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AMC10 2003 B

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AMC10 · 2003 (B)

Q1
Which of the following is the same as \[ \frac{2 -4 + 6 -8 + 10 -12 + 14}{3 -6 + 9 -12 + 15 -18 + 21} \]?
以下哪个与 \[ \frac{2 -4 + 6 -8 + 10 -12 + 14}{3 -6 + 9 -12 + 15 -18 + 21} \] 相同？
Q2
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al’s pills cost a total of $546 for the two weeks. How much does one green pill cost?
Al 得了代数炎，必须每天服用一颗绿色药丸和一颗粉色药丸，持续两周。绿色药丸的价格比粉色药丸贵 1 美元，Al 的药丸两周总共花费 546 美元。一颗绿色药丸多少钱？
Q3
The sum of 5 consecutive even integers is 4 less than the sum of the first 8 consecutive odd counting numbers. What is the smallest of the even integers?
5 个连续偶整数的和比前 8 个连续奇数的和少 4。求这些偶整数中最小的那个。
Q4
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $1 each, begonias $1.50 each, cannas $2 each, dahlias $2.50 each, and Easter lilies $3 each. What is the least possible cost, in dollars, for her garden? [Diagram shows a rectangle divided into regions with side lengths labeled 1, 5, 4, 7, 3, 3, 5, 6]
Rose 在她的矩形花坛的每个矩形区域中种不同种类的花。图中标示了花坛矩形区域的边长，单位为英尺。她每个区域每平方英尺种一朵花。紫菀每朵 1 美元，秋海棠每朵 1.50 美元，美人蕉每朵 2 美元，大丽花每朵 2.50 美元，复活节百合每朵 3 美元。她的花园的最低可能花费是多少美元？[图示一个矩形分为区域，边长标示为 1, 5, 4, 7, 3, 3, 5, 6]
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Q5
Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?
Moe 用割草机修剪他的 90 英尺乘 150 英尺的矩形草坪。割草宽度为 28 英寸，但他每条割草重叠 4 英寸以确保不漏草。他推着割草机行走速度为每小时 5000 英尺。Moe 修剪草坪需要多长时间（小时）？以下哪个最接近？
Q6
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is 4 : 3. The horizontal length of a “27-inch” television screen is closest, in inches, to which of the following? [Diagram shows a rectangle labeled “Length” (horizontal), “Height” (vertical), and “Diagonal”]
许多电视屏幕是长宽比为 4 : 3 的矩形，以对角线长度来测量尺寸。“27英寸”电视屏幕的水平长度（英寸）最接近于下列哪个选项？[图示一个矩形，标有“Length”（水平）、“Height”（垂直）和“Diagonal”]
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Q7
The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. For example, $\lfloor 3 \rfloor = 3$, and $\lfloor 9/2 \rfloor = 4$. Compute $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor$.
符号 $\lfloor x \rfloor$ 表示不大于 $x$ 的最大整数。例如，$\lfloor 3 \rfloor = 3$，$\lfloor 9/2 \rfloor = 4$。计算 $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor$。
Q8
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?
一个几何数列的第二项和第四项分别为 2 和 6。下列哪项可能是首项？
Q9
Find the value of $x$ that satisfies the equation $25^{-2} = 548/x \div 526/x \cdot 2517/x$.
求满足方程 $25^{-2} = 548/x \div 526/x \cdot 2517/x$ 的 $x$ 的值。
Q10
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased?
内布拉斯加州（AMC 的家乡）改变了其车牌方案。旧车牌由一个字母后跟四个数字组成。新车牌由三个字母后跟三个数字组成。新车牌可能数量是旧车牌的多少倍？
Q11
A line with slope 3 intersects a line with slope 5 at the point (10, 15). What is the distance between the x-intercepts of these two lines?
一条斜率为 3 的直线与一条斜率为 5 的直线相交于点 (10, 15)。这两条直线的 x 轴截距之间的距离是多少？
Q12
Al, Betty, and Clare split $1000 among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $1500. Betty and Clare have both doubled their money, whereas Al has managed to lose $100. What was Al’s original portion?
Al、Betty 和 Clare 将 1000 美元分给三人，以不同方式投资。每人初始金额不同。一年后他们总共有 1500 美元。Betty 和 Clare 的钱都翻倍了，而 Al 亏了 100 美元。Al 的初始金额是多少？
Q13
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x)) = 3$?
设 $\clubsuit(x)$ 表示正整数 $x$ 的各位数字之和。例如，$\clubsuit(8) = 8$，$\clubsuit(123) = 1 + 2 + 3 = 6$。有且仅有几个两位数 $x$ 满足 $\clubsuit(\clubsuit(x)) = 3$？
Q14
Given that $3^8 \cdot 5^2 = ab$, where both $a$ and $b$ are positive integers, find the smallest possible value for $a + b$.
已知 $3^8 \cdot 5^2 = ab$，其中 $a$ 和 $b$ 均为正整数，求 $a + b$ 的最小可能值。
Q15
There are 100 players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest 28 players are given a bye, and the remaining 72 players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is
单打网球锦标赛有 100 名选手。采用单淘汰制，输掉比赛的选手被淘汰。第一轮，最强的 28 名选手轮空，其余 72 名选手两两配对比赛。每轮后，剩余选手进入下一轮。比赛继续直到仅剩一名不败选手。总共进行了多少场比赛？
Q16
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year 2003?
一家餐厅提供三种甜点，小菜的数量恰好是主菜数量的两倍。一顿晚餐包括一道小菜、一道主菜和一道甜点。餐厅应该提供多少最少的主菜数量，使得顾客在2003年每天都能吃到不同的晚餐？
Q17
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75% of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? (Note: A cone with radius $r$ and height $h$ has volume $\pi r^2 h /3$, and a sphere with radius $r$ has volume $4\pi r^3 /3$.)
一个冰淇淋甜筒由一个香草冰淇淋球和一个直径与球相同的圆锥组成。如果冰淇淋融化，它正好充满圆锥。假设融化的冰淇淋体积占冷冻冰淇淋体积的75%。圆锥的高度与其半径的比是多少？（注：半径为$r$、高度为$h$的圆锥体积为$\pi r^2 h /3$，半径为$r$的球体积为$4\pi r^3 /3$。）
Q18
What is the largest integer that is a divisor of $(n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ for all positive even integers $n$?
对于所有正偶数$n$，什么最大的整数能整除$(n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$？
Q19
Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [Diagram shows large semicircle radius 2 on diameter AB, with three small semicircles of radius 1 inside it along AB]
在大半径为2的半圆的直径AB上，构造了三个半径为1的半圆。小半圆的圆心将AB分成四个等长的线段，如图所示。阴影区域是大半圆内但在小半圆外的面积是多少？[图示：在直径AB上的大半圆半径2，沿AB在其内部有三个半径1的小半圆]
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Q20
In rectangle ABCD, AB = 5 and BC = 3. Points F and G are on CD so that DF = 1 and GC = 2. Lines AF and BG intersect at E. Find the area of △AEB. [Diagram shows rectangle ABCD with AB = 5 (top), BC = 3 (right side), points F and G on CD with DF = 1, GC = 2]
在矩形ABCD中，AB = 5，BC = 3。点F和G在CD上，使得DF = 1，GC = 2。直线AF和BG相交于E。求△AEB的面积。[图示：矩形ABCD，AB = 5（顶部），BC = 3（右侧），点F和G在CD上，DF = 1，GC = 2]
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Q21
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?
一个袋子里有2颗红色珠子和2颗绿色珠子。你伸手进袋子取出颗珠子，无论取出的是什么颜色的珠子，都用一颗红色珠子替换回去。经过三次这样的替换后，袋子里所有珠子都是红色的概率是多少？
Q22
A clock chimes once at 30 minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February 26, 2003, on what date will the 2003rd chime occur?
一个钟在每小时30分时敲一次，在整点时按照小时数敲响。例如，下午1点敲1下，正午和午夜敲12下。从2003年2月26日上午11:15开始，第2003次敲响将在哪一天发生？
Q23
A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF? [Diagram shows regular octagon labeled A B C D E F G H clockwise]
一个正八边形ABCDEFGH的面积为一平方单位。矩形ABEF的面积是多少？[图示为顺时针标记A B C D E F G H的正八边形]
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Q24
The first four terms in an arithmetic sequence are x + y, x − y, xy, and x/y, in that order. What is the fifth term?
一个等差数列的前四项依次为 x + y, x − y, xy, 和 x/y。第几项是多少？
Q25
How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits?
有多少个不同的四位数能被3整除且末两位数是23？
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