Contest Problem Book IV: Annual High School Examinations, 1973-1982 (Anneli Lax New Mathematical Library 29)

197 Pages·1982·1.52 MB·English

by  Ralph Artino

THE CONTEST
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PROBLEM IV
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ANNUAL HIGH SCHOOL MATHEMATICS EXAMINATIONS 1973-1982
Compiled and with solutions by
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The Mathematical Association of America
New Mathematical Library
THE CONTEST
PROBLEM BOOK IV
Annual High School Examinations
1973-1982
of
The Mathematical Association of America
Society of Actuaries
Mu Alpha Theta
National Council of Teachers of Mathematics
Casualty Actuarial Society
NEW MATHEMATICAL LIBRARY
PUBLISHED BY
THE MATHEMATICAL ASSOCIATION OF AMERICA
Editorial Committee
Ivan Niven, Chairman (1981-83) Anneli Lax, Editor
University of Oregon New York University
W. G. Chinn (1980-82) City College of San Francisco
Basil Gordon (1980-82) University of California, Los Angeles
M. M. Schiffer (1979-8 1) Stanford 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 a decade the NML matured into a steadily
growing series of some twenty titles of interest not only to
the originally intended audience, but to college students
and teachers at all levels. Previously published by Ran-
dom 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.
THE CONTEST
PROBLEM BOOK IV
Annual High School Examinations
1973-1982
compiled and with solutions by
Ralph A. Artino
The City College of New York
Anthony M. Gaglione
The U.S. NavalAcademy
and
Niel Shell
The City College of New York
29
THE MATHEMATICAL ASSOCIATION
OF AMERICA
Fifth Printing
All rights reserved under International and Pan-American Copyright Conventions.
Published in Washington by the Mathematical Association of America
Library of Congress Catalog Card Number: 82-051076
Complete Set ISBN 0-88385-600-X
Vol. 29 0-88385-629-8
Manufactured in the United States of America
Contents
Preface ........................................ vii
Editors' Preface ................................. xi
List of Symbols .................................. xiii
I Problems ...... ....... 1
1973 Examination ....... 1
1974 Examination ................................ 8
1975 Examination ............................... 14
1976 Examination ............................... 20
1977 Examination ............................... 27
1978 Examination ............................... 34
1979 Examination ............................... 40
1980 Examination ............................... 46
1981 Examination ............................... 52
1982 Examination ............................... 58
II Answer Keys ................................... 65
II Solutions ...................................... 67
i973 Examination ............................... 67
1974 Examination ............................... 79
1975 Examination ............................... 88
1976 Examination ............................... 98
1977 Examination .............................. 107
1978 Examination .............................. 117
1979 Examination .............................. 126
1980 Examination .............................. 138
1981 Examination .............................. 153
1982 Examination .............................. 167
IV Classification of Problems ......................... 179
v
NEW MATHEMATICAL 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. E Beckenbach and R. Bellman
4. Geometric Inequalities by N D. Kazarinoff
5. The Contest Problem Book I Annual High School Mathematics Examinations
1950-1960. Compiled and with solutions by Charles T Salkind
6. The Lore of Large Numbers by P J. Davis
7. Uses of Infinity by Leo Zippin
8. Geometric Transformations I by L M. Yaglom, translatedb y A. Shields
9. Continued Fractions by Carl D. Olds
10. Replaced by NML-34
11. Hungarian Problem Books I and 11, Based on the Ebtvos Competitions
12. J 1894-1905 and 1906-1928, translated by E. Rapaport
13. Episodes from the Early History of Mathematics by A. Aaboe
14. Groups and Their Graphs by E. Grossman and W Magnus
15. The Mathematics of Choice by Ivan Niven
16. From Pythagoras to Einstein by K 0. Friedrichs
17. The Contest Problem Book II Annual High School Mathematics Examinations
1961-1965. Compiled and with solutions by Charles T Salkind
18. First Concepts of Topology by W G. Chinn and N E. Steenrod
19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer
20. Invitation to Number Theory by Oystein Ore
21. Geometric Transformations II by L M. Yaglom, translated by A. Shields
22. Elementary Cryptanalysis-A Mathematical Approach by A. Sinkov
23. Ingenuity in Mathematics by Ross Honsberger
24. Geometric Transformations III by L M Yaglom, translatedb y A. Shenitzer
25. The Contest Problem Book III Annual High School Mathematics Examinations
1966-1972. Compiled and with solutions by C. T Salkind and J. M Earl
26. Mathematical Methods in Science by George P6lya
27. International Mathematical Olympiads 1959-1977. Compiled and with solutions
by S. L. Greitzer
28. The Mathematics of Games and Gambling by Edward W Packel
29. The Contest Problem Book IV Annual High School Mathematics Examinations
1973-1982. Compiled and with solutions by R. A. Artino, A. M Gaglione, and N.
Shell
30. The Role of Mathematics in Science byM M Schiffer and L. Bowden
31. International Mathematical Olympiads 1978-1985 and forty supplementary
problems. Compiled and with solutions by Murray S. Klamkin
32. Riddles of the Sphinx by Martin Gardner
33. U.S.A. Mathematical Olympiads 1972-1986. Compiled and with solutions by
Murray S. Klamkin
34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson
35. Exploring Mathematics with Your Computer by Arthur Engel
Other titles in preparation.
Preface
This volume contains the Annual High School Mathematics Ex-
aminations, given 1973 through 1982. It is a continuation of Contest
Problem Books I, II, III, published as Volumes 5, 17, and 25 of the
New Mathematical Library series and which contain the first twenty-
three annual examinations. The Annual High School Mathematics
Examinations (AHSME), it is hoped, provide challenging problems
which teach, stimulate and provide enjoyment for not only the par-
ticipants, but also the readers of these volumes.
All high school students are eligible to participate in the Annual
High School Mathematics Examinations. In 1982, over 418,000 stu-
dents in the United States, Canada, Puerto Rico, Colombia, Jamaica,
Australia, Italy, England, Hungary, Ireland, Israel, Finland, Belgium
and Luxembourg participated in the examination. It was administered
also in many APO/FPO and other schools abroad. Each year a
Summary of Results and Awards is sent to all participating high
schools (in the United States and Canada). The problems are designed
so that they can be solved with a knowledge of only "pre-calculus"
mathematics, with emphasis on intermediate algebra and plane geome-
try. The subject classification at the end of the volume indicates which
questions are related to which topics.
The problems on each examination become progressively more
difficult. Between 1973 and 1977, the participants were given eighty
minutes to complete the examination, and in subsequent years they
were allowed ninety minutes. The 1973 examination consists of four
parts containing 10, 10, 10 and 5 questions respectively worth 3, 4, 5,
and 6 points each; to correct for random guessing, one fourth of the
number of points assigned to incorrectly answered problems was
deducted from the number of points assigned to correctly answered
problems. The 1974 through 1977 examinations consist of 30 questions
worth five points per question; one point was deducted for each
question answered incorrectly. Since 1978, each examination consists
of 30 questions and was scored by adding 30 points to four times the
vii
viii PREFACE
number of correct answers and then subtracting one point for each
incorrect answer.
Each year since 1972, approximately one hundred of the highest
scoring students on the AHSME and a number of members of
previous International Mathematical Olympiad training classes have
been invited to participate in the U.S.A. Mathematical Olympiad,
currently a three and one half hour essay type examination consisting
of five questions. Since 1974, a team of students has been selected to
participate in the International Mathematical Olympiad.* An Interna-
tional Mathematical Olympiad training class of approximately
twenty-four students receives an intensive problem solving course
prior to the International Olympiad.
It is a pleasure to acknowledge the contributions of the many
individuals and organizations who have made the preparation and
administration of these examinations possible. We thank the members
of the Committee on High School Contests and its Advisory Panel for
Proposing problems and suggesting many improvements in the pre-
liminary drafts of the examinations. We are grateful to Professor
Stephen B. Maurer, who succeeded us as Committee Chairman in
1981, for his assistance in the preparation of this book. We express our
appreciation to the regional examination coordinators throughout the
United States and Canada who do such an excellent job of administer-
ing the examinations in their regions, and to the members of the
Olympiad Subcommittee who administer all the Olympiad activities.
Particular thanks are due to Professor James M. Earl, who was the
chairman of the Contests Committee until his death, shortly after the
1973 examination was printed; to Professor Henry M. Cox, who was
the executive director of the Contests Committee from 1973 to 1976;
to Professor Walter E. Mientka, who has been the executive director of
the Contests Committee since September 1976; and to Professor
Samuel L. Greitzer, who has been the chairman of the Olympiad
Subcommittee since the inception of the subcommittee. We express
appreciation to our sponsors, the Mathematical Association of
America, the Society of Actuaries, Mu Alpha Theta, the National
Council of Teachers of Mathematics, and the Casualty Actuarial
Society for their financial support and guidance; we thank the City
College of New York, the University of Nebraska, Swarthmore Col-
lege and Metropolitan Life Insurance Company for the support they
have provided present and past chairmen and executive directors; and
we thank L. G. Balfour Company, W. H. Freeman and Company,
Kuhn Associates, National Semiconductor, Pickett, Inc., MAA, Mu
*The International Olympiads from 1959 to 1977 have been published in volume 27 of
the New Mathematical Library series.
THE MAA PROBLEM BOOK IV ix
Alpha Theta, NCTM and Random House for donating awards to
high-scoring individuals and schools on the AHSME.
The members of the Committee on High School Contests are
particularly pleased to acknowledge financial support for the U.S.A.
Mathematical Olympiad and the participation of the U.S. team in the
International Mathematical Olympiad. We express our gratitude to
the International Business Machines Corporation for an annual grant
to sponsor an awards ceremony in honor of the winners of the U.S.A.
Mathematical Olympiad; we thank the hosts of training sessions:
Rutgers University, United States Military Academy and United
States Naval Academy; we gratefully acknowledge financial support
of the training sessions and travel to the International Mathematical
Olympiad from the following: Army Research Office, Johnson and
Johnson Foundation, Office of Naval Research, Minnesota Mining &
Manufacturing Corporation, National Science Foundation, Spencer
Foundation, Standard Oil Company of California and Xerox Corpo-
ration.
A few minor changes in the statements of problems have been made
in this collection for the sake of greater clarity.
Ralph A. Artino
Anthony M. Gaglione
Niel Shell