Difference between revisions of "2002 AMC 10B Problems"

Latest revision as of 12:34, 23 July 2024

2002 AMC 10B (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 SAT 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.
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Problem 1

The ratio $\frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is:

$\mathrm{(A) \ } \frac{1}{6}\qquad \mathrm{(B) \ } \frac{1}{3}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{2}{3}\qquad \mathrm{(E) \ } \frac{3}{2}$

Solution

Problem 2

For the nonzero numbers $a, b,$ and $c,$ define $(a,b,c)=\frac{abc}{a+b+c}$ Find $(2,4,6)$ .

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24$

Solution

Problem 3

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 4

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 5

Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.

$[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r1=3; real r2=2; real r3=5; pair A=(-2,0), B=(3,0), C=(0,0); pair X=(1,0), Y=(5,0); path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r3); fill(circleC,gray); fill(circleA,white); fill(circleB,white); draw(circleA); draw(circleB); draw(circleC); draw(A--X); draw(B--Y); pair[] ps={A,B}; dot(ps); label("$3$",midpoint(A--X),N); label("$2$",midpoint(B--Y),N); [/asy]$

$\mathrm{(A) \ } 3\pi\qquad \mathrm{(B) \ } 4\pi\qquad \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi$

Solution

Problem 6

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 7

Let $n$ be a positive integer such that $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}$ 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 8

Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$ ? (Note: Both months have $31$ days.)

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

Solution

Problem 9

Using the letters $A$ , $M$ , $O$ , $S$ , and $U$ , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $USAMO$ occupies position

$\mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116$

Solution

Problem 10

Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$ . What is the pair $(a,b)$ ?

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

Solution

Problem 11

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 12

For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$ ?

$\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution

Problem 13

Find the value(s) of $x$ such that $8xy - 12y + 2x - 3 = 0$ is true for all values of $y$ .

$\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14$

Solution

Problem 14

The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is

$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35$

Solution

Problem 15

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

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

Solution

Problem 16

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

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

Solution

Problem 17

A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\triangle ADG$ .

$\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2$

Solution

Problem 18

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

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

Solution

Problem 19

Suppose that $\{a_n\}$ is an arithmetic sequence with $a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.$ What is the value of $a_2 - a_1 ?$

$\mathrm{(A) \ } 0.0001\qquad \mathrm{(B) \ } 0.001\qquad \mathrm{(C) \ } 0.01\qquad \mathrm{(D) \ } 0.1\qquad \mathrm{(E) \ } 1$

Solution

Problem 20

Let $a, b,$ and $c$ be real numbers such that $a-7b+8c=4$ and $8a+4b-c=7.$ Then $a^2-b^2+c^2$ is

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8$

Solution

Problem 21

Andy's lawn has twice as much area as Beth's lawn and three times as much 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 22

Let $\triangle{XOY}$ be a right-angled triangle with $\angle{XOY}=90^\circ$ . Let $M$ and $N$ be the midpoints of the legs $OX$ and $OY$ , respectively. Given $XN=19$ 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 23

Let $\{a_k\}$ be a sequence of integers such that $a_1=1$ and $a_{m+n}=a_m+a_n+mn,$ for all positive integers $m$ and $n.$ Then $a_{12}$ is

$\mathrm{(A) \ } 45\qquad \mathrm{(B) \ } 56\qquad \mathrm{(C) \ } 67\qquad \mathrm{(D) \ } 78\qquad \mathrm{(E) \ } 89$

Solution

Problem 24

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom?

$\mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15$

Solution

Problem 25

When $15$ is appended to a list of integers, the mean is increased by $2$ . When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$ . How many integers were in the original list?

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

Solution

@@ Line 1: / Line 1: @@
+{{AMC10 Problems|year=2002|ab=B}}
 ==Problem 1==
 The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is:
-<math> \mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2 </math>
+<math> \mathrm{(A) \ } \frac{1}{6}\qquad \mathrm{(B) \ } \frac{1}{3}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{2}{3}\qquad \mathrm{(E) \ } \frac{3}{2} </math>
 [[2002 AMC 10B Problems/Problem 1|Solution]]
 == Problem 2 ==
-For the nonzero numbers a, b, and c, define
+For the nonzero numbers <math>a, b,</math> and <math>c,</math> define
+<cmath>(a,b,c)=\frac{abc}{a+b+c}</cmath>
-<math>(a,b,c)=\frac{abc}{a+b+c}</math>
 Find <math>(2,4,6)</math>.
@@ Line 18: / Line 17: @@
 == Problem 3 ==
-The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-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
 <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math>
@@ Line 29: / Line 28: @@
 <math>(3x-2)(4x+1)-(3x-2)4x+1</math>
 when <math>x=4</math>?
@@ Line 37: / Line 38: @@
 == Problem 5 ==
+Circles of radius <math>2</math> and <math>3</math> are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
+<center><asy>
+unitsize(5mm);
+defaultpen(linewidth(.8pt)+fontsize(10pt));
+dotfactor=4;
+real r1=3; real r2=2; real r3=5;
+pair A=(-2,0), B=(3,0), C=(0,0);
+pair X=(1,0), Y=(5,0);
+path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r3);
+fill(circleC,gray);
+fill(circleA,white);
+fill(circleB,white);
+draw(circleA); draw(circleB); draw(circleC);
+draw(A--X); draw(B--Y);
+pair[] ps={A,B}; dot(ps);
+label("$3$",midpoint(A--X),N);
+label("$2$",midpoint(B--Y),N);
+</asy></center>
+<math> \mathrm{(A) \ } 3\pi\qquad \mathrm{(B) \ } 4\pi\qquad \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math>
 [[2002 AMC 10B Problems/Problem 5|Solution]]
@@ Line 42: / Line 68: @@
 == Problem 6 ==
-For how many positive integers n 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?
 <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>
@@ Line 58: / Line 84: @@
 == Problem 8 ==
-Suppose July of year <math>N</math> has five Mondays. Which of the following must occurs five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.)
+Suppose July of year <math>N</math> has five Mondays. Which of the following must occur five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.)
 <math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}</math>
@@ Line 74: / Line 100: @@
 == Problem 10 ==
-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 positive 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>. What is the pair <math>(a,b)</math>?
 <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>
@@ Line 82: / Line 108: @@
 == Problem 11 ==
-The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of the 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\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math>
@@ Line 96: / Line 122: @@
 [[2002 AMC 10B Problems/Problem 12|Solution]]
-== Problem 13 ==
+== Problem 13==
 Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.
-<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac34 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math>
+<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math>
@@ Line 139: / Line 165: @@
 == Problem 18 ==
+Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
+<math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16</math>
 [[2002 AMC 10B Problems/Problem 18|Solution]]
 == Problem 19 ==
+Suppose that <math>\{a_n\}</math> is an arithmetic sequence with
+<cmath> a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.</cmath>
+What is the value of <math>a_2 - a_1 ?</math>
+<math> \mathrm{(A) \ } 0.0001\qquad \mathrm{(B) \ } 0.001\qquad \mathrm{(C) \ } 0.01\qquad \mathrm{(D) \ } 0.1\qquad \mathrm{(E) \ } 1 </math>
 [[2002 AMC 10B Problems/Problem 19|Solution]]
 == Problem 20 ==
+Let <math>a, b,</math> and <math>c</math> be real numbers such that <math>a-7b+8c=4</math> and <math>8a+4b-c=7.</math> Then <math>a^2-b^2+c^2</math> is
+<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math>
 [[2002 AMC 10B Problems/Problem 20|Solution]]
 == Problem 21 ==
+Andy's lawn has twice as much area as Beth's lawn and three times as much 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 10B Problems/Problem 21|Solution]]
@@ Line 156: / Line 200: @@
 == Problem 22 ==
-Let <math>\triangle{XOY}</math> be a right-triangle with <math>m\angle{XOY}=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints of the legs <math>OX</math> and <math>OY</math>, respectively. Given <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 the legs <math>OX</math> and <math>OY</math>, respectively. Given <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>
@@ Line 163: / Line 207: @@
 == Problem 23 ==
+Let <math>\{a_k\}</math> be a sequence of integers such that <math>a_1=1</math> and <math>a_{m+n}=a_m+a_n+mn,</math> for all positive integers <math>m</math> and <math>n.</math> Then <math>a_{12}</math> is
+<math> \mathrm{(A) \ } 45\qquad \mathrm{(B) \ } 56\qquad \mathrm{(C) \ } 67\qquad \mathrm{(D) \ } 78\qquad \mathrm{(E) \ } 89 </math>
 [[2002 AMC 10B Problems/Problem 23|Solution]]
 == Problem 24 ==
+Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius <math>20</math> feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point <math>10</math> vertical feet above the bottom?
+<math> \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15 </math>
 [[2002 AMC 10B Problems/Problem 24|Solution]]
 == Problem 25 ==
+When <math>15</math> is appended to a list of integers, the mean is increased by <math>2</math>. When <math>1</math> is appended to the enlarged list, the mean of the enlarged list is decreased by <math>1</math>. How many integers were in the original list?
+<math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math>
 [[2002 AMC 10B Problems/Problem 25|Solution]]
 == See also ==
+{{AMC10 box|year=2002|ab=B|before=[[2002 AMC 10A Problems]]|after=[[2003 AMC 10A Problems]]}}
+* [[AMC 10]]
+* [[AMC 10 Problems and Solutions]]
 * [[AMC Problems and Solutions]]
+* [[Mathematics competition resources]]
+{{MAA Notice}}

2002 AMC 10B (Problems • Answer Key • Resources)
Preceded by 2002 AMC 10A Problems	Followed by 2003 AMC 10A Problems
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
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Difference between revisions of "2002 AMC 10B Problems"

Latest revision as of 12:34, 23 July 2024

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