Difference between revisions of "2002 AMC 12B Problems"

Revision as of 19:04, 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

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

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 night 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 thid as afst as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?

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

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 $(3,1)$ ?

$\mathrm{(A)}\ \qquad\mathrm{(B)}\ \qquad\mathrm{(C)}\ \qquad\mathrm{(D)}\ \qquad\mathrm{(E)}$

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

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

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 7: / Line 7: @@
 \qquad\mathrm{(D)}\ 6
 \qquad\mathrm{(E)}\ 8</math>
+[[2002 AMC 12B Problems/Problem 1|Solution]]
 == Problem 2 ==
@@ Line 18: / Line 20: @@
 \qquad\mathrm{(D)}\ 11
 \qquad\mathrm{(E)}\ 12</math>
+[[2002 AMC 12B Problems/Problem 2|Solution]]
 == Problem 3 ==
@@ Line 27: / Line 31: @@
 \qquad\mathrm{(D)}\ \text{more\ than\ two,\ but\ finitely\ many}
 \qquad\mathrm{(E)}\ \text{infinitely\ many}</math>
+[[2002 AMC 12B Problems/Problem 3|Solution]]
 == Problem 4 ==
@@ Line 36: / Line 42: @@
 \qquad\mathrm{(D)}\ 7\ \text{divides\ }n
 \qquad\mathrm{(E)}\ n > 84</math>
+[[2002 AMC 12B Problems/Problem 4|Solution]]
 == Problem 5 ==
@@ Line 45: / Line 53: @@
 \qquad\mathrm{(D)}\ 108
 \qquad\mathrm{(E)}\ 120</math>
+[[2002 AMC 12B Problems/Problem 5|Solution]]
 == Problem 6 ==
@@ Line 54: / Line 64: @@
 \qquad\mathrm{(D)}\ (2,-1)
 \qquad\mathrm{(E)}\ (4,4)</math>
+[[2002 AMC 12B Problems/Problem 6|Solution]]
 == Problem 7 ==
@@ Line 63: / Line 76: @@
 \qquad\mathrm{(D)}\ 149
 \qquad\mathrm{(E)}\ 194</math>
+[[2002 AMC 12B Problems/Problem 7|Solution]]
 == Problem 8 ==
@@ Line 72: / Line 87: @@
 \qquad\mathrm{(D)}\ \text{Thursday}
 \qquad\mathrm{(E)}\ \text{Friday}</math>
+[[2002 AMC 12B Problems/Problem 8|Solution]]
 == Problem 9 ==
@@ Line 81: / Line 98: @@
 \qquad\mathrm{(D)}\ \frac 13
 \qquad\mathrm{(E)}\ \frac 12</math>
+[[2002 AMC 12B Problems/Problem 9|Solution]]
 == Problem 10 ==
@@ Line 89: / Line 108: @@
 \qquad\mathrm{(D)}\ 30
 \qquad\mathrm{(E)}\ 35</math>
+[[2002 AMC 12B Problems/Problem 10|Solution]]
 == Problem 11 ==
@@ Line 98: / Line 119: @@
 \qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7
 \qquad\mathrm{(E)}\ \mathrm{prime}</math>
+[[2002 AMC 12B Problems/Problem 11|Solution]]
 == Problem 12 ==
@@ Line 107: / Line 130: @@
 \qquad\mathrm{(D)}\ 4
 \qquad\mathrm{(E)}\ 10</math>
+[[2002 AMC 12B Problems/Problem 12|Solution]]
 == Problem 13 ==
@@ Line 116: / Line 141: @@
 \qquad\mathrm{(D)}\ 361
 \qquad\mathrm{(E)}\ 441</math>
+[[2002 AMC 12B Problems/Problem 13|Solution]]
 == Problem 14 ==
@@ Line 125: / Line 152: @@
 \qquad\mathrm{(D)}\ 12
 \qquad\mathrm{(E)}\ 16</math>
+[[2002 AMC 12B Problems/Problem 14|Solution]]
 == Problem 15 ==
@@ Line 134: / Line 163: @@
 \qquad\mathrm{(D)}\ 7
 \qquad\mathrm{(E)}\ 8</math>
+[[2002 AMC 12B Problems/Problem 15|Solution]]
 == Problem 16 ==
@@ Line 143: / Line 174: @@
 \qquad\mathrm{(D)}\  \frac 7{12}
 \qquad\mathrm{(E)}\ \frac 23</math>
+[[2002 AMC 12B Problems/Problem 16|Solution]]
 == Problem 17 ==
@@ Line 152: / Line 185: @@
 \qquad\mathrm{(D)}\ \text{Andy\ and \ Carlos\ tie\ for\ first.}
 \qquad\mathrm{(E)}\ \text{All\ three\ tie.}</math>
+[[2002 AMC 12B Problems/Problem 17|Solution]]
 == 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 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)}\
@@ Line 161: / Line 196: @@
 \qquad\mathrm{(D)}\
 \qquad\mathrm{(E)}\ </math>
+[[2002 AMC 12B Problems/Problem 18|Solution]]
 == Problem 19 ==
@@ Line 170: / Line 207: @@
 \qquad\mathrm{(D)}\ 720
 \qquad\mathrm{(E)}\ 750</math>
+[[2002 AMC 12B Problems/Problem 19|Solution]]
 == Problem 20 ==
@@ Line 179: / Line 218: @@
 \qquad\mathrm{(D)}\ 30
 \qquad\mathrm{(E)}\ 32</math>
+[[2002 AMC 12B Problems/Problem 20|Solution]]
 == Problem 21 ==
@@ Line 201: / Line 242: @@
 \qquad\mathrm{(D)}\ 2001
 \qquad\mathrm{(E)}\ 2002</math>
+[[2002 AMC 12B Problems/Problem 21|Solution]]
 == Problem 22 ==
@@ Line 210: / Line 253: @@
 \qquad\mathrm{(D)}\ \frac{1}{1001}
 \qquad\mathrm{(E)}\ \frac 12</math>
+[[2002 AMC 12B Problems/Problem 22|Solution]]
 == Problem 23 ==
@@ Line 219: / Line 264: @@
 \qquad\mathrm{(D)}\ \frac 32
 \qquad\mathrm{(E)}\ \sqrt{3}</math>
+[[2002 AMC 12B Problems/Problem 23|Solution]]
 == Problem 24 ==
@@ Line 225: / Line 272: @@
 <math>\mathrm{(A)}\ 4\sqrt{2002}
 \qquad\mathrm{(B)}\ 2\sqrt{8465}
-\qquad\mathrm{(C)}\ 2(48+\sqrt{2002})
+\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 Problems/Problem 24|Solution]]
 == Problem 25 ==
@@ Line 239: / Line 288: @@
 \qquad\mathrm{(E)}\ 25</math>
+[[2002 AMC 12B Problems/Problem 25|Solution]]
 == See also ==
 * [[AMC 12]]

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Difference between revisions of "2002 AMC 12B Problems"

Revision as of 19:04, 18 January 2008

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