Difference between revisions of "2003 AMC 10B Problems"
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+ | {{AMC10 Problems|year=2003|ab=B}} | ||
==Problem 1== | ==Problem 1== | ||
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==Problem 4== | ==Problem 4== | ||
− | 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 <math>1</math> each, begonias <math>1.50</math> each, cannas <math>2</math> each, and Easter lilies <math>3</math> each. What is the least possible cost, in dollars, for her garden? | + | Rose fills each of the rectangular and square 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 <math> \ </math><math>1</math> each, begonias <math> \ </math><math>1.50</math> each, cannas <math> \ </math><math>2</math> each, dahlias <math> \ </math><math>2.50</math> each, and Easter lilies <math> \ </math><math>3</math> each. What is the least possible cost, in dollars, for her garden? |
+ | |||
+ | <asy> | ||
+ | unitsize(5mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | draw((6,0)--(0,0)--(0,1)--(6,1)); | ||
+ | draw((0,1)--(0,6)--(4,6)--(4,1)); | ||
+ | draw((4,6)--(11,6)--(11,3)--(4,3)); | ||
+ | draw((11,3)--(11,0)--(6,0)--(6,3)); | ||
+ | label("1",(0,0.5),W); | ||
+ | label("5",(0,3.5),W); | ||
+ | label("3",(11,1.5),E); | ||
+ | label("3",(11,4.5),E); | ||
+ | label("4",(2,6),N); | ||
+ | label("7",(7.5,6),N); | ||
+ | label("6",(3,0),S); | ||
+ | label("5",(8.5,0),S);</asy> | ||
<math>\textbf{(A) } 108 \qquad\textbf{(B) } 115 \qquad\textbf{(C) } 132 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 156 </math> | <math>\textbf{(A) } 108 \qquad\textbf{(B) } 115 \qquad\textbf{(C) } 132 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 156 </math> | ||
Line 35: | Line 52: | ||
==Problem 5== | ==Problem 5== | ||
− | Moe uses a mower to cut his rectangular <math>90</math>-foot by <math>150</math>-foot lawn. The swath he cuts is <math>28</math> inches wide, but he overlaps each cut by <math>4</math> inches to make sure that no grass is missed. He walks at the rate of <math>5000</math> 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 | + | Moe uses a mower to cut his rectangular <math>90</math>-foot by <math>150</math>-foot lawn. The swath he cuts is <math>28</math> inches wide, but he overlaps each cut by <math>4</math> inches to make sure that no grass is missed. He walks at the rate of <math>5000</math> 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? |
<math>\textbf{(A) } 0.75 \qquad\textbf{(B) } 0.8 \qquad\textbf{(C) } 1.35 \qquad\textbf{(D) } 1.5 \qquad\textbf{(E) } 3 </math> | <math>\textbf{(A) } 0.75 \qquad\textbf{(B) } 0.8 \qquad\textbf{(C) } 1.35 \qquad\textbf{(D) } 1.5 \qquad\textbf{(E) } 3 </math> | ||
Line 45: | Line 62: | ||
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 <math>4 : 3</math>. The horizontal length of a "<math>27</math>-inch" television screen is closest, in inches, to which of the following? | 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 <math>4 : 3</math>. The horizontal length of a "<math>27</math>-inch" television screen is closest, in inches, to which of the following? | ||
+ | <asy> | ||
+ | import math; | ||
+ | unitsize(7mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); | ||
+ | fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); | ||
+ | label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); | ||
+ | label(rotate(90)*"Height",(4,1.5),E); | ||
+ | label("Length",(2,0),S);</asy> | ||
<math>\textbf{(A) } 20 \qquad\textbf{(B) } 20.5 \qquad\textbf{(C) } 21 \qquad\textbf{(D) } 21.5 \qquad\textbf{(E) } 22 </math> | <math>\textbf{(A) } 20 \qquad\textbf{(B) } 20.5 \qquad\textbf{(C) } 21 \qquad\textbf{(D) } 21.5 \qquad\textbf{(E) } 22 </math> | ||
Line 77: | Line 103: | ||
==Problem 10== | ==Problem 10== | ||
− | 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 | + | 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? |
<math>\textbf{(A) } \frac{26}{10} \qquad\textbf{(B) } \frac{26^2}{10^2} \qquad\textbf{(C) } \frac{26^2}{10} \qquad\textbf{(D) } \frac{26^3}{10^3} \qquad\textbf{(E) } \frac{26^3}{10^2} </math> | <math>\textbf{(A) } \frac{26}{10} \qquad\textbf{(B) } \frac{26^2}{10^2} \qquad\textbf{(C) } \frac{26^2}{10} \qquad\textbf{(D) } \frac{26^3}{10^3} \qquad\textbf{(E) } \frac{26^3}{10^2} </math> | ||
Line 93: | Line 119: | ||
==Problem 12== | ==Problem 12== | ||
− | Al, Betty, and Clare split <math>1000</math> 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 <math>1500</math>. Betty and Clare have both doubled their money, whereas Al has managed to lose <math>100</math>. What was Al's original portion? | + | Al, Betty, and Clare split <math> \ </math><math>1000</math> 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 <math> \ </math><math>1500</math>. Betty and Clare have both doubled their money, whereas Al has managed to lose <math> \ </math><math>100</math>. What was Al's original portion? |
<math> \textbf{(A) }\ </math><math>250 \qquad\textbf{(B) }\ </math> <math>350 \qquad\textbf{(C) }\ </math><math>400\qquad\textbf{(D) }\ </math> <math>450\qquad\textbf{(E) }\ </math><math>500 </math> | <math> \textbf{(A) }\ </math><math>250 \qquad\textbf{(B) }\ </math> <math>350 \qquad\textbf{(C) }\ </math><math>400\qquad\textbf{(D) }\ </math> <math>450\qquad\textbf{(E) }\ </math><math>500 </math> | ||
Line 117: | Line 143: | ||
==Problem 15== | ==Problem 15== | ||
− | There are <math>100</math> 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 <math>28</math> players are given a bye, and the remaining <math>72</math> players are paired off to play. After each round, the remaining players play in the next round. The | + | There are <math>100</math> 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 <math>28</math> players are given a bye, and the remaining <math>72</math> players are paired off to play. After each round, the remaining players play in the next round. The tournament continues until only one player remains unbeaten. The total number of matches played is |
<math>\textbf{(A) } \text{a prime number} \qquad\textbf{(B) } \text{divisible by 2} \qquad\textbf{(C) } \text{divisible by 5} \qquad\textbf{(D) } \text{divisible by 7} \qquad\textbf{(E) } \text{divisible by 11}</math> | <math>\textbf{(A) } \text{a prime number} \qquad\textbf{(B) } \text{divisible by 2} \qquad\textbf{(C) } \text{divisible by 5} \qquad\textbf{(D) } \text{divisible by 7} \qquad\textbf{(E) } \text{divisible by 11}</math> | ||
Line 125: | Line 151: | ||
==Problem 16== | ==Problem 16== | ||
− | A restaurant offers three | + | 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 <math>2003</math>? |
<math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math> | <math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math> | ||
Line 152: | Line 178: | ||
Three semicircles of radius <math>1</math> are constructed on diameter <math>\overline{AB}</math> of a semicircle of radius <math>2</math>. The centers of the small semicircles divide <math>\overline{AB}</math> 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? | Three semicircles of radius <math>1</math> are constructed on diameter <math>\overline{AB}</math> of a semicircle of radius <math>2</math>. The centers of the small semicircles divide <math>\overline{AB}</math> 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? | ||
+ | |||
+ | <asy> | ||
+ | import graph; | ||
+ | unitsize(14mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | dashed=linetype("4 4"); | ||
+ | dotfactor=3; | ||
+ | pair A=(-2,0), B=(2,0); | ||
+ | fill(Arc((0,0),2,0,180)--cycle,mediumgray); | ||
+ | fill(Arc((-1,0),1,0,180)--cycle,white); | ||
+ | fill(Arc((0,0),1,0,180)--cycle,white); | ||
+ | fill(Arc((1,0),1,0,180)--cycle,white); | ||
+ | draw(Arc((-1,0),1,60,180)); | ||
+ | draw(Arc((0,0),1,0,60),dashed); | ||
+ | draw(Arc((0,0),1,60,120)); | ||
+ | draw(Arc((0,0),1,120,180),dashed); | ||
+ | draw(Arc((1,0),1,0,120)); | ||
+ | draw(Arc((0,0),2,0,180)--cycle); | ||
+ | dot((0,0)); | ||
+ | dot((-1,0)); | ||
+ | dot((1,0)); | ||
+ | draw((-2,-0.1)--(-2,-0.3),gray); | ||
+ | draw((-1,-0.1)--(-1,-0.3),gray); | ||
+ | draw((1,-0.1)--(1,-0.3),gray); | ||
+ | draw((2,-0.1)--(2,-0.3),gray); | ||
+ | label("$A$",A,W); | ||
+ | label("$B$",B,E); | ||
+ | label("1",(-1.5,-0.1),S); | ||
+ | label("2",(0,-0.1),S); | ||
+ | label("1",(1.5,-0.1),S);</asy> | ||
<math>\textbf{(A) } \pi - \sqrt{3} \qquad\textbf{(B) } \pi - \sqrt{2} \qquad\textbf{(C) } \frac{\pi + \sqrt{2}}{2} \qquad\textbf{(D) } \frac{\pi +\sqrt{3}}{2} \qquad\textbf{(E) } \frac{7}{6}\pi - \frac{\sqrt{3}}{2}</math> | <math>\textbf{(A) } \pi - \sqrt{3} \qquad\textbf{(B) } \pi - \sqrt{2} \qquad\textbf{(C) } \frac{\pi + \sqrt{2}}{2} \qquad\textbf{(D) } \frac{\pi +\sqrt{3}}{2} \qquad\textbf{(E) } \frac{7}{6}\pi - \frac{\sqrt{3}}{2}</math> | ||
Line 161: | Line 217: | ||
In rectangle <math>ABCD, AB=5</math> and <math>BC=3</math>. Points <math>F</math> and <math>G</math> are on <math>\overline{CD}</math> so that <math>DF=1</math> and <math>GC=2</math>. Lines <math>AF</math> and <math>BG</math> intersect at <math>E</math>. Find the area of <math>\triangle AEB</math>. | In rectangle <math>ABCD, AB=5</math> and <math>BC=3</math>. Points <math>F</math> and <math>G</math> are on <math>\overline{CD}</math> so that <math>DF=1</math> and <math>GC=2</math>. Lines <math>AF</math> and <math>BG</math> intersect at <math>E</math>. Find the area of <math>\triangle AEB</math>. | ||
+ | <asy> | ||
+ | unitsize(6mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); | ||
+ | pair E=extension(A,F,B,G); | ||
+ | draw(A--B--C--D--A--E--B); | ||
+ | label("$A$",A,SW); | ||
+ | label("$B$",B,SE); | ||
+ | label("$C$",C,NE); | ||
+ | label("$D$",D,NW); | ||
+ | label("$E$",E,N); | ||
+ | label("$F$",F,SE); | ||
+ | label("$G$",G,SW); | ||
+ | label("$B$",B,SE); | ||
+ | label("1",midpoint(D--F),N); | ||
+ | label("2",midpoint(G--C),N); | ||
+ | label("3",midpoint(B--C),E); | ||
+ | label("3",midpoint(A--D),W); | ||
+ | label("5",midpoint(A--B),S); </asy> | ||
<math>\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15</math> | <math>\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15</math> | ||
Line 185: | Line 260: | ||
A regular octagon <math>ABCDEFGH</math> has an area of one square unit. What is the area of the rectangle <math>ABEF</math>? | A regular octagon <math>ABCDEFGH</math> has an area of one square unit. What is the area of the rectangle <math>ABEF</math>? | ||
+ | <asy> | ||
+ | unitsize(10mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); | ||
+ | draw(A--B--C--D--E--F--G--H--cycle); | ||
+ | label("$A$",A,NNW); | ||
+ | label("$B$",B,NNE); | ||
+ | label("$C$",C,ENE); | ||
+ | label("$D$",D,ESE); | ||
+ | label("$E$",E,SSE); | ||
+ | label("$F$",F,SSW); | ||
+ | label("$G$",G,WSW); | ||
+ | label("$H$",H,WNW); </asy> | ||
<math>\textbf{(A) } 1 - \frac{\sqrt{2}}{2} \qquad\textbf{(B) } \frac{\sqrt{2}}{4} \qquad\textbf{(C) } \sqrt{2} - 1 \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{1+\sqrt{2}}{4}</math> | <math>\textbf{(A) } 1 - \frac{\sqrt{2}}{2} \qquad\textbf{(B) } \frac{\sqrt{2}}{4} \qquad\textbf{(C) } \sqrt{2} - 1 \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{1+\sqrt{2}}{4}</math> | ||
Line 191: | Line 279: | ||
==Problem 24== | ==Problem 24== | ||
− | The first four terms in an arithmetic sequence are <math>x + y, x - y, xy,</math> and <math>x/y,</math> in that order. What is the | + | The first four terms in an arithmetic sequence are <math>x + y, x - y, xy,</math> and <math>x/y,</math> in that order. What is the fifth term? |
− | <math>\textbf{(A) } -\frac{15}{8} \qquad\textbf{(B) } -\frac{6}{5} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \frac{27}{20} \qquad\textbf{(E) } \frac{123}{40}</math> | + | <math> \textbf{(A)}\ -\frac{15}{8}\qquad\textbf{(B)}\ -\frac{6}{5}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \frac{27}{20}\qquad\textbf{(E)}\ \frac{123}{40} </math> |
[[2003 AMC 10B Problems/Problem 24|Solution]] | [[2003 AMC 10B Problems/Problem 24|Solution]] | ||
Line 204: | Line 292: | ||
[[2003 AMC 10B Problems/Problem 25|Solution]] | [[2003 AMC 10B Problems/Problem 25|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | {{AMC10 box|year=2003|ab=B|before=[[2003 AMC 10A Problems]]|after=[[2004 AMC 10A Problems]]}} | ||
+ | * [[AMC 10]] | ||
+ | * [[AMC 10 Problems and Solutions]] | ||
+ | * [[AMC Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 19:50, 25 August 2024
2003 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Which of the following is the same as
Problem 2
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pink pill, and Al's pills cost a total of
for the two weeks. How much does one green pill cost?
Problem 3
The sum of consecutive even integers is
less than the sum of the first
consecutive odd counting numbers. What is the smallest of the even integers?
Problem 4
Rose fills each of the rectangular and square 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 each, begonias
each, cannas
each, dahlias
each, and Easter lilies
each. What is the least possible cost, in dollars, for her garden?
Problem 5
Moe uses a mower to cut his rectangular -foot by
-foot lawn. The swath he cuts is
inches wide, but he overlaps each cut by
inches to make sure that no grass is missed. He walks at the rate of
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?
Problem 6
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 . The horizontal length of a "
-inch" television screen is closest, in inches, to which of the following?
Problem 7
The symbolism denotes the largest integer not exceeding
. For example,
and
. Compute
Problem 8
The second and fourth terms of a geometric sequence are and
. Which of the following is a possible first term?
Problem 9
Find the value of that satisfies the equation
Problem 10
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?
Problem 11
A line with slope intersects a line with slope
at point
. What is the distance between the
-intercepts of these two lines?
Problem 12
Al, Betty, and Clare split 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
. Betty and Clare have both doubled their money, whereas Al has managed to lose
. What was Al's original portion?
Problem 13
Let denote the sum of the digits of the positive integer
. For example,
and
. For how many two-digit values of
is
?
Problem 14
Given that where both
and
are positive integers, find the smallest possible value for
.
Problem 15
There are 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
players are given a bye, and the remaining
players are paired off to play. After each round, the remaining players play in the next round. The tournament continues until only one player remains unbeaten. The total number of matches played is
Problem 16
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 ?
Problem 17
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 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
and height
has volume
and a sphere with radius
has volume
.)
Problem 18
What is the largest integer that is a divisor of
for all positive even integers
?
Problem 19
Three semicircles of radius are constructed on diameter
of a semicircle of radius
. The centers of the small semicircles divide
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?
Problem 20
In rectangle and
. Points
and
are on
so that
and
. Lines
and
intersect at
. Find the area of
.
Problem 21
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?
Problem 22
A clock chimes once at minutes past each hour and chimes on the hour according to the hour. For example, at
there is one chime and at noon and midnight there are twelve chimes. Starting at
on
on what date will the
chime occur?
Problem 23
A regular octagon has an area of one square unit. What is the area of the rectangle
?
Problem 24
The first four terms in an arithmetic sequence are and
in that order. What is the fifth term?
Problem 25
How many distinct four-digit numbers are divisible by and have
as their last two digits?
See also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2003 AMC 10A Problems |
Followed by 2004 AMC 10A Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.