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AMC8 2014

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AMC8 · 2014

Q1
Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$?
哈利和特里分别被要求计算 $8-(2+5)$。哈利得到了正确的答案。特里忽略了括号，计算了 $8-2+5$。如果哈利的答案是 $H$，特里的答案是 $T$，那么 $H-T$ 是多少？
Q2
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
保罗欠宝拉35美分，他口袋里有5美分、10美分和25美分的硬币，可以用来还钱。用来还钱的硬币数最大值和最小值的差是多少？
Q3
Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?
Isabella 有一周时间阅读一本书以完成学校作业。她在前三天平均每天读了 $36$ 页，接下来的三天平均每天读了 $44$ 页。最后一天她读了 $10$ 页，完成了这本书。请问这本书共有多少页？
Q4
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
两个质数的和是 $85$。这两个质数的乘积是多少？
Q5
Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $\$4$ per gallon. How many miles can Margie drive on $\$20$ worth of gas?
玛吉的车每加仑汽油可以行驶 $32$ 英里，而汽油目前每加仑价格为 $\$4$。玛吉用 $\$20$ 的汽油能行驶多少英里？
Q6
Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25$, and $36$. What is the sum of the areas of the six rectangles?
六个矩形，每个矩形的底边宽度为 $2$，长度分别为 $1, 4, 9, 16, 25$ 和 $36$。这六个矩形的面积和是多少？
Q7
There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class?
在 Raub 女士的班级中，女生比男生多 4 人，该班共有 28 名学生。班上女生与男生人数的比例是多少？
Q8
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\underline{1A2}$. What is the missing digit $A$ of this $3$-digit number?
数学俱乐部的十一名成员每人支付了相同的金额，请了一位嘉宾讲解他们数学俱乐部会议上的问题解决方法。他们支付给嘉宾讲师的费用是 $\underline{1A2}$。这个三位数中的未知数字 \(A\) 是多少？
Q9
In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
在$\bigtriangleup ABC$中，点$D$在边$\overline{AC}$上，使得$BD=DC$且$\angle BCD$的度数为$70^\circ$。求$\angle ADB$的度数。
stem
Q10
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
第一届 AMC $8$ 于 $1985$ 年举办，此后每年举办一次。Samantha 在她参加第七届 AMC $8$ 的那一年满了 $12$ 岁。Samantha 是哪一年出生的？
Q11
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
杰克想骑车从他家到吉尔家，吉尔家位于杰克家以东三条街区、以北两条街区的地方。每骑完一条街区，杰克可以选择继续向东或向北，但他需要避开一个位于他家以东一条街区、以北一条街区的危险路口。杰克骑行总共五条街区，他有多少种方式可以到达吉尔家？
Q12
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
一本杂志刊登了三位名人的照片以及三张这些名人婴儿时期的照片。婴儿照片没有标明名人身份。读者被要求将每位名人与正确的婴儿照片对应起来。随机猜测的读者全部匹配正确的概率是多少？
Q13
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
如果 $n$ 和 $m$ 是整数，且 $n^2+m^2$ 是偶数，下列哪项是不可能的？
Q14
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
矩形 $ABCD$ 和直角三角形 $DCE$ 面积相同。它们组合形成了一个梯形，如图所示。求 $DE$。
stem
Q15
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
圆心为 $O$ 的圆的圆周被分成了 $12$ 个相等的弧段，从图中所示的字母 $A$ 到 $L$。角度 $x$ 和 $y$ 的和是多少度？
stem
Q16
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
“中学八强”篮球联赛有 $8$ 支队伍。每个赛季，每支队伍要与其他每支联赛队伍进行两场比赛（一场主场、一场客场），每支队伍还要与非联赛对手进行 $4$ 场比赛。涉及“中学八强”队伍的赛季总比赛场数是多少？
Q17
George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?
乔治步行1英里去学校。他每天同一时间离开家，以3英里每小时的恒定速度走路，恰好在学校开始时到达。今天他被宜人的天气分心了，前半英里只以每小时2英里的速度走。为了今天能恰好在学校开始时到达，乔治必须以多少英里每小时的速度跑完最后的半英里？
Q18
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
昨天市医院出生了四个孩子。假设每个孩子是男孩或女孩的概率相等。以下哪个结果最可能发生？
Q19
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
一个边长为 $3$ 英寸的立方体由 $27$ 个边长为 $1$ 英寸的小立方体组成。这些小立方体中有 $21$ 个是红色的，$6$ 个是白色的。如果要构建的 $3$ 英寸立方体使得白色表面积尽可能小，那么白色表面积占总表面积的几分之几？
Q20
Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
矩形 $ABCD$ 的边长为 $CD=3$ 和 $DA=5$。以点 $A$ 为圆心，半径为 $1$ 画圆；以点 $B$ 为圆心，半径为 $2$ 画圆；以点 $C$ 为圆心，半径为 $3$ 画圆。以下哪个数值最接近矩形内部但不在这三个圆内部的区域的面积？
stem
Q21
The $7$-digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?
这两个7位数 $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ 和 $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ 都是3的倍数。以下哪个可能是 $C$ 的值？
Q22
A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?
一个两位数满足：数字的乘积加上数字的和等于该数。这个数的个位数字是多少？
Q23
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today's date. What number does Caitlin wear?
欧几里得中学女子垒球队的三名队员进行了以下对话。 Ashley：我刚意识到我们的队服号码都是两位数的质数。 Bethany：而且你们两个号码的和是我本月初生日的日期。 Caitlin：真有趣。你们两个号码的和是我本月晚些时候生日的日期。 Ashley：而你们两个号码的和是今天的日期。 Caitlin 穿的号码是多少？
Q24
One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?
一天，饮料店卖出了 $252$ 罐苏打水，共给 $100$ 位顾客，每位顾客至少买了一罐苏打水。当天每位顾客购买苏打水罐数的中位数最大可能是多少？
Q25
A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? Note: 1 mile = 5280 feet
一条直直的一英里长、高度为40英尺的公路被封闭。罗伯特骑自行车沿着如图所示由半圆组成的路径前进。如果他的骑行速度为每小时5英里，他需要多少小时才能骑完这一英里路程？ 注：1 英里 = 5280 英尺
stem
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