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AMC8 2013

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AMC8 · 2013

Q1
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange her cars in this way?
Danica 想要将她的模型车排成每行恰好 6 辆车的行。她现在有 23 辆模型车。她必须再买多少辆车（最少）才能以这种方式排列她的车？
Q2
A sign at the fish market says, “50% off, today only: half-pound packages for just $3 per package.” What is the regular price for a full pound of fish, in dollars?
鱼市上的一个标志写着：“今天仅限 50% 折扣：半磅包装仅售每包 3 美元。”一磅鱼的常规价格是多少美元？
Q3
What is the value of $4 - (-1 + 2 - 3 + 4 - 5 + 6 - 7 + \dots + 1000)$?
求 $4 - (-1 + 2 - 3 + 4 - 5 + 6 - 7 + \dots + 1000)$ 的值。
Q4
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?
八个朋友在餐厅吃饭，同意平分账单。因为 Judi 忘了带钱，她的七个朋友每人多付了 2.50 美元来支付她那份总账单。总账单是多少？
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Q5
Hammie is in the 6th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
Hammie 在六年级，重 106 磅。他的四胞胎姐妹是小婴儿，重 5、5、6 和 8 磅。这五个孩子的平均体重（均值）还是中位数体重更大，相差多少磅？
Q6
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6 \times 5$. What is the missing number in the top row?
下面每个盒子中的数字是它上方一行相邻两个盒子中数字的乘积。例如，$30 = 6 \times 5$。顶行缺失的数字是多少？
stem
Q7
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
Trey 和他妈妈在铁路道口停车让火车通过。当火车开始通过时，Trey 在前 10 秒内数到 6 节车厢。火车以恒定速度通过道口用了 2 分 45 秒。火车中最可能的车厢数是以下哪项？
stem
Q8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
公平硬币抛掷 3 次。至少两次连续正面（heads）的概率是多少？
Q9
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?
绿巨人每次跳跃的距离都能翻倍。如果第一次跳 1 米，第二次 2 米，第三次 4 米，以此类推，他在第几次跳跃中第一次能跳超过 1 公里？
Q10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
180 和 594 的最小公倍数与最大公因数的比值是多少？
Q11
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
泰德的祖父这周3天使用了跑步机。每天他走了2英里。周一他以5英里每小时的速度慢跑。周三和周五他分别以3英里每小时和4英里每小时的速度行走。如果祖父总是以4英里每小时的速度行走，他会在跑步机上花费更少的时间。他节省了多少分钟？
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Q12
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
在2013年Winnebago县博览会上，一位摊贩推出凉鞋“博览会特价”。如果你以50美元的常规价格购买一双凉鞋，你可以以40%折扣获得第二双，以半价获得第三双。Javier利用“博览会特价”购买了三双凉鞋。他节省了150美元常规价格的百分之多少？
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Q13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
当Clara计算她的总分时，她不小心将一个分数的个位数和十位数颠倒了。她的错误总分与正确总分可能相差以下哪一项？
Q14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
Abe手里拿着1颗绿色和1颗红色果冻豆。Bea手里拿着1颗绿色、1颗黄色和2颗红色果冻豆。他们各自随机挑选一颗果冻豆给对方看。颜色匹配的概率是多少？
stem
Q15
If $3^p + 3^q = 90$, $2^r + 4^s = 76$, and $5^p + 6^s = 1421$, what is the product of $p$, $r$, and $s$?
如果 $3^p + 3^q = 90$，$2^r + 4^s = 76$，和 $5^p + 6^s = 1421$，$p$、$r$ 和 $s$ 的乘积是多少？
Q16
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8th-graders to 6th-graders is 5 : 3, and the ratio of 8th-graders to 7th-graders is 8 : 5. What is the smallest number of students that could be participating in the project?
斐波那契中学的多名学生参加了一个社区服务项目。八年级学生与六年级学生的比例是 5 : 3，八年级学生与七年级学生的比例是 8 : 5。参加该项目的最少学生人数是多少？
Q17
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
六个连续正整数的和是 2013。这些六个整数中最大的是多少？
Q18
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
伊莎贝拉使用一英尺边长的立方体积木建造一个长 12 英尺、宽 10 英尺、高 5 英尺的矩形堡垒。地板和四面墙都是 1 英尺厚。这个堡垒包含多少个积木？
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Q19
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show their tests to anyone. Cassie says, "I didn't get the lowest score in our class," and Bridget adds, "I didn't get the highest score." What is the ranking of the three girls from highest to lowest?
布里奇特、卡西和汉娜在讨论她们上次数学考试的结果。汉娜向布里奇特和卡西展示了她的试卷，但布里奇特和卡西没有向任何人展示她们的试卷。卡西说：“我没有得到班上最低的分数。”布里奇特补充说：“我没有得到最高的分数。”这三个女孩从最高到最低的排名是什么？
Q20
A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
一个 $1 \times 2$ 的矩形内接于一个半圆中，长边在直径上。这个半圆的面积是多少？
Q21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner of City Park, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
Samantha 住在城市公园西南角以西 2 个街区和以南 1 个街区的地方。她的学校位于城市公园东北角以东 2 个街区和以北 2 个街区的地方。在上学日，她骑自行车沿着街道到达城市公园的西南角，然后穿过公园的斜对角路径到达东北角，然后再骑自行车沿着街道去学校。如果她的路线尽可能短，她有多少种不同的路线可以选择？
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Q22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks high. How many toothpicks are used altogether?
用牙签制作一个长 60 根牙签、高 32 根牙签的网格。总共用了多少根牙签？
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Q23
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?
$\triangle ABC$ 的 $\angle ABC$ 是直角。$\triangle ABC$ 的边是示意中的半圆的直径。半圆 $\overline{AB}$ 的面积等于 $8\pi$，半圆 $\overline{AC}$ 的弧长为 $8.5\pi$。半圆 $\overline{BC}$ 的半径是多少？
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Q24
Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AIJCB$ to the sum of the areas of the three squares?
正方形 $ABCD$、$EFGH$ 和 $GHIJ$ 面积相等。点 $C$ 和 $D$ 分别是边 $IH$ 和 $HE$ 的中点。阴影五边形 $AIJCB$ 的面积与三个正方形面积之和的比是多少？
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Q25
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance in inches the center of the ball travels over the course from A to B?
一个直径为 4 英寸的球从点 A 开始沿所示轨道滚动。轨道由 3 个半圆弧组成，半径分别为 $R_1 = 100$ 英寸、$R_2 = 60$ 英寸和 $R_3 = 80$ 英寸。球始终与轨道接触且不打滑。从 A 到 B 的过程中，球心行进的距离是多少英寸？
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