Difference between revisions of "2002 AMC 12B Problems"
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− | == Problem == | + | == 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 8: | Line 8: | ||
\qquad\mathrm{(E)}\ 8</math> | \qquad\mathrm{(E)}\ 8</math> | ||
− | == Problem == | + | == Problem 2 == |
What is the value of | What is the value of | ||
<cmath>(3x - 2)(4x + 1) - (3x - 2)4x + 1</cmath> | <cmath>(3x - 2)(4x + 1) - (3x - 2)4x + 1</cmath> | ||
Line 19: | Line 19: | ||
\qquad\mathrm{(E)}\ 12</math> | \qquad\mathrm{(E)}\ 12</math> | ||
− | == Problem == | + | == Problem 3 == |
For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a prime number? | For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a prime number? | ||
Line 28: | Line 28: | ||
\qquad\mathrm{(E)}\ \text{infinitely\ many}</math> | \qquad\mathrm{(E)}\ \text{infinitely\ many}</math> | ||
− | == Problem == | + | == 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: | ||
Line 37: | Line 37: | ||
\qquad\mathrm{(E)}\ n > 84</math> | \qquad\mathrm{(E)}\ n > 84</math> | ||
− | == Problem == | + | == Problem 5 == |
− | Let <math>v, w, x, y, </math> and <math>z</math> be the degree measures of the five angles of a pentagon. Suppose that < | + | Let <math>v, w, x, y, </math> and <math>z</math> be the degree measures of the five angles of a pentagon. Suppose that <math>v < w < x < y < z</math> and <math>v, w, x, y, </math> and <math>z</math> form an arithmetic sequence. Find the value of <math>x</math>. |
− | < | + | <math>\mathrm{(A)}\ 72 |
\qquad\mathrm{(B)}\ 84 | \qquad\mathrm{(B)}\ 84 | ||
\qquad\mathrm{(C)}\ 90 | \qquad\mathrm{(C)}\ 90 | ||
\qquad\mathrm{(D)}\ 108 | \qquad\mathrm{(D)}\ 108 | ||
− | \qquad\mathrm{(E)}\ 120<math> | + | \qquad\mathrm{(E)}\ 120</math> |
− | == Problem == | + | == Problem 6 == |
− | Suppose that < | + | 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>. Then the pair <math>(a,b)</math> is |
− | < | + | <math>\mathrm{(A)}\ (-2,1) |
\qquad\mathrm{(B)}\ (-1,2) | \qquad\mathrm{(B)}\ (-1,2) | ||
\qquad\mathrm{(C)}\ (1,-2) | \qquad\mathrm{(C)}\ (1,-2) | ||
\qquad\mathrm{(D)}\ (2,-1) | \qquad\mathrm{(D)}\ (2,-1) | ||
− | \qquad\mathrm{(E)}\ (4,4)<math> | + | \qquad\mathrm{(E)}\ (4,4)</math> |
− | == Problem == | + | == Problem 7 == |
− | The product of three consecutive positive integers is < | + | 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{(B)}\ 77 | ||
\qquad\mathrm{(C)}\ 110 | \qquad\mathrm{(C)}\ 110 | ||
\qquad\mathrm{(D)}\ 149 | \qquad\mathrm{(D)}\ 149 | ||
− | \qquad\mathrm{(E)}\ 194<math> | + | \qquad\mathrm{(E)}\ 194</math> |
− | == Problem == | + | == Problem 8 == |
− | Suppose July of year < | + | Suppose July of year <math>N</math> has five Mondays. Which of the following must occur five times in August of year <math>N</math>? (Note: Both months have 31 days.) |
− | < | + | <math>\mathrm{(A)}\ \text{Monday} |
\qquad\mathrm{(B)}\ \text{Tuesday} | \qquad\mathrm{(B)}\ \text{Tuesday} | ||
\qquad\mathrm{(C)}\ \text{Wednesday} | \qquad\mathrm{(C)}\ \text{Wednesday} | ||
\qquad\mathrm{(D)}\ \text{Thursday} | \qquad\mathrm{(D)}\ \text{Thursday} | ||
− | \qquad\mathrm{(E)}\ \text{Friday}<math> | + | \qquad\mathrm{(E)}\ \text{Friday}</math> |
− | == Problem == | + | == Problem 9 == |
− | If < | + | If <math>a,b,c,d</math> are positive real numbers such that <math>a,b,c,d</math> form an increasing arithmetic sequence and <math>a,b,d</math> form a geometric sequence, then <math>\frac ad</math> is |
− | < | + | <math>\mathrm{(A)}\ \frac 1{12} |
\qquad\mathrm{(B)}\ \frac 16 | \qquad\mathrm{(B)}\ \frac 16 | ||
\qquad\mathrm{(C)}\ \frac 14 | \qquad\mathrm{(C)}\ \frac 14 | ||
\qquad\mathrm{(D)}\ \frac 13 | \qquad\mathrm{(D)}\ \frac 13 | ||
− | \qquad\mathrm{(E)}\ \frac 12<math> | + | \qquad\mathrm{(E)}\ \frac 12</math> |
− | == Problem == | + | == Problem 10 == |
− | How many different integers can be expressed as the sum of three distinct members of the set < | + | 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 |
\qquad\mathrm{(B)}\ 16 | \qquad\mathrm{(B)}\ 16 | ||
\qquad\mathrm{(C)}\ 24 | \qquad\mathrm{(C)}\ 24 | ||
\qquad\mathrm{(D)}\ 30 | \qquad\mathrm{(D)}\ 30 | ||
− | \qquad\mathrm{(E)}\ 35<math> | + | \qquad\mathrm{(E)}\ 35</math> |
− | == Problem == | + | == Problem 11 == |
− | The positive integers < | + | The positive integers <math>A, B, A-B, </math> and <math>A+B</math> are all prime numbers. The sum of these four primes is |
− | < | + | <math>\mathrm{(A)}\ \mathrm{even} |
\qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3 | \qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3 | ||
\qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5 | \qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5 | ||
\qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7 | \qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7 | ||
− | \qquad\mathrm{(E)}\ \mathrm{prime}<math> | + | \qquad\mathrm{(E)}\ \mathrm{prime}</math> |
− | == Problem == | + | == Problem 12 == |
− | For how many integers < | + | For how many integers <math>n</math> is <math>\dfrac n{20-n}</math> the square of an integer? |
− | < | + | <math>\mathrm{(A)}\ 1 |
\qquad\mathrm{(B)}\ 2 | \qquad\mathrm{(B)}\ 2 | ||
\qquad\mathrm{(C)}\ 3 | \qquad\mathrm{(C)}\ 3 | ||
\qquad\mathrm{(D)}\ 4 | \qquad\mathrm{(D)}\ 4 | ||
− | \qquad\mathrm{(E)}\ 10<math> | + | \qquad\mathrm{(E)}\ 10</math> |
− | == Problem == | + | == Problem 13 == |
− | The sum of < | + | The sum of <math>18</math> consecutive positive integers is a perfect square. The smallest possible value of this sum is |
− | < | + | <math>\mathrm{(A)}\ 169 |
\qquad\mathrm{(B)}\ 225 | \qquad\mathrm{(B)}\ 225 | ||
\qquad\mathrm{(C)}\ 289 | \qquad\mathrm{(C)}\ 289 | ||
\qquad\mathrm{(D)}\ 361 | \qquad\mathrm{(D)}\ 361 | ||
− | \qquad\mathrm{(E)}\ 441<math> | + | \qquad\mathrm{(E)}\ 441</math> |
− | == Problem == | + | == 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? | Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? | ||
− | < | + | <math>\mathrm{(A)}\ 8 |
\qquad\mathrm{(B)}\ 9 | \qquad\mathrm{(B)}\ 9 | ||
\qquad\mathrm{(C)}\ 10 | \qquad\mathrm{(C)}\ 10 | ||
\qquad\mathrm{(D)}\ 12 | \qquad\mathrm{(D)}\ 12 | ||
− | \qquad\mathrm{(E)}\ 16<math> | + | \qquad\mathrm{(E)}\ 16</math> |
− | == Problem == | + | == Problem 15 == |
− | How many four-digit numbers < | + | How many four-digit numbers <math>N</math> have the property that the three-digit number obtained by removing the leftmost digit is one night of <math>N</math>? |
− | < | + | <math>\mathrm{(A)}\ 4 |
\qquad\mathrm{(B)}\ 5 | \qquad\mathrm{(B)}\ 5 | ||
\qquad\mathrm{(C)}\ 6 | \qquad\mathrm{(C)}\ 6 | ||
\qquad\mathrm{(D)}\ 7 | \qquad\mathrm{(D)}\ 7 | ||
− | \qquad\mathrm{(E)}\ 8<math> | + | \qquad\mathrm{(E)}\ 8</math> |
− | == Problem == | + | == Problem 16 == |
− | Juan rolls a fair regular octahedral die marked with the numbers < | + | Juan rolls a fair regular octahedral die marked with the numbers <math>1</math> through <math>8</math>. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? |
− | < | + | <math>\mathrm{(A)}\ \frac1{12} |
\qquad\mathrm{(B)}\ \frac 13 | \qquad\mathrm{(B)}\ \frac 13 | ||
\qquad\mathrm{(C)}\ \frac 12 | \qquad\mathrm{(C)}\ \frac 12 | ||
\qquad\mathrm{(D)}\ \frac 7{12} | \qquad\mathrm{(D)}\ \frac 7{12} | ||
− | \qquad\mathrm{(E)}\ \frac 23<math> | + | \qquad\mathrm{(E)}\ \frac 23</math> |
− | == Problem == | + | == 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 thid as afst as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first? | 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 thid as afst 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{(B)}\ \text{Beth} | ||
\qquad\mathrm{(C)}\ \text{Carlos} | \qquad\mathrm{(C)}\ \text{Carlos} | ||
\qquad\mathrm{(D)}\ \text{Andy\ and \ Carlos\ tie\ for\ first.} | \qquad\mathrm{(D)}\ \text{Andy\ and \ Carlos\ tie\ for\ first.} | ||
− | \qquad\mathrm{(E)}\ \text{All\ three\ tie.}<math> | + | \qquad\mathrm{(E)}\ \text{All\ three\ tie.}</math> |
− | == Problem == | + | == Problem 18 == |
− | A point < | + | A point <math>P</math> is randomly selected from the [[rectangle|rectangular]] region with vertices <math>(0,0),(2,0),(2,1),(0,1)</math>. What is the [[probability]] that <math>P</math> is closer to the origin than it is to the point <math>(3,1)</math>? |
− | < | + | <math>\mathrm{(A)}\ |
\qquad\mathrm{(B)}\ | \qquad\mathrm{(B)}\ | ||
\qquad\mathrm{(C)}\ | \qquad\mathrm{(C)}\ | ||
\qquad\mathrm{(D)}\ | \qquad\mathrm{(D)}\ | ||
− | \qquad\mathrm{(E)}\ <math> | + | \qquad\mathrm{(E)}\ </math> |
− | == Problem == | + | == Problem 19 == |
− | If < | + | If <math>a,b,</math> and <math>c</math> are positive real numbers such that <math>a(b+c) = 152, b(c+a) = 162,</math> and <math>c(a+b) = 170</math>, then <math>abc</math> is |
− | < | + | <math>\mathrm{(A)}\ 672 |
\qquad\mathrm{(B)}\ 688 | \qquad\mathrm{(B)}\ 688 | ||
\qquad\mathrm{(C)}\ 704 | \qquad\mathrm{(C)}\ 704 | ||
\qquad\mathrm{(D)}\ 720 | \qquad\mathrm{(D)}\ 720 | ||
− | \qquad\mathrm{(E)}\ 750<math> | + | \qquad\mathrm{(E)}\ 750</math> |
− | == Problem == | + | == Problem 20 == |
− | Let < | + | 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 legs <math>OX</math> and <math>OY</math>, respectively. Given that <math>XN = 19</math> and <math>YM = 22</math>, find <math>XY</math>. |
− | < | + | <math>\mathrm{(A)}\ 24 |
\qquad\mathrm{(B)}\ 26 | \qquad\mathrm{(B)}\ 26 | ||
\qquad\mathrm{(C)}\ 28 | \qquad\mathrm{(C)}\ 28 | ||
\qquad\mathrm{(D)}\ 30 | \qquad\mathrm{(D)}\ 30 | ||
− | \qquad\mathrm{(E)}\ 32<math> | + | \qquad\mathrm{(E)}\ 32</math> |
− | == Problem == | + | == Problem 21 == |
− | For all positive integers < | + | For all positive integers <math>n</math> less than <math>2002</math>, let |
<cmath>\begin{eqnarray*} | <cmath>\begin{eqnarray*} | ||
Line 194: | Line 194: | ||
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Calculate < | + | Calculate <math>\sum_{n=1}^{2001} a_n</math>. |
− | < | + | <math>\mathrm{(A)}\ 448 |
\qquad\mathrm{(B)}\ 486 | \qquad\mathrm{(B)}\ 486 | ||
\qquad\mathrm{(C)}\ 1560 | \qquad\mathrm{(C)}\ 1560 | ||
\qquad\mathrm{(D)}\ 2001 | \qquad\mathrm{(D)}\ 2001 | ||
− | \qquad\mathrm{(E)}\ 2002<math> | + | \qquad\mathrm{(E)}\ 2002</math> |
− | == Problem == | + | == Problem 22 == |
− | For all integers < | + | For all integers <math>n</math> greater than <math>1</math>, define <math>a_n = \frac{1}{\log_n 2002}</math>. Let <math>b = a_2 + a_3 + a_4 + a_5</math> and <math>c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}</math>. Then <math>b- c</math> equals |
− | < | + | <math>\mathrm{(A)}\ -2 |
\qquad\mathrm{(B)}\ -1 | \qquad\mathrm{(B)}\ -1 | ||
\qquad\mathrm{(C)}\ \frac{1}{2002} | \qquad\mathrm{(C)}\ \frac{1}{2002} | ||
\qquad\mathrm{(D)}\ \frac{1}{1001} | \qquad\mathrm{(D)}\ \frac{1}{1001} | ||
− | \qquad\mathrm{(E)}\ \frac 12<math> | + | \qquad\mathrm{(E)}\ \frac 12</math> |
− | == Problem == | + | == Problem 23 == |
− | In < | + | In <math>\triangle ABC</math>, we have <math>AB = 1</math> and <math>AC = 2</math>. Side <math>\overline{BC}</math> and the median from <math>A</math> to <math>\overline{BC}</math> have the same length. What is <math>BC</math>? |
− | < | + | <math>\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} |
\qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 | \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 | ||
\qquad\mathrm{(C)}\ \sqrt{2} | \qquad\mathrm{(C)}\ \sqrt{2} | ||
\qquad\mathrm{(D)}\ \frac 32 | \qquad\mathrm{(D)}\ \frac 32 | ||
− | \qquad\mathrm{(E)}\ \sqrt{3}<math> | + | \qquad\mathrm{(E)}\ \sqrt{3}</math> |
− | == Problem == | + | == Problem 24 == |
− | A convex quadrilateral < | + | A convex quadrilateral <math>ABCD</math> with area <math>2002</math> contains a point <math>P</math> in its interior such that <math>PA = 24, PB = 32, PC = 28, PD = 45</math>. Find the perimeter of <math>ABCD</math>. |
− | < | + | <math>\mathrm{(A)}\ 4\sqrt{2002} |
\qquad\mathrm{(B)}\ 2\sqrt{8465} | \qquad\mathrm{(B)}\ 2\sqrt{8465} | ||
\qquad\mathrm{(C)}\ 2(48+\sqrt{2002}) | \qquad\mathrm{(C)}\ 2(48+\sqrt{2002}) | ||
\qquad\mathrm{(D)}\ 2\sqrt{8633} | \qquad\mathrm{(D)}\ 2\sqrt{8633} | ||
− | \qquad\mathrm{(E)}\ 4(36 + \sqrt{113})<math> | + | \qquad\mathrm{(E)}\ 4(36 + \sqrt{113})</math> |
− | == Problem == | + | == Problem 25 == |
− | Let < | + | Let <math>f(x) = x^2 + 6x + 1</math>, and let <math>R</math> denote the set of points <math>(x,y)</math> in the coordinate plane such that |
<cmath>f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0</cmath> | <cmath>f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0</cmath> | ||
− | The area of < | + | The area of <math>R</math> is closest to |
− | < | + | <math>\mathrm{(A)}\ 21 |
\qquad\mathrm{(B)}\ 22 | \qquad\mathrm{(B)}\ 22 | ||
\qquad\mathrm{(C)}\ 23 | \qquad\mathrm{(C)}\ 23 | ||
\qquad\mathrm{(D)}\ 24 | \qquad\mathrm{(D)}\ 24 | ||
− | \qquad\mathrm{(E)}\ 25 | + | \qquad\mathrm{(E)}\ 25</math> |
== See also == | == See also == |
Revision as of 18:57, 18 January 2008
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 night 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 thid as afst 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