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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 <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>.
+
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
+
<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 </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
+
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)
+
<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 </math>8<math> times their sum. What is the sum of their squares?
+
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
+
<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 </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.)
+
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}
+
<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 </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
+
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}
+
<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 </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
 
\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 </math>A, B, A-B, <math> and </math>A+B<math> are all prime numbers. The sum of these four primes is
+
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}
+
<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 </math>n<math> is </math>\dfrac n{20-n}<math> the square of an integer?
+
For how many integers <math>n</math> is <math>\dfrac n{20-n}</math> the square of an integer?
  
</math>\mathrm{(A)}\ 1
+
<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 </math>18<math> consecutive positive integers is a perfect square. The smallest possible value of this sum is
+
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
+
<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
+
<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 </math>N<math> have the property that the three-digit number obtained by removing the leftmost digit is one night of </math>N<math>?
+
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
+
<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 </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?
+
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}
+
<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}
+
<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 </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>?
+
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)}\  
+
<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 </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
+
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
+
<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 </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>.
+
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
+
<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 </math>n<math> less than </math>2002<math>, let  
+
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 </math>\sum_{n=1}^{2001} a_n<math>.
+
Calculate <math>\sum_{n=1}^{2001} a_n</math>.
  
</math>\mathrm{(A)}\ 448
+
<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 </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
+
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
+
<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 </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>?
+
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}
+
<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 </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>.
+
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}
+
<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 </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  
+
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 </math>R<math> is closest to  
+
The area of <math>R</math> is closest to  
</math>\mathrm{(A)}\ 21
+
<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

Problem 1

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 2

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 3

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 4

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 5

Let $v, w, x, y,$ and $z$ be the degree measures of the five angles of a pentagon. Suppose that $v < w < x < y < z$ 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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 16

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 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?

$\mathrm{(A)}\ \text{Andy} \qquad\mathrm{(B)}\ \text{Beth} \qquad\mathrm{(C)}\ \text{Carlos} \qquad\mathrm{(D)}\ \text{Andy\ and \ Carlos\ tie\ for\ first.} \qquad\mathrm{(E)}\ \text{All\ three\ tie.}$

Problem 18

A point $P$ is randomly selected from the 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 19

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 20

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 21

For all positive integers $n$ less than $2002$, let

\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*}

Calculate $\sum_{n=1}^{2001} a_n$.

$\mathrm{(A)}\ 448 \qquad\mathrm{(B)}\ 486 \qquad\mathrm{(C)}\ 1560 \qquad\mathrm{(D)}\ 2001 \qquad\mathrm{(E)}\ 2002$

Problem 22

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 23

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 24

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 25

Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that \[f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0\] 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