Difference between revisions of "2002 AMC 12B Problems"

Latest revision as of 03:20, 9 October 2022

2002 AMC 12B (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. 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). Figures are not necessarily drawn to scale. 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

@@ Line 1: / Line 1: @@
+{{AMC12 Problems|year=2002|ab=B}}
 == 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
@@ Line 15: / Line 16: @@
 when <math>x=4</math>?
 <math>\mathrm{(A)}\ 0
 \qquad\mathrm{(B)}\ 1
@@ Line 35: / Line 37: @@
 == 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:
 <math>\mathrm{(A)}\ 2\ \text{divides\ }n
@@ Line 41: / Line 43: @@
 \qquad\mathrm{(C)}\ 6\ \text{divides\ }n
 \qquad\mathrm{(D)}\ 7\ \text{divides\ }n
-\qquad\mathrm{(E)}\ n > 84</math>
+\qquad\mathrm{(E)}\ {n > 84}</math>
 [[2002 AMC 12B Problems/Problem 4|Solution]]
@@ Line 103: / Line 105: @@
 == 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
@@ Line 156: / Line 159: @@
 == 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 ninth of <math>N</math>?
 <math>\mathrm{(A)}\ 4
@@ Line 178: / Line 181: @@
 == 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 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}
@@ Line 191: / Line 194: @@
 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)}\
-\qquad\mathrm{(B)}\
+<math>\mathrm{(A)}\ \frac 12
-\qquad\mathrm{(C)}\
+\qquad\mathrm{(B)}\ \frac 23
-\qquad\mathrm{(D)}\
+\qquad\mathrm{(C)}\ \frac 34
-\qquad\mathrm{(E)}\ </math>
+\qquad\mathrm{(D)}\ \frac 45
+\qquad\mathrm{(E)}\ 1</math>
 [[2002 AMC 12B Problems/Problem 18|Solution]]
@@ Line 211: / Line 215: @@
 == 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>\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
@@ Line 282: / Line 286: @@
 <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
@@ Line 289: / Line 294: @@
 [[2002 AMC 12B Problems/Problem 25|Solution]]
 == See also ==
+{{AMC12 box|year=2002|ab=B|before=[[2002 AMC 12A Problems]]|after=[[2003 AMC 12A Problems]]}}
 * [[AMC 12]]
 * [[AMC 12 Problems and Solutions]]
 * [[2002 AMC 12A]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

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

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Difference between revisions of "2002 AMC 12B Problems"

Latest revision as of 03:20, 9 October 2022

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also