Difference between revisions of "2002 AMC 10B Problems"
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What is the value of | What is the value of | ||
− | < | + | <math>(3x-2)(4x+1)-(3x-2)4x+1</math> |
+ | |||
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when <math>x=4</math>? | when <math>x=4</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 solutions <math>a</math> and <math>b</math>. | + | 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 22 == | == Problem 22 == | ||
− | Let <math>\triangle{XOY}</math> be a right-angled triangle with <math> | + | 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> | <math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math> |
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.