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Difference between revisions of "2002 AMC 12B Problems"

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{{AMC12 Problems|year=2002|ab=B}}
 
== 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
  
−
[[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>
 +
 
 +
[[2002 AMC 12B Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
What is the value of
 +
<cmath>(3x - 2)(4x + 1) - (3x - 2)4x + 1</cmath>
 +
 +
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>
  
−
[[2002 AMC 12B Problem 2|Solution]]
+
[[2002 AMC 12B Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a prime number?
 +
 +
<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>
  
−
[[2002 AMC 12B Problem 3|Solution]]
+
[[2002 AMC 12B Problems/Problem 3|Solution]]
  
 
== 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:
 +
 +
<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>
  
−
[[2002 AMC 12B Problem 4|Solution]]
+
[[2002 AMC 12B Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== 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>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{(C)}\ 90
 +
\qquad\mathrm{(D)}\ 108
 +
\qquad\mathrm{(E)}\ 120</math>
  
−
[[2002 AMC 12B Problem 5|Solution]]
+
[[2002 AMC 12B Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== 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
 +
 +
<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>
 +
 +
[[2002 AMC 12B Problems/Problem 6|Solution]]
  
−
[[2002 AMC 12B Problem 6|Solution]]
 
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
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{(C)}\ 110
 +
\qquad\mathrm{(D)}\ 149
 +
\qquad\mathrm{(E)}\ 194</math>
  
−
[[2002 AMC 12B Problem 7|Solution]]
+
[[2002 AMC 12B Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== 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.)
 +
 +
<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>
  
−
[[2002 AMC 12B Problem 8|Solution]]
+
[[2002 AMC 12B Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== 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
 +
 +
<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>
  
−
[[2002 AMC 12B Problem 9|Solution]]
+
[[2002 AMC 12B Problems/Problem 9|Solution]]
  
 
== 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>?
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== 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
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
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{(C)}\ 3
 +
\qquad\mathrm{(D)}\ 4
 +
\qquad\mathrm{(E)}\ 10</math>
  
−
[[2002 AMC 12B Problem 12|Solution]]
+
[[2002 AMC 12B Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
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{(C)}\ 289
 +
\qquad\mathrm{(D)}\ 361
 +
\qquad\mathrm{(E)}\ 441</math>
  
−
[[2002 AMC 12B Problem 13|Solution]]
+
[[2002 AMC 12B Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== 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?
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 14|Solution]]
  
 
== 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 ninth of <math>N</math>?
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== 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?
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== 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?
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
A point <math>P</math> is randomly selected from the 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)}\ \frac 12
 +
\qquad\mathrm{(B)}\ \frac 23
 +
\qquad\mathrm{(C)}\ \frac 34
 +
\qquad\mathrm{(D)}\ \frac 45
 +
\qquad\mathrm{(E)}\ 1</math>
  
−
[[2002 AMC 12B Problem 18|Solution]]
+
[[2002 AMC 12B Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== 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
 +
 +
<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]]
+
[[2002 AMC 12B Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
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 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{(C)}\ 28
 +
\qquad\mathrm{(D)}\ 30
 +
\qquad\mathrm{(E)}\ 32</math>
  
−
[[2002 AMC 12B Problem 20|Solution]]
+
[[2002 AMC 12B Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
For all positive integers <math>n</math> less than <math>2002</math>, 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 <math>\sum_{n=1}^{2001} a_n</math>.
 +
 +
<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 21|Solution]]
+
[[2002 AMC 12B Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== 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
 +
 +
<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 22|Solution]]
+
[[2002 AMC 12B Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== 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>?
 +
 +
<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 23|Solution]]
+
[[2002 AMC 12B Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== 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>.
 +
 +
<math>\mathrm{(A)}\ 4\sqrt{2002}
 +
\qquad\mathrm{(B)}\ 2\sqrt{8465}
 +
\qquad\mathrm{(C)}\ 2(48+ </math> <math>\sqrt{2002})
 +
\qquad\mathrm{(D)}\ 2\sqrt{8633}
 +
\qquad\mathrm{(E)}\ 4(36 + \sqrt{113})</math>
  
−
[[2002 AMC 12B Problem 24|Solution]]
+
[[2002 AMC 12B Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== 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
 +
<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</math>
  
−
[[2002 AMC 12B 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 12B]]
+
* [[2002 AMC 12A]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 03:20, 9 October 2022

2002 AMC 12B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

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

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

Solution

Problem 2

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

when $x=4$?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 11 \qquad\mathrm{(E)}\ 12$

Solution

Problem 3

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

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

Solution

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:

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

Solution

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 $x$.

$\mathrm{(A)}\ 72 \qquad\mathrm{(B)}\ 84 \qquad\mathrm{(C)}\ 90 \qquad\mathrm{(D)}\ 108 \qquad\mathrm{(E)}\ 120$

Solution

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

$\mathrm{(A)}\ (-2,1) \qquad\mathrm{(B)}\ (-1,2) \qquad\mathrm{(C)}\ (1,-2) \qquad\mathrm{(D)}\ (2,-1) \qquad\mathrm{(E)}\ (4,4)$

Solution


Problem 7

The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?

$\mathrm{(A)}\ 50 \qquad\mathrm{(B)}\ 77 \qquad\mathrm{(C)}\ 110 \qquad\mathrm{(D)}\ 149 \qquad\mathrm{(E)}\ 194$

Solution

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.)

$\mathrm{(A)}\ \text{Monday} \qquad\mathrm{(B)}\ \text{Tuesday} \qquad\mathrm{(C)}\ \text{Wednesday} \qquad\mathrm{(D)}\ \text{Thursday} \qquad\mathrm{(E)}\ \text{Friday}$

Solution

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

$\mathrm{(A)}\ \frac 1{12} \qquad\mathrm{(B)}\ \frac 16 \qquad\mathrm{(C)}\ \frac 14 \qquad\mathrm{(D)}\ \frac 13 \qquad\mathrm{(E)}\ \frac 12$

Solution

Problem 10

How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$?

$\mathrm{(A)}\ 13 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 24 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 35$

Solution

Problem 11

The positive integers $A, B, A-B,$ and $A+B$ are all prime numbers. The sum of these four primes is

$\mathrm{(A)}\ \mathrm{even} \qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3 \qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5 \qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7 \qquad\mathrm{(E)}\ \mathrm{prime}$

Solution

Problem 12

For how many integers $n$ is $\dfrac n{20-n}$ the square of an integer?

$\mathrm{(A)}\ 1 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 3 \qquad\mathrm{(D)}\ 4 \qquad\mathrm{(E)}\ 10$

Solution

Problem 13

The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is

$\mathrm{(A)}\ 169 \qquad\mathrm{(B)}\ 225 \qquad\mathrm{(C)}\ 289 \qquad\mathrm{(D)}\ 361 \qquad\mathrm{(E)}\ 441$

Solution

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?

$\mathrm{(A)}\ 8 \qquad\mathrm{(B)}\ 9 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 16$

Solution

Problem 15

How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$?

$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7 \qquad\mathrm{(E)}\ 8$

Solution

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?

$\mathrm{(A)}\ \frac1{12} \qquad\mathrm{(B)}\ \frac 13 \qquad\mathrm{(C)}\ \frac 12 \qquad\mathrm{(D)}\ \frac 7{12} \qquad\mathrm{(E)}\ \frac 23$

Solution

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?

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

Solution

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 $(3,1)$?


$\mathrm{(A)}\ \frac 12 \qquad\mathrm{(B)}\ \frac 23 \qquad\mathrm{(C)}\ \frac 34 \qquad\mathrm{(D)}\ \frac 45 \qquad\mathrm{(E)}\ 1$

Solution

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

$\mathrm{(A)}\ 672 \qquad\mathrm{(B)}\ 688 \qquad\mathrm{(C)}\ 704 \qquad\mathrm{(D)}\ 720 \qquad\mathrm{(E)}\ 750$

Solution

Problem 20

Let $\triangle XOY$ be a right-angled triangle with $\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find $XY$.

$\mathrm{(A)}\ 24 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 28 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 32$

Solution

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

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

Solution

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

$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ \frac{1}{2002} \qquad\mathrm{(D)}\ \frac{1}{1001} \qquad\mathrm{(E)}\ \frac 12$

Solution

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 $BC$?

$\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 \qquad\mathrm{(C)}\ \sqrt{2} \qquad\mathrm{(D)}\ \frac 32 \qquad\mathrm{(E)}\ \sqrt{3}$

Solution

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 $ABCD$.

$\mathrm{(A)}\ 4\sqrt{2002} \qquad\mathrm{(B)}\ 2\sqrt{8465} \qquad\mathrm{(C)}\ 2(48+$ $\sqrt{2002}) \qquad\mathrm{(D)}\ 2\sqrt{8633} \qquad\mathrm{(E)}\ 4(36 + \sqrt{113})$

Solution

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

$\mathrm{(A)}\ 21 \qquad\mathrm{(B)}\ 22 \qquad\mathrm{(C)}\ 23 \qquad\mathrm{(D)}\ 24 \qquad\mathrm{(E)}\ 25$

Solution

See also

2002 AMC 12B (Problems • Answer Key • Resources)
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2002 AMC 12A Problems 		Followed by
2003 AMC 12A Problems
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All AMC 12 Problems and Solutions

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