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>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is: | The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is: | ||
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== Problem 8 == | == Problem 8 == | ||
− | Suppose July of year <math>N</math> has five Mondays. Which of the following must | + | 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> | <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> | ||
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== 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 | + | 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> | <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> | ||
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== Problem 11 == | == Problem 11 == | ||
− | The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of | + | 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> | <math> \mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math> | ||
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== Problem 22 == | == Problem 22 == | ||
− | Let <math>\triangle{XOY}</math> be a right-triangle with <math>m\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>. | + | Let <math>\triangle{XOY}</math> be a right-angled triangle with <math>m\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> | <math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math> | ||
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[[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 13:07, 19 February 2020
2002 AMC 10B (Answer Key) Printable version: | 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.
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.