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AMC8 2020

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AMC8 · 2020

Q1
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?
Luka 正在制作柠檬水在学校筹款活动上出售。他的配方要求水的量是糖的 4 倍，糖的量是柠檬汁的 2 倍。他使用了 3 杯柠檬汁。他需要多少杯水？
Q2
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?
四个朋友周末为邻居做院子工作，分别赚了 $\$$15、$\$$20、$\$$25 和 $\$$40。他们决定平分他们的收入。总共，赚了 $\$$40 的那个朋友需要给其他人多少钱？
Q3
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?
Carrie 有一个 6 英尺乘 8 英尺的长方形花园。她在整个花园里种满了草莓植株。Carrie 每平方英尺能种 4 株草莓植株，她每株植株平均收获 10 个草莓。她能期望收获多少草莓？
Q4
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?
下面展示了三个大小递增的六边形。假设点图案继续，使得每个连续的六边形包含一圈更多的点。下一个六边形有多少个点？
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Q5
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?
一个水壶有四分之三装满了菠萝汁。水壶通过将等量的汁倒入 5 个杯子中而被清空。每个杯子接收了水壶总容量的百分之多少？
Q6
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?
Aaron、Darren、Karen、Maren 和 Sharon 乘坐一列小型火车，这列火车有五个座位，每个座位容纳一人。Maren 坐在最后一节车厢。Aaron 坐在 Sharon 正后面。Darren 坐在 Aaron 前面的某个车厢。Karen 和 Darren 之间至少坐着一个人。谁坐在中间车厢？
Q7
How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.).
2020 到 2400 之间有多少个四位不同数字且按升序排列的整数？（例如，2347 就是一个这样的整数。）
Q8
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?
Ricardo 有 2020 枚硬币，其中一些是便士（1 分硬币），其余是镍币（5 分硬币）。他至少有一枚便士和一枚镍币。Ricardo 可能拥有的最大金额和最小金额之间的差值是多少分？
Q9
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ 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 $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
Akash 的生日蛋糕是一个 $4 \times 4 \times 4$ 英寸的立方体。蛋糕顶部和四个侧面有糖霜，底部没有糖霜。假设蛋糕被切成 64 个 $1 \times 1 \times 1$ 英寸的小立方体，如下图所示。有多少个小块正好有两个面有糖霜？
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Q10
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?
Zara 有 4 颗弹珠：一颗 Aggie、一颗 Bumblebee、一颗 Steelie 和一颗 Tiger。她想把它们排成一排放在架子上，但不想让 Steelie 和 Tiger 紧挨着。有多少种方式可以做到这一点？
Q11
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?
放学后，Maya 和 Naomi 前往 6 英里外的海滩。Maya 决定骑自行车，而 Naomi 坐公交车。下图显示了她们的旅程，标明了时间和行驶距离。Naomi 和 Maya 的平均速度差多少英里每小时？
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Q12
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? \[5!\cdot 9!=12\cdot N!\]
对于正整数 $n$，阶乘记号 $n!$ 表示从 $n$ 到 $1$ 的整数乘积。哪一个 $N$ 满足以下方程？\[5!\cdot 9!=12\cdot N!\]
Q13
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?
Jamal 的抽屉里有 6 双绿色袜子、18 双紫色袜子和 12 双橙色袜子。添加更多紫色袜子后，Jamal 注意到现在随机抽取一只袜子是紫色的概率为 60%。Jamal 添加了多少紫色袜子？
Q14
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?
Newton 县有 20 个城市。下图显示了它们的人口。所有城市的平均人口由水平虚线表示。以下哪项最接近所有 20 个城市的总人口？
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Q15
Suppose $15\%$ of $x$ equals $20\%$ of $y.$ What percentage of $x$ is $y?$
假设 $x$ 的 15% 等于 $y$ 的 20%。$y$ 是 $x$ 的百分之多少？
Q16
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 $B?$
图中有点 $A,B,C,D,E,$ 和 $F$，每个点代表从 $1$ 到 $6$ 的不同数字。图中显示的五条直线每条都经过其中一些点。每条直线上的数字相加得到五个和，这五个和的总和是 $47$。$B$ 代表的数字是多少？
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Q17
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 $12.$)
$2020$ 有多少个正整数因数具有超过 $3$ 个因数？（例如，$12$ 有 $6$ 个因数，即 $1,2,3,4,6,$ 和 $12$。）
Q18
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 $ABCD?$
矩形 $ABCD$ 铭刻在一个以直径 $\overline{FE}$ 为直径的半圆中，如图所示。设 $DA=16$，且 $FD=AE=9$。$ABCD$ 的面积是多少？
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Q19
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 $15?$
如果一个数的数字在两个不同的数字之间交替出现，则称其为翻转数。例如，$2020$ 和 $37373$ 是翻转数，但 $3883$ 和 $123123$ 不是。多少个五位翻转数能被 $15$ 整除？
Q20
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?
一位科学家在森林中行走，记录了 $5$ 棵成排站立的树的整数高度。她观察到每棵树要么是其右侧树的两倍高，要么是其一半高。不幸的是，她的笔记本被雨淋湿，有些数据丢失了。她的笔记如下，空白处表示缺失的数字。根据她的观察，科学家能够重建丢失的数据。树的平均高度是多少米？
Q21
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.)
一个游戏棋盘由 $64$ 个方格组成，方格颜色在黑白之间交替。下面的图显示了底排的方格 $P$ 和顶排的方格 $Q$。标记放置在 $P$ 上。一歩操作是将标记移动到上方一行相邻的白方格之一。从 $P$ 到 $Q$ 有多少条 $7$ 步路径？（图中显示了一条示例路径。）
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Q22
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
当一个正整数 $N$ 输入机器时，输出是一个根据下面规则计算的数字。
Q23
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?
要将五个不同的奖项颁发给三个学生。每个学生至少获得一个奖项。奖项可以以多少种不同的方式分配？
Q24
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 $n?$
一个大正方形区域铺有 $n^2$ 块边长 $s$ 英寸的灰色正方形瓷砖。每块瓷砖周围有一条宽度 $d$ 英寸的边框。下图显示 $n=3$ 的情况。当 $n=24$ 时，$576$ 块灰色瓷砖覆盖了大正方形区域的 $64\%$ 面积。对于这个较大的 $n$ 值，$\frac{d}{s}$ 的比值为多少？
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Q25
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?
下面的矩形 $R_1$ 和 $R_2$ 以及正方形 $S_1,\,S_2,\,$ 和 $S_3$ 组合成一个宽 $3322$ 单位、高 $2020$ 单位的矩形。$S_2$ 的边长是多少单位？
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