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AMC8 2004

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AMC8 · 2004

Q1
On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
在地图上，12厘米长度代表72公里。那么17厘米长度代表多少公里？
Correct Answer: B
We set up the proportion $\frac{12 \text{cm}}{72 \text{km}}=\frac{17 \text{cm}}{x \text{km}}$. Thus $x=102 \Rightarrow \boxed{\textbf{(B)}\ 102}$
我们建立比例 $\frac{12 \text{cm}}{72 \text{km}}=\frac{17 \text{cm}}{x \text{km}}$。因此 $x=102 \Rightarrow \boxed{\textbf{(B)}\ 102}$
Q2
How many different four-digit numbers can be formed by rearranging the four digits in 2004?
通过重新排列2004中的四个数字，可以组成多少个不同的四位数？
Correct Answer: B
(B) To form a four-digit number using 2, 0, 0 and 4, the digit in the thousands place must be 2 or 4. There are three places available for the remaining nonzero digit, whether it is 4 or 2. So the final answer is 6.
（B）要用 2、0、0 和 4 组成一个四位数，千位上的数字必须是 2 或 4。对于剩下的那个非零数字（无论是 4 还是 2），都有三个位置可放。因此最终答案是 6。
Q3
Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they share, how many meals should they have ordered to have just enough food for the 12 of them?
十二个朋友在Oscar's Overstuffed Oyster House聚餐，每人点了一份餐点。份量太大，足够18个人吃。如果他们分享，为12个人刚好够吃，应该点多少份餐点？
Correct Answer: A
(A) If 12 people order $\frac{18}{12}=1\frac{1}{2}$ times too much food, they should have ordered $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=8$ meals.
（A）如果 12 个人点了 $\frac{18}{12}=1\frac{1}{2}$ 倍过多的食物，他们本应点 $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=8$ 份餐。
Q4
Lance, Sally, Joy and Fred are chosen for the team. In how many ways can the three starters be chosen?
Lance、Sally、Joy和Fred被选入球队。有多少种方法可以选择三个首发？
Correct Answer: B
(B) When three players start, one is the alternate. Because any of the four players might be the alternate, there are four ways to select a starting team: Lance-Sally-Joy, Lance-Sally-Fred, Lance-Joy-Fred and Sally-Joy-Fred.
（B）当三名球员首发时，其中一人是替补。由于四名球员中的任何一位都可能是替补，因此有四种方式选择首发阵容：Lance-Sally-Joy、Lance-Sally-Fred、Lance-Joy-Fred 以及 Sally-Joy-Fred。
Q5
The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
每场比赛的失败队伍将被淘汰。如果有十六支队伍参赛，为了决出冠军，将进行多少场比赛？
Correct Answer: D
(D) It takes 15 games to eliminate 15 teams.
(D) 淘汰15支队伍需要15场比赛。
Q6
After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots did she make?
Sally 投了 20 次后，她的命中率为 55%。再投 5 次后，她的命中率提高到 56%。她在最后 5 次投篮中命中了几次？
Correct Answer: C
(C) If Sally makes 55% of her 20 shots, she makes $0.55 \times 20 = 11$ shots. If Sally makes 56% of her 25 shots, she makes $0.56 \times 25 = 14$ shots. So she makes $14 - 11 = 3$ of the last 5 shots.
（C）如果 Sally 在 20 次投篮中命中 55%，那么她命中 $\0.55 \times 20 = 11$ 次。如果 Sally 在 25 次投篮中命中 56%，那么她命中 $\0.56 \times 25 = 14$ 次。因此她在最后 5 次投篮中命中 $14 - 11 = 3$ 次。
Q7
An athlete's target heart rate, in beats per minute, is 80% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220. To the nearest whole number, what is the target heart rate of an athlete who is 26 years old?
运动员的目标心率（每分钟跳动次数）为其理论最大心率的 80%。最大心率等于 220 减去运动员的年龄（年）。对于一名 26 岁的运动员，目标心率取整到最接近的整数是多少？
Correct Answer: B
(B) A 26-year-old’s target heart rate is $0.8(220-26)=155.2$ beats per minute. The nearest whole number is 155.
（B）一位 26 岁人的目标心率是 $0.8(220-26)=155.2$ 次/分钟。最接近的整数是 155。
Q8
Find the number of two-digit positive integers whose digits total 7.
找出两位正整数各位数字之和为 7 的个数。
Correct Answer: B
(B) There are 7 two-digit numbers whose digits sum to 7: 16, 61, 25, 52, 34, 43 and 70.
（B）有 7 个两位数，它们的各位数字之和为 7：16、61、25、52、34、43 和 70。
Q9
The average of the five numbers in a list is 54. The average of the first two numbers is 48. What is the average of the last three numbers?
一个列表中五个数的平均数是 54。前两个数的平均数是 48。后三个数的平均数是多少？
Correct Answer: D
(D) The sum of all five numbers is $5 \times 54 = 270$. The sum of the first two numbers is $2 \times 48 = 96$, so the sum of the last three numbers is $270 - 96 = 174$. The average of the last three numbers is $\frac{174}{3} = 58$.
（D）五个数的总和是 $5 \times 54 = 270$。前两个数的和是 $2 \times 48 = 96$，所以后三个数的和是 $270 - 96 = 174$。后三个数的平均数是 $\frac{174}{3} = 58$。
Q10
Handy Aaron helped a neighbor $1 \frac{1}{4}$ hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid \$3 per hour. How much did he earn for the week?
Handy Aaron 周一帮助邻居 $1 \frac{1}{4}$ 小时，周二 50 分钟，周三早上从 8:20 到 10:45，周五半小时。他每小时得 3 美元。这周他赚了多少钱？
Correct Answer: E
(E) Aaron worked 75 minutes on Monday, 50 on Tuesday, 145 on Wednesday and 30 on Friday, for a total of 300 minutes or 5 hours. He earned \(5 \times \$3 = \$15\).
（E）Aaron 周一工作了 75 分钟，周二 50 分钟，周三 145 分钟，周五 30 分钟，总计 300 分钟，即 5 小时。他赚了 \(5 \times \$3 = \$15\)。
Q11
The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules: 1. The largest isn't first, but it is in one of the first three places. 2. The smallest isn't last, but it is in one of the last three places. 3. The median isn't first or last. What is the average of the first and last numbers?
数字 -2、4、6、9 和 12 按照以下规则重新排列：1. 最大的数不是第一个，但它在头三个位置之一。2. 最小的数不是最后一个，但它在最后三个位置之一。3. 中位数不是第一个也不是最后一个。第一个和最后一个数字的平均数是多少？
Correct Answer: C
(C) The largest, smallest and median occupy the three middle places, so the other two numbers, 9 and 4, are in the first and last places. The average of 9 and 4 is $\frac{9+4}{2}=6.5$.
（C）最大值、最小值和中位数占据中间三个位置，因此另外两个数 9 和 4 在第一个和最后一个位置。9 和 4 的平均数是 $\frac{9+4}{2}=6.5$。
Q12
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
Niki 通常保持她的手机开着。如果她的手机开着但她没有实际使用它，电池能持续 24 小时。如果她持续使用它，电池只能持续 3 小时。自上次充电以来，她的手机已经开机 9 小时，在此期间她使用了 60 分钟。如果她不再通话但保持手机开着，电池还能持续多少小时？
Correct Answer: B
(B) The phone has been used for 60 minutes, or 1 hour, to talk, during which time it has used $\frac{1}{3}$ of the battery. In addition, the phone has been on for 8 hours without talking, which used an additional $\frac{8}{24}$ or $\frac{1}{3}$ of the battery. Consequently, $\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$ of the battery has been used, meaning that $\frac{1}{3}$ of the battery, or $\frac{1}{3}\times 24=8$ hours remain if Niki does not talk on her phone.
（B）手机已通话60分钟，即1小时，在此期间消耗了电量的$\frac{1}{3}$。此外，手机在不通话的情况下开机了8小时，又额外消耗了$\frac{8}{24}$，即电量的$\frac{1}{3}$。因此，$\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$的电量已经被消耗，这意味着还剩下$\frac{1}{3}$的电量；如果Niki不再用手机通话，则剩余时间为$\frac{1}{3}\times 24=8$小时。
Q13
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. I. Bill is the oldest. II. Amy is not the oldest. III. Celine is not the youngest. Rank the friends from the oldest to the youngest.
Amy、Bill 和 Celine 是不同年龄的朋友。以下三个陈述中恰好有一个是真的。I. Bill 是最年长的。II. Amy 不是最年长的。III. Celine 不是最年轻的。将朋友们从最年长到最年轻排名。
Correct Answer: E
(E) Bill is not the oldest, because if he were, the first two statements would be true. Celine is not the oldest, because if she were, the last two statements would be true. Therefore, Amy is the oldest. So the first two statements are false. The last statement must be true. This means that Celine is not the youngest, so Bill is the youngest. The correct order from oldest to youngest is Amy, Celine, Bill.
（E）Bill 不是最年长的，因为如果他是，前两条陈述就会为真。Celine 也不是最年长的，因为如果她是，最后两条陈述就会为真。因此，Amy 是最年长的。所以前两条陈述是假的。最后一条陈述必须为真。这意味着 Celine 不是最年轻的，因此 Bill 是最年轻的。从年长到年幼的正确顺序是 Amy、Celine、Bill。
Q14
What is the area enclosed by the geoboard quadrilateral below?
下面几何板上的四边形围成的面积是多少？
stem
Correct Answer: C
Square: $10 \times 10 = 100$ Region $A$: $3 \times 5 = 15$ Region $B$: $\frac{1}{2} \times 6 \times 7 = 21$ Region $C$: $\frac{1}{2} \times 1 \times 3 = 1\frac{1}{2}$ Region $D$: $\frac{1}{2} \times 4 \times 5 = 10$ Region $E$: $\frac{1}{2} \times 6 \times 10 = 30$ The area is $100 - (15 + 21 + 1\frac{1}{2} + 10 + 30) = 100 - 77\frac{1}{2} = 22\frac{1}{2}$ square units.
正方形：$10 \times 10 = 100$ 区域 $A$：$3 \times 5 = 15$ 区域 $B$：$\frac{1}{2} \times 6 \times 7 = 21$ 区域 $C$：$\frac{1}{2} \times 1 \times 3 = 1\frac{1}{2}$ 区域 $D$：$\frac{1}{2} \times 4 \times 5 = 10$ 区域 $E$：$\frac{1}{2} \times 6 \times 10 = 30$ 面积为：$100 - (15 + 21 + 1\frac{1}{2} + 10 + 30) = 100 - 77\frac{1}{2} = 22\frac{1}{2}$ 平方单位。
solution
Q15
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
使用 13 块黑色和 6 块白色六边形瓷砖创建了下面的图形。如果通过附加一层与其它相同大小和形状的白色瓷砖边框创建新图形，新图形中白色瓷砖总数与黑色瓷砖总数的差是多少？
stem
Correct Answer: C
(C) The next border requires an additional $6 \times 3 = 18$ white tiles. A total of 24 white and 13 black tiles will be used, so the difference is $24 - 13 = 11$.
（C）下一圈边框需要额外的白色瓷砖数量为 $6 \times 3 = 18$。总共将使用 24 块白色瓷砖和 13 块黑色瓷砖，因此差值为 $24 - 13 = 11$。
solution
Q16
Two 600 ml pitchers contain orange juice. One pitcher is $\frac{1}{3}$ full and the other pitcher is $\frac{2}{3}$ full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
有两个600毫升的橙汁罐。一个罐子装有\frac{1}{3}的橙汁，另一个装有\frac{2}{3}的橙汁。向每个罐子中加水直到装满，然后将两个罐子倒入一个大容器中。大容器中的混合物中有多少分数是橙汁？
Correct Answer: C
(C) Because the first pitcher was $\frac{1}{3}$ full of orange juice, after filling with water it contains 200 ml of juice and 400 ml of water. Because the second pitcher was $\frac{2}{5}$ full of orange juice, after filling it contains 240 ml of orange juice and 360 ml of water. In all, the amount of orange juice is 440 ml out of a total of 1200 ml or $\frac{440}{1200}=\frac{11}{30}$ of the mixture.
（C）因为第一个壶里原本有 $\frac{1}{3}$ 是橙汁，加满水后它包含 200 ml 橙汁和 400 ml 水。因为第二个壶里原本有 $\frac{2}{5}$ 是橙汁，加满后它包含 240 ml 橙汁和 360 ml 水。总共橙汁是 440 ml，总量是 1200 ml，因此所占比例为 $\frac{440}{1200}=\frac{11}{30}$。
Q17
Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?
三个朋友总共有6支相同的铅笔，每人至少有一支。这种情况有多少种分配方式？
Correct Answer: D
(D) The largest number of pencils that any friend can have is four. There are 3 ways that this can happen: (4, 1, 1), (1, 4, 1) and (1, 1, 4). There are 6 ways one person can have 3 pencils: (3, 2, 1), (3, 1, 2), (2, 3, 1), (2, 1, 3), (1, 2, 3) and (1, 3, 2). There is only one way all three can have two pencils each: (2, 2, 2). The total number of possibilities is $3 + 6 + 1 = 10$.
（D）任何一个朋友最多能有四支铅笔。出现这种情况有 3 种方式：（4, 1, 1）、（1, 4, 1）和（1, 1, 4）。有 6 种方式使得某一个人有 3 支铅笔：（3, 2, 1）、（3, 1, 2）、（2, 3, 1）、（2, 1, 3）、（1, 2, 3）和（1, 3, 2）。三个人各有两支铅笔只有一种方式：（2, 2, 2）。可能性的总数是 $3 + 6 + 1 = 10$。
Q18
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a different value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points?
五个朋友参加飞镖投掷比赛。每人投两支飞镖击中同一个圆形靶，每个人的得分是击中靶区得分的总和。靶区得分为1到10的整数。每支飞镖击中的区域得分不同。得分分别是：Alice 16分，Ben 4分，Cindy 7分，Dave 11分，Ellen 17分。谁击中了6分的区域？
Correct Answer: A
(A) Ben must hit 1 and 3. This means Cindy must hit 5 and 2, because she scores 7 using two different numbers, neither of which is 1 or 3. By similar reasoning, Alice hits 10 and 6, Dave hits 7 and 4, and Ellen hits 9 and 8. Alice hits the 6.
（A）Ben 必须击中 1 和 3。这意味着 Cindy 必须击中 5 和 2，因为她用两个不同的数字得到 7 分，而这两个数字都不是 1 或 3。用类似的推理，Alice 击中 10 和 6，Dave 击中 7 和 4，Ellen 击中 9 和 8。Alice 击中 6。
Q19
A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5 and 6. The smallest such number lies between which two numbers?
一个大于2的整数，除以3、4、5、6各数时余数均为2。最小这样的数在哪两个数之间？
Correct Answer: B
(B) The numbers that leave a remainder of 2 when divided by 4 and 5 are 22, 42, 62 and so on. Checking these numbers for a remainder of 2 when divided by both 3 and 6 yields 62 as the smallest.
（B）当被 4 和 5 除时余 2 的数是 22、42、62 等等。检验这些数在被 3 和 6 同时除时是否也余 2，可得最小的是 62。
Q20
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
房间里的人中有三分之二坐在四分之三的椅子上。其余人站着。如果有6把空椅子，房间里有多少人？
Correct Answer: D
(D) Because the 6 empty chairs are $\frac{1}{4}$ of the chairs in the room, there are $6 \times 4 = 24$ chairs in all. The number of seated people is $\left(\frac{3}{4}\right)24 = 18$, and this is $\frac{2}{3}$ of the people present. It follows that \[ \frac{18}{\text{people present}}=\frac{2}{3}. \] So there are 27 people in the room.
（D）因为 6 把空椅子是房间里椅子总数的 $\frac{1}{4}$，所以一共有 $6 \times 4 = 24$ 把椅子。坐着的人数是 $\left(\frac{3}{4}\right)24 = 18$，而这又是在场人数的 $\frac{2}{3}$。因此 \[ \frac{18}{\text{在场人数}}=\frac{2}{3}. \] 所以房间里有 27 个人。
Q21
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?
转盘 A 和 B 同时旋转。每个转盘上的箭头等可能地落在每个数字上。两个转盘数字的乘积为偶数的概率是多少？
stem
Correct Answer: D
(D) In eight of the twelve outcomes the product is even: $1\times 2,\ 2\times 1,\ 2\times 2,\ 2\times 3,\ 3\times 2,\ 4\times 1,\ 4\times 2,\ 4\times 3.$ In four of the twelve, the product is odd: $1\times 1,\ 1\times 3,\ 3\times 1,\ 3\times 3.$ So the probability that the product is even is $\frac{8}{12}$ or $\frac{2}{3}.$
（D）在十二种结果中，有八种结果的乘积是偶数：$1\times 2,\ 2\times 1,\ 2\times 2,\ 2\times 3,\ 3\times 2,\ 4\times 1,\ 4\times 2,\ 4\times 3.$ 在十二种结果中，有四种结果的乘积是奇数：$1\times 1,\ 1\times 3,\ 3\times 1,\ 3\times 3.$ 因此乘积为偶数的概率是 $\frac{8}{12}$ 或 $\frac{2}{3}.$
Q22
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men?
在一个派对上，只有单身女性和带着妻子的已婚男性。随机选择的女性是单身的概率为 $\frac{2}{5}$。房间里的人中有多少分数是已婚男性？
Correct Answer: B
(B) Because $\frac{2}{5}$ of all the women in the room are single, there are two single women for every three married women in the room. There are also two single women for every three married men in the room. So out of every $2 + 3 + 3 = 8$ people, 3 are men. The fraction of the people who are married men is $\frac{3}{8}$.
（B）因为房间里所有女性中有$\frac{2}{5}$是单身，所以房间里每三位已婚女性对应两位单身女性。并且，房间里每三位已婚男性也对应两位单身女性。因此，在每$2 + 3 + 3 = 8$个人中，有3个人是男性。已婚男性占所有人的比例是$\frac{3}{8}$。
Q23
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?
Tess 逆时针绕矩形街区 JKLM 跑步。她住在角 J。哪张图可能表示她与家的直线距离？
stem
Correct Answer: D
(D) The distance increases as Tess moves from $J$ to $K$, and continues at perhaps a different rate as she moves from $K$ to $L$. The greatest distance from home will occur at $L$. The distance decreases as she runs from $L$ to $M$ and continues at perhaps a different rate as she moves from $M$ to $J$. Graph D shows these changes.
（D）当 Tess 从 $J$ 移动到 $K$ 时，距离增加；当她从 $K$ 移动到 $L$ 时，距离继续增加，但速率可能不同。离家最远的距离将出现在 $L$。当她从 $L$ 跑到 $M$ 时，距离减少；当她从 $M$ 移动到 $J$ 时，距离继续减少，但速率可能不同。图 D 展示了这些变化。
Q24
In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $\overline{HE}$ and $\overline{FG}$?
在图中，$ABCD$ 是矩形，$EFGH$ 是平行四边形。使用图中给出的测量值，垂直于 $\overline{HE}$ 和 $\overline{FG}$ 的线段 $d$ 的长度是多少？
stem
Correct Answer: C
(C) By the Pythagorean Theorem, $HE = 5$. Rectangle $ABCD$ has area $10 \times 8 = 80$, and the corner triangles have areas $\frac{1}{2} \times 3 \times 4 = 6$ and $\frac{1}{2} \times 6 \times 5 = 15$. So the area of $EFGH$ is $80 - (2)(6) - (2)(15) = 38$. Because the area of $EFGH$ is $EH \times d$ and $EH = 5$, $38 = 5 \times d$, so $d = 7.6$.
（C）由勾股定理，$HE = 5$。矩形 $ABCD$ 的面积为 $10 \times 8 = 80$，四角的三角形面积分别为 $\frac{1}{2} \times 3 \times 4 = 6$ 和 $\frac{1}{2} \times 6 \times 5 = 15$。因此 $EFGH$ 的面积是 $80 - (2)(6) - (2)(15) = 38$。因为 $EFGH$ 的面积为 $EH \times d$ 且 $EH = 5$，所以 $38 = 5 \times d$，从而 $d = 7.6$。
Q25
Two $4 \times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?
两个 $4 \times 4$ 正方形垂直相交，平分它们的相交边，如图所示。圆的直径是两个交点之间的线段。从正方形中移除圆后形成的阴影区域的面积是多少？
stem
Correct Answer: D
(D) The overlap of the two squares is a smaller square with side length 2, so the area of the region covered by the squares is $2(4\times 4)-(2\times 2)=32-4=28.$ The diameter of the circle has length $\sqrt{2^2+2^2}=\sqrt{8}$, the length of the diagonal of the smaller square. The shaded area created by removing the circle from the squares is $28-\pi\left(\frac{\sqrt{8}}{2}\right)^2=28-2\pi.$
（D）两个正方形的重叠部分是一个边长为 2 的小正方形，因此正方形覆盖的区域面积为 $2(4\times 4)-(2\times 2)=32-4=28.$ 圆的直径长度为 $\sqrt{2^2+2^2}=\sqrt{8}$，即小正方形对角线的长度。将圆从正方形覆盖区域中去除后得到的阴影面积为 $28-\pi\left(\frac{\sqrt{8}}{2}\right)^2=28-2\pi.$