x = t$ , where $t$ is the total number of points scored and $x\le 14$ is the number of points scored by the rem
24
第 24 题
几何·面积
In triangle \triangle ABC , point D divides side \overline{AC} so that AD:DC=1:2 . Let E be the midpoint of \overline{BD} and let F be the point of intersection of line \overline{BC} and line \overline{AE} . Given that the area of \triangle ABC is 360 , what is the area of \triangle EBF ? [图]
💡 解题思路
We use the line-segment ratios to infer area ratios and height ratios.
25
第 25 题
计数
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
💡 解题思路
Note: This solution uses the non-negative version for stars and bars. A solution using the positive version of stars is similar (first removing an apple from each person instead of 2).