The Contest Problem Book IV The Annual High School Mathematics Examination (AHSME) began as a local contest in New York City in 1950. C By 1960, 150,000 students throughout the United States and Canada took the AHSME. The 1982 Examination was administered to 418,000 participants in the United States C O C and Canada and to 20,000 students in various other coun- T tries. In the United States and Canada, one use of AHSME h e C O N O C is to select approximately one hundred participants in the C o USA Mathematical Olympiad, and the Olympiads are used in the selection of a student team to represent the United nte TheC COoNnteTstN O C s States in the International Mathematical Olympiad. Since t P PCroOblem Book IOVC the difficulty of problems appearing in the AHSME varies r E o over a wide range, they are a valuable teaching aid for all bl C O O C high school students interested in mathematics. e m Annual High School MSathematics O C This volume contains the 1973–1982 Examinations and C B OExaminations 1973–1982 solutions. Earlier Examinations and solutions were pub- lished in NML volumes 5, 17, and 25. The problems and oo S T S k solutions are prepared by the Committee on High School I V Contests, whose members are representatives of the five organizations that jointly sponsor the AHSME: the Math- For how many paths consisting of a sequence of ematical Association of America, Society of Actuaries, Mu Alpha Theta, Casualty Actuarial Society, and National horizontal and/or vertical line segments, with Council of Teachers of Mathematics. each segment connecting a pair of adjacent The questions in this volume were compiled by Profes- sors R. A. Artino, A. M. Gaglione and N. Shell. Professor letters in the diagram above, is the word S. B Maurer (Committee Chair since 1981) worked with A r t CONTEST spelled out as the path is them and the editors of the NML series in the preparation i n of this volume. They made some changes in the originally o traversed from beginning , published solutions and added some alternate solutions G a to end? and some explanatory notes. g l i o n Compiled and with solutions by e , Ralph A. Artino, a ISBN 978-0-88385-629-1 n Anthony M. Gaglione, d S h and Niel Shell e 9 780883 856291 l l Anneli Lax New Mathematical Library | Vol. 29 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 Illustrations by George H. Buehler ©Copyright 1982 by The Mathematical Association of America (Inc.) All rights reserved under International and Pan-American Copyright Conventions. Published in Washington, D.C. by The Mathematical Association of America Library of Congress Catalog Card Number: 82-051076 Print ISBN 978-0-88385-629-1 Electronic ISBN 978-0-88385-944-5 Manufactured in the United States of America 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. Naval Academy and Niel Shell The City College of New York 29 MATHEMATICAALSS OCIATION THE 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 ............................... 46 198 1 Examination ............................... 52 1982 Examination ............................... 58 I1 Answer Keys ................................... 65 ...................................... I1 Solutions 67 1973 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 Classification of Problems ......................... 179 IV ANNELI LAX NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by M! 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 1 Annual High School Mathematics Examinations 1950-1960. Cornpiled 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 I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. OIds 10. Redaced NML-34 by Hungarian Problem Books I and 11, Based on the Eatvos Competitions 1894-1905 and 1906-1928, transkated 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 11 Annual High School Mathematics Examinations 1961-1965. Compiled and with solutions by Charles 7: 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. Creitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations I1 by 1. 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 I11 by 1. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book 111 Annual High School Mathematics Examinations 1966-1972. Compiled and with solutions by C. 7: Salkind and J. M. Earl 26. Mathematical Methods in Science by George Pdya 27. International Mathematical Olympiads 1959-1977. Compiled and with solutions by L. Greitzer S. 28. The Mathematics of Games and Gambling by Edward W! fackel 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 by M. M. Schiffer and L Bowden 31. International Mathematical Olympiads 1978-1985 and forty supplementary problems. Compiled and with solutions by Murray S. Kkamkin 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 Exploring Mathematics with Your Computer by Arthur Engel 35. 36. Game Theory and Strategy by Philip D. Strafln, Js 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger 38. The Contest Problem Book V American High School Mathematics Examinations and American Invitational Mathematics Examinations 1983-1988. Compiled and augmented by George Berzsenyi and Stephen B. Maurer 39. Over and Over Again by Gengzhe Chaiig atid Thomas W. Sederberg 40. The Contest Problem Book VI American High School Mathematics Examinations 1989-1994. Compiled and augmented by Leo J. Schneider 41. The Geometry of Numbers by C.D. Olds. Anneli Lax, and Giuliana Davidoff 42. Hungarian Problem Book 111 Based on the Eatvos competitions 1929-1943 translated by Andy Liu Other titles in preparation. Books may be ordered from: MAA Service Center P. 0. Box 91112 Washington, DC 20090- I1 12 1-800-33 1-1622 fax: 301-206-9789 Preface This volume contains the Annual High School Mathematics Ex- aminations, given 1973 through 1982. It is a continuation of Contest Problem Books I, 11, 111, 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 ... vul 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 8c 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