A Gentle Introduction to the American Invitational Mathematics Exam

362 Pages·2015·1.733 MB·English

by  Scott Annin

A Gentle Introduction to the American Invitational
Mathematics Examination
Scott A. Annin
California State University, Fullerton
March 16, 2014
ii
Contents
1 Algebraic Equations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Distance-Rate-Time Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Combinatorics 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Permutations with Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Permutations without Repetition . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Combinations without Repetition . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Combinations with Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 More Challenging Combinatorics Problems . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Some Combinatorial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 The Inclusion-Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Probability 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Properties of Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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3.3 Examples: Tournaments, Socks, and Dice . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Number Theory 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Greatest Common Divisor and Least Common Multiple . . . . . . . . . . . . . . . . 62
4.4 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Sequences and Series 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.2 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.1 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.2 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Some Other Useful Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Additional Examples of Sequences and Series . . . . . . . . . . . . . . . . . . . . . . 90
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Logarithmic and Trigonometric Functions 97
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.4 Putting Logarithmic and Trigonometric Functions Together . . . . . . . . . . . . . . 116
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Complex Numbers and Polynomials 121
CONTENTS v
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 The Algebra of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3 The Geometry of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Basic Definitions and Facts about Polynomials . . . . . . . . . . . . . . . . . . . . . 133
7.5 Polynomials with Complex Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 Plane Geometry 149
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2.1 Congruent and Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2.2 Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.4 Geometrical Concepts in the Complex Plane. . . . . . . . . . . . . . . . . . . . . . . 176
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9 Spatial Geometry 183
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 Rectangular Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.3 Cylinders, Cones, and Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.4 Tetrahedra and Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10 Hints for the Exercises 193
10.1 Hints for Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.1.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.1.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.2 Hints for Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.2.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.2.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.3 Hints for Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
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10.3.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.3.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.4 Hints for Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.4.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.4.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.5 Hints for Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.5.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.5.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.6 Hints for Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.6.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.6.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.7 Hints for Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.7.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.7.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.8 Hints for Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.8.1 Hints to Get Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.8.2 More Extensive Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10.9 Hints for Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11 Solutions to Exercise Sets 217
11.1 Chapter 1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.2 Chapter 2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
11.3 Chapter 3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
11.4 Chapter 4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
11.5 Chapter 5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
11.6 Chapter 6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11.7 Chapter 7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11.8 Chapter 8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
11.9 Chapter 9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Preface
A Gentle Approach to the American Invitational Mathematics Examinationisacelebrationofmath-
ematical problem solving at the level of the high school American Invitational Mathematics Exami-
nation (AIME). The book is intended, in part, as a resource for comprehensive study for the AIME
competition to be used by students, coaches, teachers, and mentors preparing for the exam.
However, this manuscript is also written for anyone who enjoys solving problems, exploring recre-
ational mathematics, and sharpening their critical thinking skills. Readers who study this book are
destined to see their problem solving skills enhanced. It is not intended just for the exceptionally
brilliantorforthosewhoareintenselycompetitive,eventhoughtheAIMEisacompetition. Rather,
anyone who enjoys thinking about interesting problems in mathematics should enjoy this book.
One of the author’s aims in writing this book is to make the problems accessible and interesting to
a wide audience and to lure in those who feel perhaps that the AIME problems are not for them.
Therefore,evenifyouarenotinvolvedintheAIMEasastudentormentorordonothaveconfidence
in your own skills in mathematical problem solving, as long as you have an interest mathematical
problem solving and a desire to explore fun mathematical problems at the high school level, you
should find this book engaging. You may surprise yourself with what you can do!
The AIME is the second in a series of exams known collectively as the American Mathematics
Competitions (AMC). In the next couple of pages, we will describe the series of exams comprising
the AMC, including the focus of this manuscript, the AIME, and offer some words of advice and
encouragement. After that, the book will survey a large compendium of topics and problems from
past AIMEs that will form the basis for learning, discussion, and discovery.
The American Mathematics Competitions
In 1950, the Mathematical Association of America launched the AMC as a means of “strengthening
the mathematical capabilities of our nation’s youth”. In 2008, nearly 60 years later, almost a
half-million middle and high school students across the United States participated in the program,
representing over 5,000 schools.
The AMC consists of a series of examinations covering a broad spectrum of topics. The problems
appearingintheseexaminationscanallbesolvedbyusingprecalculusmethods. Theexamsbecome
progressively more difficult, beginning with the AMC 10 and AMC 12 in February each year1,
1There is also an AMC 8 exam given each year in November that is only open to students in eighth grade or
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followedbytheAIMEinMarchorearlyApril,theUnitedStatesofAmericaMathematicalOlympiad
(USAMO) later in April, and the International Mathematics Olympiad (IMO) in July. With the
help of a three-week Mathematical Olympiad Summer Program (MOSP) hosted at the University
of Nebraska each June for the six members of the United States’ IMO team, the United States has
performed exceptionally well in the IMO. The United States fielded its first IMO team in 1974, and
placedsecond. ThenumberofcountriesparticipatingintheIMOisusuallybetween80and90. The
United States has placed first in total team score at the IMO four times and, on 26 occasions, has
finished in the top three.
The goals of the AMC, however, reach far beyond success in the IMO. There have been disturbing
trends in the United States in recent years in mathematics education at the middle and high school
levels. Many students in the U.S. under-perform in mathematics compared to their counterparts in
othercountries. TheAMCexaminationsaredesignedtoinvolvestudentsofallmathematicalability,
not just the brightest few. The AMC 10 and AMC 12 exams, in particular, contain some problems
that do not require extraordinarily creative thought. The hope is that such problems can serve as
a confidence boost to many students, and that participation in the exam can foster enthusiasm for
mathematics in general. The problems, by and large, are fun and engaging. They usually do not
requireheavymachineryorexcessivecomputations. Instead,theycallonastudenttoanalyzeideas,
to think logically, and in the case of some of the harder problems, to find a clever approach.
About the AMC 10, AMC 12, and the American Invitational Mathematics Exam
The AMC 10 exam is open to all students in 10th grade or below, while the AMC 12 is open to all
students in 12th grade or below. Both exams consist of 25 multiple-choice problems to be solved
without a calculator in 75 minutes. Students receive 6 points for each correct answer, 1.5 points for
each blank answer, and no points for a wrong answer. The AMC 10 and AMC 12 exams are both
offered twice every year. Therefore, an 11th or 12th grader can attempt the AMC 12 exam twice,
while a 10th grader could attempt the AMC 10 twice, the AMC 12 twice, or take each exam once.
Students who score at least 120 points or finish in the top 2.5% on the AMC 10 exam or who
score at least 100 points or finish in the top 5% on the AMC 12 exam are invited to take the
American Invitational Mathematics Exam. The AIME is a 15 problem exam in which each answer
is an integer in the range 000-999, inclusive. As with the AMC 10 and AMC 12 exams, the AIME
competition is offered twice each year. Throughout this book, problems from the first version of a
given year’s exam with be denoted by “AIME” and problems from the second version of a given
year’s exam will be denoted by “AIME-2”. Since students mark their answers on a scanned answer
sheet, it is imperative that all three digits of the answer are accurately recorded. This includes the
possibility of leading zeros in the answer. For instance, the answer 46 should be marked as 046 on
the answer sheet. Students have three hours to complete the exam, and no calculators are allowed.
The problems appearing on the AIME are noticeably more challenging than those on the AMC
10 and AMC 12 exams overall, but still, they do not require knowledge of calculus. Some of the
mostcommonthemesintheAIMEproblemsarealgebra, combinatorics(i.e. counting), probability,
number theory, complex numbers, polynomials trigonometry, and geometry. However, any topic at
the high school level is fair game, and problems on the AIME often require students to use ideas
from multiple mathematics disciplines to obtain the solution.
younger. Thatexamisnotthesubjectofthisvolume.
CONTENTS ix
Students who achieve high scores on both the AMC and AIME may then be selected to compete in
the United States of America Mathematical Olympiad (USAMO). Selection criteria are not fixed,
but roughly 500 students nationwide are chosen each year. The USAMO is a six problem essay
competition spanning two days. The problems contained in the USAMO are extremely challenging
andrequiregreatingenuityandcreativethinking. FromtheUSAMOparticipants, sixarechosento
representtheUnitedStatesattheInternationalMathematicsOlympiad(IMO)eachsummer. Several
textsandresourceshavebeendevelopedonthesubjectoftheUSAMOandIMOcompetitions. The
AIME, however, has been comparatively under-represented in materials and resources for coaches,
teachers, and students. The present volume aims to make some headway in this area by providing
a resource for teachers and students preparing for the AIME.
An Integer Answer from 000 to 999
We indicated that every answer to every question appearing on the AIME is a three digit integer in
the range 000 to 999. At first, this may seem to place severe limitations on the breadth of questions
that are suitable for the exam. However, there are several standard ways that questions whose
answers do not meet this criterion are typically modified so that the answers do comply. Here are
just a few:
• If the actual answer x is a whole number larger than 999, the question can be modified to ask
for the left-most three digits of x.
• Alternatively, if the actual answer x is a whole number larger than 999, the question might
ask for the remainder when x is divided by 1000.
• For some negative answers x, the question can simply ask for the absolute value, |x|.
• If the actual answer x is a real number in the interval [0,1000], the question might ask for
either the ceiling of x, (cid:100)x(cid:101), or the floor of x, (cid:98)x(cid:99). Recall that
(cid:100)x(cid:101) denotes the smallest integer greater than or equal to x
and
(cid:98)x(cid:99) denotes the largest integer less than or equal to x.
x
• Iftheactualanswerisafraction,say ,thequestionwillusuallyaskthesolvertofirstsimplify
y
x m
thefractionto“lowestterms”, = ,wheremandnarerelativelyprimeintegerswithn>0.
y n
This is accomplished by removing all common factors that arise in both the numerator x and
denominator y. Finally, the question can ask for some integer combination of m and n (such
as m+n). The subject of relatively prime integers is taken up in Chapter 4 of the book (see
Section 4.3). Reducing the fraction as described above is necessary in order to ensure the
problem has a unique answer.
√
• Some AIME questions will ask for the value of m appearing under a radical, such as m. In
such cases, it is customary for the question to require us to pull out from under the radical
√ √
any square factors that occur. For instance, we can rewrite 75 as 5 3. The same applies to
x CONTENTS
√ √ √
any root. For instance, we can rewrite 4162 = 481·2 = 342. In general, by extracting all
factorsoftheformpk (foraprimepandpositiveintegerk ≥2)fromanexpressionoftheform
√
kN, we can obtain a unique form for the answer to an AIME problem. Here is an excerpt
√
m−n
from an example from Chapter 5: “...can be written in the form , where m, n, and p
p
are positive integers and m is not divisible by the square of any prime. Find 100m+10n+p.”
The requirement that m not be divisible by the square of any prime ensures its unique value,
as any square factors must be extracted and then simplified with n and p.
Even though the answer to every AIME problem is an integer in the range 000–999, AIME
competitorsshouldstillbepreparedtodohandcalculationsinvolvingapproximationsofcom-
monly seen mathematical values. While it is impossible to give a complete list, we briefly
enumerate a few of these values here:
x Approximate Value of x
e 2.71828
π 3.14159
ln2 0.69315
ln3 1.09861
√
2 1.41421
√
3 1.73205
Thenumberofdecimalplacesrequiredtoobtainasufficientlyaccurateapproximationdepends
on the problem, but certainly two or three digits to the right of the decimal point would be
an excellent start. This is by no means a complete list, but it is a start and can help to
√
approximate other needed values as well, such as π2 or 6 or ln8, and so on.
The Structure of this Book
Thefirstninechaptersinthebookarearrangedaccordingtomathematicaltopic,sometimesbroken
into sections, each of which begins with some of the important facts and theorems about that topic
that are commonly required to solve AIME problems. These summaries are not intended to serve
as a review of everything important about the topic, but rather, attention has been focused on
those aspects that are likely to arise in solving AIME problems. Moreover, in an effort to keep the
discoursefocusedonproblemsolvingandavoidextendingthelengthofthemanuscript,itisnotour
intentiontoproverigorouslyalloftheresultsthataresummarizedherein. Ofcourse,understanding
whyaresultistruecanoftenassisttheproblemsolverinapplyingiteffectively,andsofarasthiscan
be achieved, some informal discussion of the results will usually be given. Throughout the chapters,
some problems that relate to the topic at hand are provided, along with solutions. Most of these
problems are taken directly from past AIMEs. They have been carefully chosen to display a wide
range of strategies, subtopics, and difficulty. It is worth noting that many AIME problems require
ideas from multiple branches of mathematics. We have made every effort in this book, however, to
place each problem in the chapter of the topic deemed most crucial in understanding the solution.
Finally, each chapter concludes with a collection of additional problems on the topic from past
AIMEs. Again, a variety of exercises have been chosen, generally of gradually increasing difficulty.
Readers are encouraged to try to work out solutions to these problems.