2020 AMC 8 — Official Competition Problems (November 2020)
📅 2020年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 742 人已练习
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1
第 1 题
行程问题
Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?
💡 解题思路
We have $\text{water} : \text{sugar} : \text{lemon juice} = 4\cdot 2 : 2 : 1 = 8 : 2 : 1,$ so Luka needs $3 \cdot 8 = \boxed{\textbf{(E) }24}$ cups.
2
第 2 题
工程问题
Four friends do yardwork for their neighbors over the weekend, earning \15, \20, \25, and \40, respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned \40$ give to the others?
💡 解题思路
The friends earn $\$\left(15+20+25+40\right)=\$100$ in total. Since they decided to split their earnings equally, it follows that each person will get $\$\left(\frac{100}{4}\right)=\$25$ . Since the f
3
第 3 题
几何·面积
Carrie has a rectangular garden that measures 6 feet by 8 feet. She plants the entire garden with strawberry plants. Carrie is able to plant 4 strawberry plants per square foot, and she harvests an average of 10 strawberries per plant. How many strawberries can she expect to harvest?
💡 解题思路
Note that the unit of the answer is strawberries , which is the product of
4
第 4 题
规律与数列
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon? [图]
💡 解题思路
Looking at the rows of each hexagon, we see that the first hexagon has $1$ dot, the second has $2+3+2$ dots, and the third has $3+4+5+4+3$ dots. Given the way the hexagons are constructed, it is clear
5
第 5 题
分数与比例
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?
💡 解题思路
Each cup is filled with $\frac{3}{4} \cdot \frac{1}{5} = \frac{3}{20}$ of the amount of juice in the pitcher, so the percentage is $\frac{3}{20} \cdot 100 = \boxed{\textbf{(C) }15}$ .
6
第 6 题
综合
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
💡 解题思路
Write the order of the cars as $\square\square\square\square\square$ , where the left end of the row represents the back of the train and the right end represents the front. Call the people $A$ , $D$
7
第 7 题
统计
How many integers between 2020 and 2400 have four distinct digits arranged in increasing order? (For example, 2347 is one integer.).
💡 解题思路
Firstly, we can observe that the second digit of such a number cannot be $1$ or $2$ because the digits must be distinct and in increasing order. The second digit also cannot be $4$ as the number must
8
第 8 题
概率
Ricardo has 2020 coins, some of which are pennies ( 1 -cent coins) and the rest of which are nickels ( 5 -cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have?
💡 解题思路
Clearly, the amount of money Ricardo has will be maximized when he has the maximum number of nickels. Since he must have at least one penny, the greatest number of nickels he can have is $2019$ , givi
9
第 9 题
行程问题
Akash's birthday cake is in the form of a 4 × 4 × 4 inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring 1 × 1 × 1 inch, as shown below. How many of the small pieces will have icing on exactly two sides?
💡 解题思路
Notice that, for a small cube which does not form part of the bottom face, it will have exactly $2$ faces with icing on them only if it is one of the $2$ center cubes of an edge of the larger cube. Th
10
第 10 题
计数
Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
💡 解题思路
Let the Aggie, Bumblebee, Steelie, and Tiger, be referred to by $A,B,S,$ and $T$ , respectively. If we ignore the constraint that $S$ and $T$ cannot be next to each other, we get a total of $4!=24$ wa
11
第 11 题
坐标几何
After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? [图]
💡 解题思路
Naomi travels $6$ miles in a time of $10$ minutes, which is equivalent to $\dfrac{1}{6}$ of an hour. Since $\text{speed} = \frac{\text{distance}}{\text{time}}$ , her speed is $\frac{6}{\left(\frac{1}{
12
第 12 题
数论
For a positive integer n , the factorial notation n! represents the product of the integers from n to 1 . What value of N satisfies the following equation? \[5!· 9!=12· N!\]
💡 解题思路
We have $5! = 2 \cdot 3 \cdot 4 \cdot 5$ , and $2 \cdot 5 \cdot 9! = 10 \cdot 9! = 10!$ . Therefore, the equation becomes $3 \cdot 4 \cdot 10! = 12 \cdot N!$ , and so $12 \cdot 10! = 12 \cdot N!$ . Ca
13
第 13 题
概率
Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60\% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
💡 解题思路
After Jamal adds $x$ purple socks, he has $(18+x)$ purple socks and $6+18+12+x=(36+x)$ total socks. This means the probability of drawing a purple sock is $\frac{18+x}{36+x}$ , so we obtain \[\frac{18
14
第 14 题
统计
There are 20 cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all 20 cities? [图]
💡 解题思路
We can see that the dotted line is exactly halfway between $4{,}500$ and $5{,}000$ , so it is at $4{,}750$ . As this is the average population of all $20$ cities, the total population is simply $4{,}7
15
第 15 题
分数与比例
Suppose 15\% of x equals 20\% of y. What percentage of x is y?
💡 解题思路
Since $20\% = \frac{1}{5}$ , multiplying the given condition by $5$ shows that $y$ is $15 \cdot 5 = \boxed{\textbf{(C) }75}$ percent of $x$ .
16
第 16 题
规律与数列
Each of the points A,B,C,D,E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B? [图]
💡 解题思路
We can form the following expressions for the sum along each line: \[\begin{dcases}A+B+C\\A+E+F\\C+D+E\\B+D\\B+F\end{dcases}\] Adding these together, we must have $2A+3B+2C+2D+2E+2F=47$ , i.e. $2(A+B+
17
第 17 题
数论
How many positive integer factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1,2,3,4,6, and 12. )
💡 解题思路
Since $2020 = 2^2 \cdot 5 \cdot 101$ , we can simply list its factors: \[1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.\] There are $12$ factors; only $1, 2, 4, 5, 101$ don't have over $3$ factor
18
第 18 题
几何·面积
Rectangle ABCD is inscribed in a semicircle with diameter \overline{FE}, as shown in the figure. Let DA=16, and let FD=AE=9. What is the area of ABCD?
A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15?
💡 解题思路
A number is divisible by $15$ precisely if it is divisible by $3$ and $5$ . The latter means the last digit must be either $5$ or $0$ , and the former means the sum of the digits must be divisible by
20
第 20 题
统计
A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
💡 解题思路
We will show that $22$ , $11$ , $22$ , $44$ , and $22$ meters are the heights of the trees from left to right. We are given that all tree heights are integers, so since Tree 2 has height $11$ meters,
21
第 21 题
几何·面积
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square P in the bottom row and square Q in the top row. A marker is placed at P. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7 -step paths are there from P to Q? (The figure shows a sample path.) [图]
💡 解题思路
Notice that, from one of the $1$ or $2$ white squares immediately beneath it (since the marker can only move on white squares). This means that the number of ways to move from $P$ to that square is th
22
第 22 题
整数运算
When a positive integer N is fed into a machine, the output is a number calculated according to the rule shown below.
💡 解题思路
We start with final output of $1$ and work backward, taking cares to consider all possible inputs that could have resulted in any particular output. This produces following set of possibilities each s
23
第 23 题
计数
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
💡 解题思路
Firstly, observe that a single student can't receive $4$ or $5$ awards because this would mean that one of the other students receives no awards. Thus, each student must receive either $1$ , $2$ , or
24
第 24 题
几何·面积
A large square region is paved with n^2 gray square tiles, each measuring s inches on a side. A border d inches wide surrounds each tile. The figure below shows the case for n=3 . When n=24 , the 576 gray tiles cover 64\% of the area of the large square region. What is the ratio \frac{d}{s} for this larger value of n? [图]
💡 解题思路
The area of the shaded region is $(24s)^2$ . To find the area of the large square, we note that there is a $d$ -inch border between each of the $23$ pairs of consecutive squares, as well as from betwe
25
第 25 题
几何·面积
Rectangles R_1 and R_2 and squares S_1, S_2, and S_3, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of S_2 in units? [图]
💡 解题思路
Let the side length of each square $S_k$ be $s_k$ . Then, from the diagram, we can line up the top horizontal lengths of $S_1$ , $S_2$ , and $S_3$ to cover the top side of the large rectangle, so $s_{