Table Of ContentThe Contest Problem Book V
This is the fifth book of problems and solutions from the
American High School Mathematics Examinations (AHSME),
covering the six examinations in 1983–88. It is also the first
compilation of the follow-up American Invitational Math-
ematics Examination (AIME), which began in 1983 as an
intermediate step between the AHSME and the USA Math- T The Contest
h
ematical Olympiad (USAMO). The AIME has a unique answer
e
format — every answer is an integer from 0 to 999. C Problem Book V
o
n
The AHSME, AIME, and USAMO comprise the high-school t
e
level of the American Mathematics Competitions (AMC), s
t
and they presume only pre-calculus material. Over 400,000 P American High School Mathematics Examinations
r
students, in many countries, participated yearly in these o and
b
exams. l American Invitational Mathematics Examinations
e
m 1983–1988
This book will be of interest to anyone of any age who likes
B
good problems. Students practicing for AMC exams will o
o
want to solve these problems under test conditions, but k
others can work through them in any way. V
This volume was prepared by George Berzsenyi, Chairman
of the AIME Committee for the years covered by this volume,
and Stephen B Maurer, then Chairman of both the AHSME
Committee and the overall AMC governing committee. In
addition to collecting the problems and the careful solu-
tions written by the examination committees, Berzsenyi
and Maurer have provided a thorough index, a comprehen-
sive guide to other problem materials worldwide, addition-
al solutions, some additional problems that didn’t make the
B
final cut, statistical information on problem difficulty, and
e
r
infomation on test development and history. z
s
e Problems and solutions
n
y compiled and augmented by
i
a George Berzsenyi
n
ISBN 978-0-88385-640-6 d
Stephen B Maurer
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9 780883 856406 e
r
Anneli Lax New Mathematical Library | Vol. 38
The Contest
Problem Book V
American High School Mathematics Examinations
and
American Invitational Mathematics Examinations
198S1988
Problems and solutions compiled and augmented by
George Berzsenyi
Rose-Hulman Institute of Technology
Stephen B Maurer
Swarthmore College
THE MATHEMATICAL ASSOCIATION OF AMERICA
NEW MATHEMATICAL LIBRARY
published by
The Mathematical Association of America
Editorial Committee
Underwood Dudley, Editor
DePauw University
Ross Honsberger, University of Waterloo
Daniel Kennedy, Baylor School
Michael J. McAsey, Bradley University
Mark E. Saul, Bronxville Schools
Peter Ungar
Anneli Lax, Consulting Editor
New York University
The New Mathematical Library (NML) was begun in 1961 by the School
Mathematics Study Group to make available to high school students
short expository books on various topics not usually covered in the high
school syllabus. In three decades the NML has matured into a steadily
growing series of over thirty titles of interest not only to the originally
intended audience, but to college students and teachers at all levels. Pre-
viously published by Random House and L. W. Singer, the NML became
a publication series of the Mathematical Association of America (MAA)
in 1975. Under the auspices of the MAA the NML will continue to grow
and will remain dedicated to its original and expanded purposes.
© 1997 by the Mathematical Association of America
Library of Congress Catalog Number 97-70508
Print ISBN 978-0-88385-640-6
Electronic ISBN 978-0-88385-952-0
Printed in the United States of America
The Contest
Problem Book V
American High School Mathematics Examinations
and
American Invitational Mathematics Examinations
1983-1988
MATHEMATICAL
NEW LIBRARY
1. Numbers: Rational and Irrational by Ivan Niven
2. What is Calculus About? by W W Sawyer
3. An Introduction to Inequalities by E. R Beckenbach and R. Bellman
4. Geometric Inequalities by N. D. Kazatimff
5. The Contest Roblem Book I Anad High School Mlthanotics Examinations 1950-1960.
Compiled and with sohtions by Charles L Salkind
6. The Lon of Large Numbers by PI J. Davis
7. Uses of Inhity by Leo Zippin
8. Gmmctric Transfonnations 1 by I. M. Yaglom, translated by A. Shieldr
9. Continued Fractions by corl D. OIdp
10. Replaced by Nh4L.-34
}
11. Hungarian probian Books I and Il, Based on the =tv& Competitions
12. 1894-1905 and 1906-1928. tmnslatedby E. Ramport
13. Episodes from the Early Histury of kth&cs by A. Aaboe
14. Chup and Their Graphs by E. Gtussinan and W: Magnus
15. of Choice by Ivan Niwn
The Mathematics
16. F mP ythagoras to Einstein by K 0. Friedrich
17. The Conteat Roblem Book I1 Ann4 High School Mathematics Examinations 1%1-1%5.
Cornpiled and with solutions by Charla L Sa&d
18. First Coacepts of Topology by W: G. Chinn and N. E. Steenrod
19. Geomerry Revisited by H. S. M,C oxefer and S. L. Gmiiser
20. Invitation to Numbcr Theory by @stein Om
21. Cmmctric Transfonnatiioos Il by I. M. Yaglom, translated by A. Shieldr
22. Elc~ncnta~Cyr yptanalysis-A Mathumb' cal Approach by A. Sinkw
23. Ingenuity in Mathematics by Ross Honsbetger
24. Geometric Transfrmnations III by I. M. hglom. translated by A. Sheniaer
25. The Conkst Roblem Bodr IIl Annd High School Muhcmuics Exlmirutions 1-1972.
Compiled with by L Salkind and J. M. &I
and solutions C.
26. Mathematical Methods in Science by Geotge Pdlya
27. Intcmational Olympiade1959-1977. Compiled with
Mathematical and solutions
by L. Gmiaer
S.
28. The Mathematics of Games and Gambling by Edwanj IK Packel
29. The Contad Roblem Bodr lV Anad High School 'cs Eurm'o M' ons 1973-1982.
Compiled and with solutions by R A. Arfino, A. M. Gaglione, and N. Shell
30. The Role of Mathematics in Scieacc by M. M. Schiffer and L. Bowden
31. Intanatid Mathematical Olympiads 1978-1985 and forty supplemntary
problems. Compiled and with solutions by Murray S. Klamkin
32. Riddles of the Sphinx by Mcvrin Gadner
33. U.S.A. Olympiads 1972-1986. Compiled and with by
Math-tical solutions
Murray S. Klamkin
34. Graphs and Their Uses by @stein Om. Rcvised and updated by Robin J. Wilson
35. Exploring Mathematics with Your Gnnputcr by Arthur Engel
36. Game Theory and Strategy by Philip D. SfrgBn, JI:
37. Episodes in Ninetemth and lbenthicth Cenhuy Euclidean Geometry by Ross
Honsbetger
38. The Contest Roblcm Book V Amrican High School Mathematics Examinations sad
American Invitational hbtbanatics Euminrtions 1983-1988. compiled and augmented by
George Bemenyi and Stephen B. Maumr
Ofher titles in pmpmtion.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . vii
ASHME . . . . . . . . . . . . . . . . . . . . . . . . 1
Problems . . . . . . . . . . . . . . . . . . . . . . 1
1983AHSME . . . . . . . . . . . . . . . . . . 1
1984AHSME . . . . . . . . . . . . . . . . . . 7
1985 AHSME . . . . . . . . . . . . . . . . . 13
1986AHSME . . . . . . . . . . . . . . . . . 19
1987 AHSME . . . . . . . . . . . . . . . . . 26
1988 AHSME . . . . . . . . . . . . . . . . . 33
Dropped AHSME Problems . . . . . . . . . . . 40
AnswerKey . . . . . . . . . . . . . . . . . . . 51
Response F’requency Tables . . . . . . . . . . . . . 52
Solutions . . . . . . . . . . . . . . . . . . : . . 59
1983 AHSME Solutions . . . . . . . . . . . . 59
1984 AHSME Solutions . . . . . . . . . . . . 73
1985 AHSME Solutions . . . . . . . . . . . . 85
1986 AHSME Solutions . . . . . . . . . . . . 1 00
1987 AHSME Solutions . . . . . . . . . . . . 113
1988 AHSME Solutions . . . . . . . . . . . . 1 28
Solutions to Dropped AHSME Problems . . . . . 139
AIME . . . . . . . . . . . . . . . . . . . . . . . . 157
Problems . . . . . . . . . . . . . . . . . . . . . 157
1983AIME . . . . . . . . . . . . . . . . . . 1 57
1984AIME . . . . . . . . . . . . . . . . . . 1 60
1985AIME . . . . . . . . . . . . . . . . . . 163
1986AIME . . . . . . . . . . . . . . . . . . 166
1987AIME . . . . . . . . . . . . . . . . . . 1 69
1988AIME . . . . . . . . . . . . . . . . . . 1 72
Dropped AIME Problems . . . . . . . . . . . . 1 75
AnswerKey . . . . . . . . . . . . . . . . . . . 1 78
Solutions . . . . . . . . . . . . . . . . . . . . . 179
1983 AIME Solutions . . . . . . . . . . . . . 179
1984 AIME Solutions . . . . . . . . . . . . . 1 92
V
1985 AIME Solutions . . . . . . . . . . . . . 2 04
1986 AIME Solutions . . . . . . . . . . . . . 216
1987 AIME Solutions . . . . . . . . . . . . . 2 30
1988 AIME Solutions . . . . . . . . . . . . . 240
Solutions to Dropped AIME Problems . . . . . . 2 50
A Guide to the Problem Literature . . . . . . . . . . . 261
Classification of Problems . . . . . . . . . . . . . . . 277
vi
Preface
This is the fifth book of problems from the American High
School Mathematics Examination (AHSME), covering the
six
examinations from 1983-88. It is the first book of prob-
also
lems for the follow-up American Invitational Mathematics Ex-
amination (AIME), which began in 1983. These are two
of
the four examinations in the American Mathematics Competi-
tions (AMC); the others are the USA Mathematical Olympiad
(USAMO) and the American Junior High School Mathematics
Examination (AJHSME).
Each AHSME consists of 30 multiplechoice questions, and each
AIME consists of 15 questions, with each answer an integer from 0
to 999. Both examinations cover precalculus material. Contestants
have minutes for the AHSME, 3 hours for the AIME (2.5 hours
90
in 1983-85). During the period covered by this book, about
400,OOO students took the AHSME each Spring and 1000-4000
were invited to the AIME (based on AHSME score). both
On
exams, problems are roughly in increasing order difficulty, with
of
the AIME questions, average, much harder. The AHSME is
on
one of the largest mathematics competitions in the world, and all
the AMC exams are and respected worldwide.
known
Why Buy this Book?
The immediate reason is to practice in order to do better on
future offerings of these competitions. And there is doubt that
no
practice helps. The AHSME is a hard exam: typically the best
score in a school will be less than points out of 150 possible,
100
so there is plenty of room for improvement.
However, the fundamental goal the mathematics community
of
in providing these exams is to pique interest in mathematics and
develop talent. Problems are at the heart of mathematics, and
experience greatly sharpens problem-solving skills. We believe that
some the problems the AHSME and AIME are intriguing
of on
in themselves, and/or they bring out important mathematical
points. Thus, one doesn’t ever have to participate in the AHSME
or AIME contests to something from this book - older readers
gain
aren’t eligible anyway! Indeed, working these problems from this
book you need not stick to the artificial time limits that contests
impose. Often the best solutions, and the deepest learning, come
when there is time to reflect.
vii
viii V
CONTESTP ROBLEMB OOK
While supplies last (many years), each exam and its solu-
tion pamphlet are available individually. This book offers the
convenience and lower cost of bundling - and much more:
1. Additional solutions. Although most solutions in this book are
the same provided in the complete solution manual made
as
available right after each exam is given, there have been some
editorial improvements in them and some additional solutions
have been added. Some of these additional solutions weren't in
the solution pamphlets due to space limits, some are solutions
mailed to by students and teachers, and some were provided
us
by members of the MAA Publications committee when they
reviewed our manuscript.
2. An index. If a reader wishes to work specific types of
on
problems, this index makes it possible.
3. Pointers to other material. We have added some further
references to related problems and to articles and books that
expand upon ideas in some of the problems.
4. "Dropped Problems". In order to get good exams, one must
begin with many more problems than will fit. Invariably, some
very good problems don't make the final cut. We provide two
sets of dropped problems (heretofore never revealed!), one for
the AHSME and one for the AIME.
5. Statistical information. The AHSME is a multiple-choice
exam. We provide frequency-response tables that show how
popular each answer was for each question. The introduction
to that section explains how one can these tables to
use
calibrate one's work and identify important mathematical
errors.
How to Use This Book
can learn from this book through both the problems you
You
get right and the problems you get wrong. After doing problem
a
(or a set of problems - but you don't need to stick to the time
limits), first look at the answer key. If you got the right answer,
then look at our solution(s) to compare with yours. If you are
fortunate, our solution will be very different from yours and
will thus introduce you to an alternative approach. If you got
the wrong go back to the problem before looking at the
answer,
solution. Just knowing that your answer was wrong (or sometimes
knowing the right answer) will jog you into seeing your error and
PREFACE ix
lead you to a correct solution. If not, then look at the solution.
Ask yourself: What is the key idea in this solution that I missed
(or misunderstood)?
Another good thing to do with any problem is to change the
hypotheses and see how the conclusions change. More generally,
see if you can use ideas from this book to make up and solve new
problems at the right challenge level for you and your friends.
Problem posing can be good a learning experience problem
as as
solving.
We hope this book will be not just by individuals, but
used
by math clubs or groups of students working together in class
also
or elsewhere. Sharing ideas - from initial false starts through
to a joint solution - is sometimes the most exciting way to do
mathematics.
Do not be diswumged if there are many problems you cannot
solve. These are hard exams and are meant to challenge almost
everybody. Keep in mind that during 1983-88 at most a few
thousand students obtained an Honor score on the AHSME
Roll
(100 out of 150). The fewest Honor Roll students was 624 in
1984, the most 8050 in 1988. Also, during these 6 years only 7
students obtained a perfect AHSME score of 150. The AIME is
even more difficult. On average, the extremely able students who
took it scored about 5 out of 15, and only 11 students obtained
a perfect 15 during these If you find, for instance, that
years.
the last 10 problems on the 1983 AHSME are beyond you, then
on later AHSMEs concentrate on problems just before and after
this cutoff point. Stretching yourself just beyond what you can
do easily brings about the best growth.
In particular, younger students especially should not be con-
cerned that they cannot do all the problems. They haven't even
studied some of the topics covered yet. If a problem treats a topic
you don't know, just skip it.
History and Changes
The AHSME was first given in 1950, under the name Annual
High School Contest; it was then sponsored by the New York
Metropolitan Section of the Mathematical Association of America
(MAA) and was offered in that region only. In 1957 it became a
national competition, cosponsored by the MAA and the Society
of Actuaries. By 1982 it was sponsored by five organizations: The
MAA, the Society of Actuaries, Mu Alpha Theta, the National
