Difference between revisions of "2002 AMC 10B Problems"
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+ | {{AMC10 Problems|year=2002|ab=B}} | ||
==Problem 1== | ==Problem 1== | ||
− | The ratio <math>2^{2001}\ | + | The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is: |
− | (A) 1 | + | <math> \mathrm{(A) \ } \frac{1}{6}\qquad \mathrm{(B) \ } \frac{1}{3}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{2}{3}\qquad \mathrm{(E) \ } \frac{3}{2} </math> |
[[2002 AMC 10B Problems/Problem 1|Solution]] | [[2002 AMC 10B Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | For the nonzero numbers a, b, and c, define | + | For the nonzero numbers <math>a, b,</math> and <math>c,</math> define |
− | + | <cmath>(a,b,c)=\frac{abc}{a+b+c}</cmath> | |
− | < | ||
− | |||
Find <math>(2,4,6)</math>. | Find <math>(2,4,6)</math>. | ||
− | (A) 1 (B) 2 (C) 4 (D) 6 (E) 24 | + | <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math> |
[[2002 AMC 10B Problems/Problem 2|Solution]] | [[2002 AMC 10B Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
+ | The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit | ||
+ | |||
+ | <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math> | ||
[[2002 AMC 10B Problems/Problem 3|Solution]] | [[2002 AMC 10B Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
+ | |||
+ | What is the value of | ||
+ | |||
+ | <math>(3x-2)(4x+1)-(3x-2)4x+1</math> | ||
+ | |||
+ | |||
+ | |||
+ | when <math>x=4</math>? | ||
+ | |||
+ | <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math> | ||
[[2002 AMC 10B Problems/Problem 4|Solution]] | [[2002 AMC 10B Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
+ | |||
+ | Circles of radius <math>2</math> and <math>3</math> are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. | ||
+ | |||
+ | <center><asy> | ||
+ | unitsize(5mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||
+ | dotfactor=4; | ||
+ | |||
+ | real r1=3; real r2=2; real r3=5; | ||
+ | pair A=(-2,0), B=(3,0), C=(0,0); | ||
+ | pair X=(1,0), Y=(5,0); | ||
+ | path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r3); | ||
+ | fill(circleC,gray); | ||
+ | fill(circleA,white); | ||
+ | fill(circleB,white); | ||
+ | draw(circleA); draw(circleB); draw(circleC); | ||
+ | draw(A--X); draw(B--Y); | ||
+ | |||
+ | pair[] ps={A,B}; dot(ps); | ||
+ | |||
+ | label("$3$",midpoint(A--X),N); | ||
+ | label("$2$",midpoint(B--Y),N); | ||
+ | </asy></center> | ||
+ | |||
+ | <math> \mathrm{(A) \ } 3\pi\qquad \mathrm{(B) \ } 4\pi\qquad \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math> | ||
[[2002 AMC 10B Problems/Problem 5|Solution]] | [[2002 AMC 10B Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
+ | |||
+ | For how many positive integers <math>n</math> is <math>n^2-3n+2</math> a prime number? | ||
+ | |||
+ | <math> \mathrm{(A) \ } \text{none}\qquad \mathrm{(B) \ } \text{one}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math> | ||
[[2002 AMC 10B Problems/Problem 6|Solution]] | [[2002 AMC 10B Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | |||
+ | Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true? | ||
+ | |||
+ | <math> \mathrm{(A) \ } 2\text{ divides }n\qquad \mathrm{(B) \ } 3\text{ divides }n\qquad \mathrm{(C) \ } 6\text{ divides }n\qquad \mathrm{(D) \ } 7\text{ divides }n\qquad \mathrm{(E) \ } n>84 </math> | ||
[[2002 AMC 10B Problems/Problem 7|Solution]] | [[2002 AMC 10B Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | |||
+ | Suppose July of year <math>N</math> has five Mondays. Which of the following must occur five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.) | ||
+ | |||
+ | <math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}</math> | ||
[[2002 AMC 10B Problems/Problem 8|Solution]] | [[2002 AMC 10B Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | |||
+ | Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" <math>USAMO</math> occupies position | ||
+ | |||
+ | <math> \mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116 </math> | ||
[[2002 AMC 10B Problems/Problem 9|Solution]] | [[2002 AMC 10B Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
+ | |||
+ | Suppose that <math>a</math> and <math>b</math> are nonzero real numbers, and that the equation <math>x^2+ax+b=0</math> has solutions <math>a</math> and <math>b</math>. What is the pair <math>(a,b)</math>? | ||
+ | |||
+ | <math> \mathrm{(A) \ } (-2,1)\qquad \mathrm{(B) \ } (-1,2)\qquad \mathrm{(C) \ } (1,-2)\qquad \mathrm{(D) \ } (2,-1)\qquad \mathrm{(E) \ } (4,4) </math> | ||
[[2002 AMC 10B Problems/Problem 10|Solution]] | [[2002 AMC 10B Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | |||
+ | The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of their squares? | ||
+ | |||
+ | <math> \mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math> | ||
[[2002 AMC 10B Problems/Problem 11|Solution]] | [[2002 AMC 10B Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | |||
+ | For which of the following values of <math>k</math> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math> | ||
[[2002 AMC 10B Problems/Problem 12|Solution]] | [[2002 AMC 10B Problems/Problem 12|Solution]] | ||
− | == Problem 13 == | + | == Problem 13== |
+ | |||
+ | Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>. | ||
+ | |||
+ | <math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math> | ||
+ | |||
[[2002 AMC 10B Problems/Problem 13|Solution]] | [[2002 AMC 10B Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | |||
+ | The number <math>25^{64}\cdot 64^{25}</math> is the square of a positive integer <math>N</math>. In decimal representation, the sum of the digits of <math>N</math> is | ||
+ | |||
+ | <math> \mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35 </math> | ||
[[2002 AMC 10B Problems/Problem 14|Solution]] | [[2002 AMC 10B Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | |||
+ | The positive integers <math>A</math>, <math>B</math>, <math>A-B</math>, and <math>A+B</math> are all prime numbers. The sum of these four primes is | ||
+ | |||
+ | <math> \mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}</math> | ||
[[2002 AMC 10B Problems/Problem 15|Solution]] | [[2002 AMC 10B Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | |||
+ | For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer? | ||
+ | |||
+ | <math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math> | ||
+ | |||
[[2002 AMC 10B Problems/Problem 16|Solution]] | [[2002 AMC 10B Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
+ | |||
+ | A regular octagon <math>ABCDEFGH</math> has sides of length two. Find the area of <math>\triangle ADG</math>. | ||
+ | |||
+ | <math>\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2</math> | ||
[[2002 AMC 10B Problems/Problem 17|Solution]] | [[2002 AMC 10B Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | |||
+ | Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? | ||
+ | |||
+ | <math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16</math> | ||
[[2002 AMC 10B Problems/Problem 18|Solution]] | [[2002 AMC 10B Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
+ | |||
+ | Suppose that <math>\{a_n\}</math> is an arithmetic sequence with | ||
+ | <cmath> a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.</cmath> | ||
+ | What is the value of <math>a_2 - a_1 ?</math> | ||
+ | |||
+ | <math> \mathrm{(A) \ } 0.0001\qquad \mathrm{(B) \ } 0.001\qquad \mathrm{(C) \ } 0.01\qquad \mathrm{(D) \ } 0.1\qquad \mathrm{(E) \ } 1 </math> | ||
[[2002 AMC 10B Problems/Problem 19|Solution]] | [[2002 AMC 10B Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | Let <math>a, b,</math> and <math>c</math> be real numbers such that <math>a-7b+8c=4</math> and <math>8a+4b-c=7.</math> Then <math>a^2-b^2+c^2</math> is | ||
+ | |||
+ | <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math> | ||
[[2002 AMC 10B Problems/Problem 20|Solution]] | [[2002 AMC 10B Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | |||
+ | Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? | ||
+ | |||
+ | <math> \mathrm{(A) \ } \text{Andy}\qquad \mathrm{(B) \ } \text{Beth}\qquad \mathrm{(C) \ } \text{Carlos}\qquad \mathrm{(D) \ } \text{Andy and Carlos tie for first.}\qquad \mathrm{(E) \ } \text{All three tie.} </math> | ||
[[2002 AMC 10B Problems/Problem 21|Solution]] | [[2002 AMC 10B Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
+ | |||
+ | Let <math>\triangle{XOY}</math> be a right-angled triangle with <math>\angle{XOY}=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints of the legs <math>OX</math> and <math>OY</math>, respectively. Given <math>XN=19</math> and <math>YM=22</math>, find <math>XY</math>. | ||
+ | |||
+ | <math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math> | ||
[[2002 AMC 10B Problems/Problem 22|Solution]] | [[2002 AMC 10B Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
+ | |||
+ | Let <math>\{a_k\}</math> be a sequence of integers such that <math>a_1=1</math> and <math>a_{m+n}=a_m+a_n+mn,</math> for all positive integers <math>m</math> and <math>n.</math> Then <math>a_{12}</math> is | ||
+ | |||
+ | <math> \mathrm{(A) \ } 45\qquad \mathrm{(B) \ } 56\qquad \mathrm{(C) \ } 67\qquad \mathrm{(D) \ } 78\qquad \mathrm{(E) \ } 89 </math> | ||
[[2002 AMC 10B Problems/Problem 23|Solution]] | [[2002 AMC 10B Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
+ | |||
+ | Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius <math>20</math> feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point <math>10</math> vertical feet above the bottom? | ||
+ | |||
+ | <math> \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15 </math> | ||
[[2002 AMC 10B Problems/Problem 24|Solution]] | [[2002 AMC 10B Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
+ | |||
+ | When <math>15</math> is appended to a list of integers, the mean is increased by <math>2</math>. When <math>1</math> is appended to the enlarged list, the mean of the enlarged list is decreased by <math>1</math>. How many integers were in the original list? | ||
+ | |||
+ | <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math> | ||
[[2002 AMC 10B Problems/Problem 25|Solution]] | [[2002 AMC 10B Problems/Problem 25|Solution]] | ||
+ | |||
== See also == | == See also == | ||
+ | {{AMC10 box|year=2002|ab=B|before=[[2002 AMC 10A Problems]]|after=[[2003 AMC 10A Problems]]}} | ||
+ | * [[AMC 10]] | ||
+ | * [[AMC 10 Problems and Solutions]] | ||
* [[AMC Problems and Solutions]] | * [[AMC Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 13:34, 23 July 2024
2002 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
The ratio is:
Problem 2
For the nonzero numbers and
define
Find
.
Problem 3
The arithmetic mean of the nine numbers in the set is a
-digit number
, all of whose digits are distinct. The number
does not contain the digit
Problem 4
What is the value of
when ?
Problem 5
Circles of radius and
are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
![[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r1=3; real r2=2; real r3=5; pair A=(-2,0), B=(3,0), C=(0,0); pair X=(1,0), Y=(5,0); path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r3); fill(circleC,gray); fill(circleA,white); fill(circleB,white); draw(circleA); draw(circleB); draw(circleC); draw(A--X); draw(B--Y); pair[] ps={A,B}; dot(ps); label("$3$",midpoint(A--X),N); label("$2$",midpoint(B--Y),N); [/asy]](http://latex.artofproblemsolving.com/2/c/5/2c5a7bbafa30af5c988825a58d564ad2037aaa7b.png)
Problem 6
For how many positive integers is
a prime number?
Problem 7
Let be a positive integer such that
is an integer. Which of the following statements is not true?
Problem 8
Suppose July of year has five Mondays. Which of the following must occur five times in the August of year
? (Note: Both months have
days.)
Problem 9
Using the letters ,
,
,
, and
, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word"
occupies position
Problem 10
Suppose that and
are nonzero real numbers, and that the equation
has solutions
and
. What is the pair
?
Problem 11
The product of three consecutive positive integers is times their sum. What is the sum of their squares?
Problem 12
For which of the following values of does the equation
have no solution for
?
Problem 13
Find the value(s) of such that
is true for all values of
.
Problem 14
The number is the square of a positive integer
. In decimal representation, the sum of the digits of
is
Problem 15
The positive integers ,
,
, and
are all prime numbers. The sum of these four primes is
Problem 16
For how many integers is
the square of an integer?
Problem 17
A regular octagon has sides of length two. Find the area of
.
Problem 18
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
Problem 19
Suppose that is an arithmetic sequence with
What is the value of
Problem 20
Let and
be real numbers such that
and
Then
is
Problem 21
Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?
Problem 22
Let be a right-angled triangle with
. Let
and
be the midpoints of the legs
and
, respectively. Given
and
, find
.
Problem 23
Let be a sequence of integers such that
and
for all positive integers
and
Then
is
Problem 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point
vertical feet above the bottom?
Problem 25
When is appended to a list of integers, the mean is increased by
. When
is appended to the enlarged list, the mean of the enlarged list is decreased by
. How many integers were in the original list?
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2002 AMC 10A Problems |
Followed by 2003 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.