Difference between revisions of "2002 AMC 12B Problems"
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== Problem 1 == | == Problem 1 == | ||
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 | 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 | ||
Line 15: | Line 16: | ||
when <math>x=4</math>? | when <math>x=4</math>? | ||
+ | |||
<math>\mathrm{(A)}\ 0 | <math>\mathrm{(A)}\ 0 | ||
\qquad\mathrm{(B)}\ 1 | \qquad\mathrm{(B)}\ 1 | ||
Line 35: | Line 37: | ||
== Problem 4 == | == Problem 4 == | ||
− | Let <math>n</math> be a positive integer such that <math>\frac 12 + \frac 13 + \frac 17 + \frac 1n</math> is an integer. Which of the following statements is '''not ''' true: | + | Let <math>n</math> be a positive integer such that <math>\frac 12 + \frac 13 + \frac 17 + \frac 1n</math> is an integer. Which of the following statements is '''not''' true: |
<math>\mathrm{(A)}\ 2\ \text{divides\ }n | <math>\mathrm{(A)}\ 2\ \text{divides\ }n | ||
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\qquad\mathrm{(C)}\ 6\ \text{divides\ }n | \qquad\mathrm{(C)}\ 6\ \text{divides\ }n | ||
\qquad\mathrm{(D)}\ 7\ \text{divides\ }n | \qquad\mathrm{(D)}\ 7\ \text{divides\ }n | ||
− | \qquad\mathrm{(E)}\ n > 84</math> | + | \qquad\mathrm{(E)}\ {n > 84}</math> |
[[2002 AMC 12B Problems/Problem 4|Solution]] | [[2002 AMC 12B Problems/Problem 4|Solution]] | ||
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== Problem 10 == | == Problem 10 == | ||
How many different integers can be expressed as the sum of three distinct members of the set <math>\{1,4,7,10,13,16,19\}</math>? | How many different integers can be expressed as the sum of three distinct members of the set <math>\{1,4,7,10,13,16,19\}</math>? | ||
+ | |||
<math>\mathrm{(A)}\ 13 | <math>\mathrm{(A)}\ 13 | ||
\qquad\mathrm{(B)}\ 16 | \qquad\mathrm{(B)}\ 16 | ||
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== Problem 15 == | == Problem 15 == | ||
− | How many four-digit numbers <math>N</math> have the property that the three-digit number obtained by removing the leftmost digit is one | + | How many four-digit numbers <math>N</math> have the property that the three-digit number obtained by removing the leftmost digit is one ninth of <math>N</math>? |
<math>\mathrm{(A)}\ 4 | <math>\mathrm{(A)}\ 4 | ||
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[[2002 AMC 12B Problems/Problem 25|Solution]] | [[2002 AMC 12B Problems/Problem 25|Solution]] | ||
== See also == | == See also == | ||
+ | |||
+ | {{AMC12 box|year=2002|ab=B|before=[[2002 AMC 12A Problems]]|after=[[2003 AMC 12A Problems]]}} | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[2002 AMC 12A]] | * [[2002 AMC 12A]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 12:06, 19 February 2020
2002 AMC 12B (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 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 2
What is the value of
when ?
Problem 3
For how many positive integers is a prime number?
Problem 4
Let be a positive integer such that is an integer. Which of the following statements is not true:
Problem 5
Let and be the degree measures of the five angles of a pentagon. Suppose that and and form an arithmetic sequence. Find the value of .
Problem 6
Suppose that and are nonzero real numbers, and that the equation has solutions and . Then the pair is
Problem 7
The product of three consecutive positive integers is times their sum. What is the sum of their squares?
Problem 8
Suppose July of year has five Mondays. Which of the following must occur five times in August of year ? (Note: Both months have 31 days.)
Problem 9
If are positive real numbers such that form an increasing arithmetic sequence and form a geometric sequence, then is
Problem 10
How many different integers can be expressed as the sum of three distinct members of the set ?
Problem 11
The positive integers and are all prime numbers. The sum of these four primes is
Problem 12
For how many integers is the square of an integer?
Problem 13
The sum of consecutive positive integers is a perfect square. The smallest possible value of this sum is
Problem 14
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
Problem 15
How many four-digit numbers have the property that the three-digit number obtained by removing the leftmost digit is one ninth of ?
Problem 16
Juan rolls a fair regular octahedral die marked with the numbers through . Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?
Problem 17
Andy’s lawn has twice as much area as Beth’s lawn and three times as much area 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 18
A point is randomly selected from the rectangular region with vertices . What is the probability that is closer to the origin than it is to the point ?
Problem 19
If and are positive real numbers such that and , then is
Problem 20
Let be a right-angled triangle with . Let and be the midpoints of legs and , respectively. Given that and , find .
Problem 21
For all positive integers less than , let
Calculate .
Problem 22
For all integers greater than , define . Let and . Then equals
Problem 23
In , we have and . Side and the median from to have the same length. What is ?
Problem 24
A convex quadrilateral with area contains a point in its interior such that . Find the perimeter of .
Problem 25
Let , and let denote the set of points in the coordinate plane such that The area of is closest to
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2002 AMC 12A Problems |
Followed by 2003 AMC 12A 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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.