2003 AMC 10B — Official Competition Problems (January 2003)
📅 2003 B 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
综合
Which of the following is the same as \[\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?\] \text {(A) } -1 \text {(B) } -\frac{2}{3} \text {(C) } \frac{2}{3} \text {(D) } 1 \text {(E) } \frac{14}{3}
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 \textdollar 546 for the two weeks. How much does one green pill cost?
💡 解题思路
Because there are $14$ days in two weeks, Al spends $546/14 = 39$ dollars per day for the cost of a green pill and a pink pill. If the green pill costs $x$ dollars and the pink pill $x-1$ dollars, the
3
第 3 题
计数
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?
💡 解题思路
It is a well-known fact that the sum of the first $n$ odd numbers is $n^2$ . This makes the sum of the first $8$ odd numbers equal to $64$ .
4
第 4 题
几何·面积
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? [图]
💡 解题思路
The areas of the five regions from greatest to least are $21,20,15,6$ and $4$ .
5
第 5 题
统计
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 the lawn? Easy problem if you understand what swath means, not hard at all
6
第 6 题
几何·面积
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?
💡 解题思路
If you divide the television screen into two right triangles, the legs are in the ratio of $4 : 3$ , and we can let one leg be $4x$ and the other be $3x$ . Then we can use the Pythagorean Theorem.
7
第 7 题
整数运算
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 \[\lfloor √(1) \rfloor + \lfloor √(2) \rfloor + \lfloor √(3) \rfloor + ·s + \lfloor √(16) \rfloor.\]
💡 解题思路
The first three values in the sum are equal to $1,$ the next five equal to $2,$ the next seven equal to $3,$ and the last one equal to $4.$ For example, since $2^2=4$ any square root of a number less
8
第 8 题
规律与数列
The second and fourth terms of a geometric sequence are 2 and 6 . Which of the following is a possible first term?
💡 解题思路
Let the first term be $a$ and the common ratio be $r$ . Therefore,
9
第 9 题
方程
Find the value of x that satisfies the equation 25^{-2} = \frac{5^{48/x}}{5^{26/x} · 25^{17/x}}.
💡 解题思路
Manipulate the powers of $5$ in order to get a clean expression.
10
第 10 题
行程问题
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 has the number of possible license plates increased?
💡 解题思路
There are $26$ letters and $10$ digits. There were $26 \cdot 10^4$ old license plates. There are $26^3 \cdot 10^3$ new license plates. The number of license plates increased by
11
第 11 题
坐标几何
A line with slope 3 intersects a line with slope 5 at point (10,15) . What is the distance between the x -intercepts of these two lines?
💡 解题思路
Using the point-slope form, the equation of each line is
12
第 12 题
计数
Al, Betty, and Clare split \textdollar 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 \textdollar 1500 dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose \textdollar100 dollars. What was Al's original portion?
💡 解题思路
For this problem, we will have to write a three-variable equation, but not necessarily solve it. Let $a, b,$ and $c$ represent the original portions of Al, Betty, and Clare, respectively. At the end o
13
第 13 题
规律与数列
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 ?
💡 解题思路
Let $y=\clubsuit (x)$ . Since $x \leq 99$ , we have $y \leq 18$ . Thus if $\clubsuit (y)=3$ , then $y=3$ or $y=12$ . The 3 values of $x$ for which $\clubsuit (x)=3$ are 12, 21, and 30, and the 7 value
14
第 14 题
整数运算
Given that 3^8·5^2=a^b, where both a and b are positive integers, find the smallest possible value for a+b .
There are 100 players in a single 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 (A) a prime number(B) divisible by 2(C) divisible by 5(D) divisible by 7(E) divisible by 11
💡 解题思路
Notice that $99$ players need to be eliminated for there to be declared a winner. Notice also that every match eliminates exactly one person. Therefore, $99$ matches are needed to eliminate $99$ peopl
16
第 16 题
综合
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 a restaurant should offer so that a customer could have a different dinner each night in the year 2003 ?
💡 解题思路
Let $m$ be the number of main courses the restaurant serves, so $2m$ is the number of appetizers. Then the number of dinner combinations is $2m\times m\times3=6m^2$ . Since the customer wants to eat a
17
第 17 题
分数与比例
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 π r^2 h / 3 and a sphere with radius r has volume 4 π r^3 / 3 .)
💡 解题思路
Let $r$ be the radius of both the cone and the sphere of ice cream, and let $h$ be the height of the cone. The cone can hold a maximum of $\pi r^2h/3$ melted ice cream. The volume of the frozen ice cr
18
第 18 题
整数运算
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 ? \text {(A) } 3 \text {(B) } 5 \text {(C) } 11 \text {(D) } 15 \text {(E) } 165
💡 解题思路
For all consecutive odd integers, one of every five is a multiple of 5 and one of every three is a multiple of 3. The answer is $3 \cdot 5 = 15$ , so ${\boxed{\textbf{(D)15}}}$ is the correct answer.
19
第 19 题
几何·面积
Three semicircles of radius 1 are constructed on diameter \overline{AB} of a semicircle of radius 2 . The centers of the small semicircles divide \overline{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? [图]
In rectangle ABCD, AB=5 and BC=3 . Points F and G are on \overline{CD} so that DF=1 and GC=2 . Lines AF and BG intersect at E . Find the area of \triangle AEB .
💡 解题思路
$\triangle EFG \sim \triangle EAB$ because $FG \parallel AB.$ The ratio of $\triangle EFG$ to $\triangle EAB$ is $2:5$ since $AB=5$ and $FG=2$ from subtraction. If we let $h$ be the height of $\triang
21
第 21 题
概率
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?
💡 解题思路
We can divide the case of all beads in the bag being red after three replacements into three cases.
22
第 22 题
时间问题
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 2003^{rd} chime occur?
💡 解题思路
First, find how many chimes will have already happened before midnight (the beginning of the day) of $\text{February 27, 2003}.$ $13$ half-hours have passed, and the number of chimes according to the
23
第 23 题
几何·面积
A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF ? [图] https://www.youtube.com/watch?v=LREcUjK-56U&feature=youtu.be
💡 解题思路
Here is an easy way to look at this, where $p$ is the perimeter, and $a$ is the apothem :
24
第 24 题
规律与数列
The first four terms in an arithmetic sequence are x+y , x-y , xy , and \frac{x}{y} , in that order. What is the fifth term?
💡 解题思路
The difference between consecutive terms is $(x-y)-(x+y)=-2y.$ Therefore we can also express the third and fourth terms as $x-3y$ and $x-5y.$ Then we can set them equal to $xy$ and $\frac{x}{y}$ becau
25
第 25 题
数论
How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits?
💡 解题思路
To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repe