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Difference between revisions of "2002 AMC 12B Problems"
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== Problem 1 ==
+
== Problem ==
 +
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
  
−
[[2002 AMC 12B/Problem 1|Solution]]
+
<math>\mathrm{(A)}\ 0
 +
\qquad\mathrm{(B)}\ 2
 +
\qquad\mathrm{(C)}\ 4
 +
\qquad\mathrm{(D)}\ 6
 +
\qquad\mathrm{(E)}\ 8</math>
  
−
== Problem 2 ==
+
== Problem ==
 +
What is the value of
 +
<cmath>(3x - 2)(4x + 1) - (3x - 2)4x + 1</cmath>
  
−
[[2002 AMC 12B/Problem 2|Solution]]
+
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>
  
−
== Problem 3 ==
+
== Problem ==
 +
For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a prime number?
  
−
[[2002 AMC 12B/Problem 3|Solution]]
+
<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>
  
−
== Problem 4 ==
+
== Problem ==
 +
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:
  
−
[[2002 AMC 12B/Problem 4|Solution]]
+
<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>
  
−
== Problem 5 ==
+
== Problem ==
 +
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></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>.
  
−
[[2002 AMC 12B/Problem 5|Solution]]
+
</math>\mathrm{(A)}\ 72
 +
\qquad\mathrm{(B)}\ 84
 +
\qquad\mathrm{(C)}\ 90
 +
\qquad\mathrm{(D)}\ 108
 +
\qquad\mathrm{(E)}\ 120<math>
  
−
== Problem 6 ==
+
== Problem ==
 +
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
  
−
[[2002 AMC 12B/Problem 6|Solution]]
+
</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>
  
−
== Problem 7 ==
+
== Problem ==
 +
The product of three consecutive positive integers is </math>8<math> times their sum. What is the sum of their squares?
  
−
[[2002 AMC 12B/Problem 7|Solution]]
+
</math>\mathrm{(A)}\ 50
 +
\qquad\mathrm{(B)}\ 77
 +
\qquad\mathrm{(C)}\ 110
 +
\qquad\mathrm{(D)}\ 149
 +
\qquad\mathrm{(E)}\ 194<math>
  
−
== Problem 8 ==
+
== Problem ==
 +
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.)
  
−
[[2002 AMC 12B/Problem 8|Solution]]
+
</math>\mathrm{(A)}\ \text{Monday}
 +
\qquad\mathrm{(B)}\ \text{Tuesday}
 +
\qquad\mathrm{(C)}\ \text{Wednesday}
 +
\qquad\mathrm{(D)}\ \text{Thursday}
 +
\qquad\mathrm{(E)}\ \text{Friday}<math>
  
−
== Problem 9 ==
+
== Problem ==
 +
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
  
−
[[2002 AMC 12B/Problem 9|Solution]]
+
</math>\mathrm{(A)}\ \frac 1{12}
 +
\qquad\mathrm{(B)}\ \frac 16
 +
\qquad\mathrm{(C)}\ \frac 14
 +
\qquad\mathrm{(D)}\ \frac 13
 +
\qquad\mathrm{(E)}\ \frac 12<math>
  
−
== Problem 10 ==
+
== Problem ==
 +
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{(C)}\ 24
 +
\qquad\mathrm{(D)}\ 30
 +
\qquad\mathrm{(E)}\ 35<math>
  
−
[[2002 AMC 12B/Problem 10|Solution]]
+
== Problem ==
 +
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
  
−
== Problem 11 ==
+
</math>\mathrm{(A)}\ \mathrm{even}
 +
\qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3
 +
\qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5
 +
\qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7
 +
\qquad\mathrm{(E)}\ \mathrm{prime}<math>
  
−
[[2002 AMC 12B/Problem 11|Solution]]
+
== Problem ==
 +
For how many integers </math>n<math> is </math>\dfrac n{20-n}<math> the square of an integer?
  
−
== Problem 12 ==
+
</math>\mathrm{(A)}\ 1
 +
\qquad\mathrm{(B)}\ 2
 +
\qquad\mathrm{(C)}\ 3
 +
\qquad\mathrm{(D)}\ 4
 +
\qquad\mathrm{(E)}\ 10<math>
  
−
[[2002 AMC 12B/Problem 12|Solution]]
+
== Problem ==
 +
The sum of </math>18<math> consecutive positive integers is a perfect square. The smallest possible value of this sum is
  
−
== Problem 13 ==
+
</math>\mathrm{(A)}\ 169
 +
\qquad\mathrm{(B)}\ 225
 +
\qquad\mathrm{(C)}\ 289
 +
\qquad\mathrm{(D)}\ 361
 +
\qquad\mathrm{(E)}\ 441<math>
  
−
[[2002 AMC 12B/Problem 13|Solution]]
+
== Problem ==
 +
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
  
−
== Problem 14 ==
+
</math>\mathrm{(A)}\ 8
 +
\qquad\mathrm{(B)}\ 9
 +
\qquad\mathrm{(C)}\ 10
 +
\qquad\mathrm{(D)}\ 12
 +
\qquad\mathrm{(E)}\ 16<math>
  
−
[[2002 AMC 12B/Problem 14|Solution]]
+
== Problem ==
 +
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>?
  
−
== Problem 15 ==
+
</math>\mathrm{(A)}\ 4
 +
\qquad\mathrm{(B)}\ 5
 +
\qquad\mathrm{(C)}\ 6
 +
\qquad\mathrm{(D)}\ 7
 +
\qquad\mathrm{(E)}\ 8<math>
  
−
[[2002 AMC 12B/Problem 15|Solution]]
+
== Problem ==
 +
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?
  
−
== Problem 16 ==
+
</math>\mathrm{(A)}\ \frac1{12}
 +
\qquad\mathrm{(B)}\ \frac 13
 +
\qquad\mathrm{(C)}\ \frac 12
 +
\qquad\mathrm{(D)}\  \frac 7{12}
 +
\qquad\mathrm{(E)}\ \frac 23<math>
  
−
[[2002 AMC 12B/Problem 16|Solution]]
+
== Problem ==
 +
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 17 ==
+
</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 12B/Problem 17|Solution]]
+
== Problem ==
 +
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>?
  
−
== Problem 18 ==
+
</math>\mathrm{(A)}\
 +
\qquad\mathrm{(B)}\
 +
\qquad\mathrm{(C)}\
 +
\qquad\mathrm{(D)}\
 +
\qquad\mathrm{(E)}\ <math>
  
−
[[2002 AMC 12B/Problem 18|Solution]]
+
== Problem ==
 +
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
  
−
== Problem 19 ==
+
</math>\mathrm{(A)}\ 672
 +
\qquad\mathrm{(B)}\ 688
 +
\qquad\mathrm{(C)}\ 704
 +
\qquad\mathrm{(D)}\ 720
 +
\qquad\mathrm{(E)}\ 750<math>
  
−
[[2002 AMC 12B/Problem 19|Solution]]
+
== Problem ==
 +
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>.
  
−
== Problem 20 ==
+
</math>\mathrm{(A)}\ 24
 +
\qquad\mathrm{(B)}\ 26
 +
\qquad\mathrm{(C)}\ 28
 +
\qquad\mathrm{(D)}\ 30
 +
\qquad\mathrm{(E)}\ 32<math>
  
−
[[2002 AMC 12B/Problem 20|Solution]]
+
== Problem ==
 +
For all positive integers </math>n<math> less than </math>2002<math>, let
  
−
== Problem 21 ==
+
<cmath>\begin{eqnarray*}
 +
a_n =\left\{
 +
\begin{array}{lr}
 +
11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\
 +
13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\
 +
14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\
 +
0, & \text{otherwise}.
 +
\end{array}
 +
\right.
 +
\end{eqnarray*}</cmath>
  
−
[[2002 AMC 12B/Problem 21|Solution]]
+
Calculate </math>\sum_{n=1}^{2001} a_n<math>.
  
−
== Problem 22 ==
+
</math>\mathrm{(A)}\ 448
 +
\qquad\mathrm{(B)}\ 486
 +
\qquad\mathrm{(C)}\ 1560
 +
\qquad\mathrm{(D)}\ 2001
 +
\qquad\mathrm{(E)}\ 2002<math>
  
−
[[2002 AMC 12B/Problem 22|Solution]]
+
== Problem ==
 +
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
  
−
== Problem 23 ==
+
</math>\mathrm{(A)}\ -2
 +
\qquad\mathrm{(B)}\ -1
 +
\qquad\mathrm{(C)}\ \frac{1}{2002}
 +
\qquad\mathrm{(D)}\ \frac{1}{1001}
 +
\qquad\mathrm{(E)}\ \frac 12<math>
  
−
[[2002 AMC 12B/Problem 23|Solution]]
+
== Problem ==
 +
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>?
  
−
== Problem 24 ==
+
</math>\mathrm{(A)}\ \frac{1+\sqrt{2}}{2}
 +
\qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2
 +
\qquad\mathrm{(C)}\ \sqrt{2}
 +
\qquad\mathrm{(D)}\ \frac 32
 +
\qquad\mathrm{(E)}\ \sqrt{3}<math>
  
−
[[2002 AMC 12B/Problem 24|Solution]]
+
== Problem ==
 +
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>.
  
−
== Problem 25 ==
+
</math>\mathrm{(A)}\ 4\sqrt{2002}
 +
\qquad\mathrm{(B)}\ 2\sqrt{8465}
 +
\qquad\mathrm{(C)}\ 2(48+\sqrt{2002})
 +
\qquad\mathrm{(D)}\ 2\sqrt{8633}
 +
\qquad\mathrm{(E)}\ 4(36 + \sqrt{113})<math>
  
−
[[2002 AMC 12B/Problem 25|Solution]]
+
== Problem ==
 +
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>
 +
The area of </math>R<math> is closest to
 +
</math>\mathrm{(A)}\ 21
 +
\qquad\mathrm{(B)}\ 22
 +
\qquad\mathrm{(C)}\ 23
 +
\qquad\mathrm{(D)}\ 24
 +
\qquad\mathrm{(E)}\ 25$
  
 
== See also ==
 
== See also ==
 
* [[AMC 12]]
 
* [[AMC 12]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[AMC 12 Problems and Solutions]]
−
* [[2002 AMC 12B]]
+
* [[2002 AMC 12A]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]

Revision as of 17:54, 18 January 2008

Problem

The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ does not contain the digit

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 4 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 8$

Problem

What is the value of \[(3x - 2)(4x + 1) - (3x - 2)4x + 1\]

when $x=4$? $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 11 \qquad\mathrm{(E)}\ 12$

Problem

For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?

$\mathrm{(A)}\ \text{none} \qquad\mathrm{(B)}\ \text{one} \qquad\mathrm{(C)}\ \text{two} \qquad\mathrm{(D)}\ \text{more\ than\ two,\ but\ finitely\ many} \qquad\mathrm{(E)}\ \text{infinitely\ many}$

Problem

Let $n$ be a positive integer such that $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ is an integer. Which of the following statements is not true:

$\mathrm{(A)}\ 2\ \text{divides\ }n \qquad\mathrm{(B)}\ 3\ \text{divides\ }n \qquad\mathrm{(C)}\ 6\ \text{divides\ }n \qquad\mathrm{(D)}\ 7\ \text{divides\ }n \qquad\mathrm{(E)}\ n > 84$

Problem

Let $v, w, x, y,$ and $z$ be the degree measures of the five angles of a pentagon. Suppose that $$ (Error compiling LaTeX. Unknown error_msg)v < w < x < y < z$and$v, w, x, y, $and$z$form an arithmetic sequence. Find the value of$x$.$\mathrm{(A)}\ 72 \qquad\mathrm{(B)}\ 84 \qquad\mathrm{(C)}\ 90 \qquad\mathrm{(D)}\ 108 \qquad\mathrm{(E)}\ 120$== Problem == Suppose that$a$and$b$are nonzero real numbers, and that the equation$x^2 + ax + b = 0$has solutions$a$and$b$. Then the pair$(a,b)$is$\mathrm{(A)}\ (-2,1) \qquad\mathrm{(B)}\ (-1,2) \qquad\mathrm{(C)}\ (1,-2) \qquad\mathrm{(D)}\ (2,-1) \qquad\mathrm{(E)}\ (4,4)$== Problem == The product of three consecutive positive integers is$8$times their sum. What is the sum of their squares?$\mathrm{(A)}\ 50 \qquad\mathrm{(B)}\ 77 \qquad\mathrm{(C)}\ 110 \qquad\mathrm{(D)}\ 149 \qquad\mathrm{(E)}\ 194$== Problem == Suppose July of year$N$has five Mondays. Which of the following must occur five times in August of year$N$? (Note: Both months have 31 days.)$\mathrm{(A)}\ \text{Monday} \qquad\mathrm{(B)}\ \text{Tuesday} \qquad\mathrm{(C)}\ \text{Wednesday} \qquad\mathrm{(D)}\ \text{Thursday} \qquad\mathrm{(E)}\ \text{Friday}$== Problem == If$a,b,c,d$are positive real numbers such that$a,b,c,d$form an increasing arithmetic sequence and$a,b,d$form a geometric sequence, then$\frac ad$is$\mathrm{(A)}\ \frac 1{12} \qquad\mathrm{(B)}\ \frac 16 \qquad\mathrm{(C)}\ \frac 14 \qquad\mathrm{(D)}\ \frac 13 \qquad\mathrm{(E)}\ \frac 12$== Problem == How many different integers can be expressed as the sum of three distinct members of the set$\{1,4,7,10,13,16,19\}$?$\mathrm{(A)}\ 13 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 24 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 35$== Problem == The positive integers$A, B, A-B, $and$A+B$are all prime numbers. The sum of these four primes is$\mathrm{(A)}\ \mathrm{even} \qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3 \qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5 \qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7 \qquad\mathrm{(E)}\ \mathrm{prime}$== Problem == For how many integers$n$is$\dfrac n{20-n}$the square of an integer?$\mathrm{(A)}\ 1 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 3 \qquad\mathrm{(D)}\ 4 \qquad\mathrm{(E)}\ 10$== Problem == The sum of$18$consecutive positive integers is a perfect square. The smallest possible value of this sum is$\mathrm{(A)}\ 169 \qquad\mathrm{(B)}\ 225 \qquad\mathrm{(C)}\ 289 \qquad\mathrm{(D)}\ 361 \qquad\mathrm{(E)}\ 441$== Problem == Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?$\mathrm{(A)}\ 8 \qquad\mathrm{(B)}\ 9 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 16$== Problem == How many four-digit numbers$N$have the property that the three-digit number obtained by removing the leftmost digit is one night of$N$?$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7 \qquad\mathrm{(E)}\ 8$== Problem == Juan rolls a fair regular octahedral die marked with the numbers$1$through$8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?$\mathrm{(A)}\ \frac1{12} \qquad\mathrm{(B)}\ \frac 13 \qquad\mathrm{(C)}\ \frac 12 \qquad\mathrm{(D)}\ \frac 7{12} \qquad\mathrm{(E)}\ \frac 23$== Problem == 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?$\mathrm{(A)}\ \text{Andy} \qquad\mathrm{(B)}\ \text{Beth} \qquad\mathrm{(C)}\ \text{Carlos} \qquad\mathrm{(D)}\ \text{Andy\ and \ Carlos\ tie\ for\ first.} \qquad\mathrm{(E)}\ \text{All\ three\ tie.}$== Problem == A point$P$is randomly selected from the [[rectangle|rectangular]] region with vertices$(0,0),(2,0),(2,1),(0,1)$. What is the [[probability]] that$P$is closer to the origin than it is to the point$(3,1)$?$\mathrm{(A)}\ \qquad\mathrm{(B)}\ \qquad\mathrm{(C)}\ \qquad\mathrm{(D)}\ \qquad\mathrm{(E)}\ $== Problem == If$a,b,$and$c$are positive real numbers such that$a(b+c) = 152, b(c+a) = 162,$and$c(a+b) = 170$, then$abc$is$\mathrm{(A)}\ 672 \qquad\mathrm{(B)}\ 688 \qquad\mathrm{(C)}\ 704 \qquad\mathrm{(D)}\ 720 \qquad\mathrm{(E)}\ 750$== Problem == Let$\triangle XOY$be a right-angled triangle with$m\angle XOY = 90^{\circ}$. Let$M$and$N$be the midpoints of legs$OX$and$OY$, respectively. Given that$XN = 19$and$YM = 22$, find$XY$.$\mathrm{(A)}\ 24 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 28 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 32$== Problem == For all positive integers$n$less than$2002$, let

<cmath>\begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\ 13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\ 14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\ 0, & \text{otherwise}. \end{array} \right. \end{eqnarray*}</cmath>

Calculate$ (Error compiling LaTeX. Unknown error_msg)\sum_{n=1}^{2001} a_n$.$\mathrm{(A)}\ 448 \qquad\mathrm{(B)}\ 486 \qquad\mathrm{(C)}\ 1560 \qquad\mathrm{(D)}\ 2001 \qquad\mathrm{(E)}\ 2002$== Problem == For all integers$n$greater than$1$, define$a_n = \frac{1}{\log_n 2002}$. Let$b = a_2 + a_3 + a_4 + a_5$and$c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Then$b- c$equals$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ \frac{1}{2002} \qquad\mathrm{(D)}\ \frac{1}{1001} \qquad\mathrm{(E)}\ \frac 12$== Problem == In$\triangle ABC$, we have$AB = 1$and$AC = 2$. Side$\overline{BC}$and the median from$A$to$\overline{BC}$have the same length. What is$BC$?$\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 \qquad\mathrm{(C)}\ \sqrt{2} \qquad\mathrm{(D)}\ \frac 32 \qquad\mathrm{(E)}\ \sqrt{3}$== Problem == A convex quadrilateral$ABCD$with area$2002$contains a point$P$in its interior such that$PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of$ABCD$.$\mathrm{(A)}\ 4\sqrt{2002} \qquad\mathrm{(B)}\ 2\sqrt{8465} \qquad\mathrm{(C)}\ 2(48+\sqrt{2002}) \qquad\mathrm{(D)}\ 2\sqrt{8633} \qquad\mathrm{(E)}\ 4(36 + \sqrt{113})$== Problem == Let$f(x) = x^2 + 6x + 1$, and let$R$denote the set of points$(x,y)$in the coordinate plane such that <cmath>f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0</cmath> The area of$R$is closest to$\mathrm{(A)}\ 21 \qquad\mathrm{(B)}\ 22 \qquad\mathrm{(C)}\ 23 \qquad\mathrm{(D)}\ 24 \qquad\mathrm{(E)}\ 25$

See also

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