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AMC8 2022

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AMC8 · 2022

Q1
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
数学队设计了一个乘号形状的标志，如下图所示，位于1英寸方格网格上。标志的面积是多少平方英寸？
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Q2
Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the output of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$
考虑以下两个运算： \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} 什么是 $(5 \, \blacklozenge \, 3) \, \bigstar \, 6$ 的输出？
Q3
When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?
当三个正整数 $a$、$b$ 和 $c$ 相乘时，它们的乘积是 $100$。假设 $a < b < c$。可以选择这些数字有多少种方式？
Q4
The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
下图中的字母 M 先映关于直线 $q$，然后映关于直线 $p$。结果图像是什么？
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Q5
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?
安娜和贝拉一起庆祝她们的生日。五年前，当贝拉6岁生日时，她收到了一只新生小猫作为生日礼物。今天，两个孩子和小猫的年龄之和是30岁。安娜比贝拉大几岁？
Q6
Three positive integers are equally spaced on a number line. The middle number is $15,$ and the largest number is $4$ times the smallest number. What is the smallest of these three numbers?
三个正整数在数轴上等距分布。中间的数是$15$，最大的数是最小的数的$4$倍。这三个数中最小的数是多少？
Q7
When the World Wide Web first became popular in the $1990$s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$-megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.)
当万维网在$1990$年代首次流行时，下载速度最高约为每秒$56$千比特。一个$4.2$兆字节的歌曲在这个速度下大约需要下载多少分钟？（注意1兆字节有$8000$千比特。）
Q8
What is the value of \[\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?\]
计算\[\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}\]的值。
Q9
A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?
一杯沸水（$212^{\circ}\text{F}$）被放置在室温恒定为$68^{\circ}\text{F}$的房间中冷却。假设水温与室温的差每$5$分钟减半。$15$分钟后水温是多少华氏度？
Q10
One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?
在一个晴朗的日子，Ling决定去山里徒步。她早上$8 \, \textsc{am}$离开家，以每小时$45$英里的恒定速度开车，$10 \, \textsc{am}$到达徒步小径。徒步$3$小时后，Ling以每小时$60$英里的恒定速度开车回家。以下哪个图最好地展示了Ling的车与她家之间的距离在她整个行程中的变化？
Q11
Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with?
亨利驴有一根很长的意大利面。他咬了好几口，每次从一根面的中间吃掉3英寸。在最后，他有10根意大利面，总长度是17英寸。他开始的那根意大利面有多长（英寸）？
Q12
The arrows on the two spinners shown below are spun. Let the number $N$ equal $10$ times the number on Spinner $\text{A}$, added to the number on Spinner $\text{B}$. What is the probability that $N$ is a perfect square number?
下面两个转盘上的箭头转动。让数字$N$等于转盘A上的数字乘以10，加上转盘B上的数字。$N$是完全平方数的概率是多少？
Q13
How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice another, and the sum of the two numbers is $28$.”
下面句子中的空白可以填入多少个正整数？ “一个正整数比另一个正整数多_____，并且两个数的和是28。”
Q14
In how many ways can the letters in $\textbf{BEEKEEPER}$ be rearranged so that two or more $E$s do not appear together
字母 $\textbf{BEEKEEPER}$ 可以重新排列多少种方式，使得两个或更多 $E$s 不同时出现
Q15
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?
拉斯洛上网购买黑胡椒，发现三十种不同的黑胡椒选项，重量和价格不同，如下面的散点图所示。提供最低每盎司价格的胡椒重量是多少盎司？
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Q16
Four numbers are written in a row. The average of the first two is $21,$ the average of the middle two is $26,$ and the average of the last two is $30.$ What is the average of the first and last of the numbers?
四个数字排成一行。前两个数的平均数是$21$，中间两个数的平均数是$26$，后两个数的平均数是$30$。这四个数的第一个和最后一个的平均数是多少？
Q17
If $n$ is an even positive integer, the $\text{double factorial}$ notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? \[2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!\]
如果$n$是偶正整数，双阶乘记号$n!!$表示从$2$到$n$的所有偶整数的乘积。例如，$8!! = 2 \cdot 4 \cdot 6 \cdot 8$。以下和的单位数是多少？ \[2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!\]
Q18
The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle?
一个矩形的四边中点是$(-3,0), (2,0), (5,4)$和$(0,4)$。这个矩形的面积是多少？
Q19
Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points? (Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)
拉莫斯先生给他的20名学生出了一次测试。下方的点图显示了测试分数的分布。 后来拉莫斯先生发现有一道题评分错误。他重新评分，给一些学生加了5分，这使得中位数测试分数增加到$85$。至少有多少名学生获得了额外分数？ （注意，如果将20个测试分数按升序排列，中位数等于中间两个分数的平均值。）
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Q20
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$?
下面的网格要填入整数，使得每行每列的数字和相同。有四个数字缺失。左下角的数字$x$比其他三个缺失数字都大。$x$的最小可能值是多少？
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Q21
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
Steph在上半场20次尝试中投中了15个篮，在下半场10次尝试中投中了10个篮。Candace在上半场尝试了12次，下半场尝试了18次。在每半场，Steph的命中率都高于Candace。令人惊讶的是她们最终的总命中率相同。Candace在下半场比上半场多投中了多少个篮？
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Q22
A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus $3$ stops behind. After how many minutes will Zia board the bus?
一辆公交车从一站到下一站需要2分钟，并在每站停留1分钟让乘客上车。Zia步行从一站到下一站需要5分钟。当Zia到达一站时，如果公交车在前一站或已经离开前一站，她就等待公交车；否则她开始向下一站步行。假设公交车和Zia同时出发向图书馆前进，公交车落后3站。Zia何时上公交车？
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Q23
A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a $3$-by-$3$ grid. Shown below is a sample configuration with three $\triangle$s in a line. How many configurations will have three $\triangle$s in a line and three $\bigcirc$s in a line?
在3×3网格的九个方格中每个放置一个$\triangle$或$\bigcirc$。下面是一个示例配置，有一行三个$\triangle$。 有多少种配置既有行三个$\triangle$又有行三个$\bigcirc$？
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Q24
The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?
下面的图形显示了一个由矩形和直角三角形组成的八边形$ABCDEFGH$。沿虚线剪下并折叠后，形成一个三棱柱。已知$AH = EF = 8$且$GH = 14$。该棱柱的体积是多少？
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Q25
A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops what is the probability that the cricket has returned to the leaf where it started?
一只蟋蟀在4片叶子上随机跳跃，每次跳到其他3片叶子之一，概率相等。经过4次跳跃后，它回到起始叶子的概率是多少？
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