TEXTS AND READINGS 56 IN MATHEMATICS Iranian Mathematics Competitions 1973-2007 Texts and Readings in Mathematics Advisory Editor C. S. Seshadri, Chennai Mathematical Institute, Chennai. Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi. Editors R. B. Bapat, Indian Statistical Institute, New Delhi. V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai. Probal Chaudhuri, Indian Statistical Institute, Kolkata. V. S. Sunder, Inst. of Mathematical Sciences, Chennai. M. Vanninathan, TIFR Centre, Bangalore. Iranian Mathematics Competitions 1973-2007 Bamdad R. Yahaghi University of Golestan, Iran HINDUSTAN [ldgl@ U L!LJ UB OOK AGENCY Published in India by Hindustan Book Agency (India) P 19 Green Park Extension New Delhi 110 016 lndia email: [email protected] Http://www.hindbook.com Copyright © 2010, Hindustan Book Agency (India) No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest exclusively with Hindustan Book Agency (India) . Unauthorized export is a violation of Copyright Law and is subject to legal action. ISBN 978-81-85931-99-9 ISBN 978-93-86279-48-4 (eBook) DOI 10.1007/978-93-86279-48-4 beh vajh-e marhamat ey saakenaane sadr-e jalaal, ze rooy-e HaJez 0 een aastaaneh yaad aareed. Dedicated to all those who teach people to think. Contents Foreword Xl Preface Xlll Acknowledgement XV Part 1. Problems 1 1.1. First Competition, University of Tehran, March 1973 2 1.2. Second Competition, Shiraz (former Pahlavi) University, March 1974 4 1.3. Third Competition, (former Jondi Shapour) University of Ahwaz, March 1975 7 1.4. Fourth Competition, University of Tabriz, March 1976 9 1.5. Fifth Competition, Sharif (former Aryamehr) University of Technology, March 1977 10 1.6. Sixth Competition, The University of Isfahan, March 1978 12 1.7. Seventh Competition, Ferdowsi University of Mashhad, March 1980 13 1.8. Eighth Competition, Shiraz University, March 1984 14 1.9. Ninth Competition, Tehran Teacher Training (Tarbiat Moallem) University, September 1985 15 1.10. Tenth Competition, University of Sistan and Baluchestan, March 1986 17 1.11. Eleventh Competition, The University of Birjand, March 1987 19 1.12. Twelfth Competition, Guilan University, March 1988 20 1.13. Thirteenth Competition, University of Tehran, March 1989 21 1.14. Fourteenth Competition, The University of Isfahan, March 1990 23 1.15. Fifteenth Competition, Ferdowsi University of Mashhad, March 1991 24 1.16. Sixteenth Competition, Razi (Rhazes or Rasis) University 9f Kermanshah, March 1992 26 1.17. Seventeenth Competition, Shahid Beheshti (former National) University, March 1993 27 1.18. Eighteenth Competition, Sharif University of Technology, March 1994 28 1.19. Nineteenth Competition, University of Kerman, March 1995 29 1.20. Twentieth Competition, Sharif University of Technology, February 1996 30 vii viii CONTENTS 1.21. Twenty-First Competition, University of Tehran, March 1997 30 1.22. Twenty-Second Competition, University of Ahwaz, March 1998 31 1.23. Twenty-Third Competition, Sharif University of Technology, March 1999 39 1.24. Twenty-Fourth Competition, Khajeh Nasir Toosi University of Technology, May 2000 40 1.25. Twenty-Fifth Competition, IKIU of Qazvin, May 2001 41 1.26. Twenty-Sixth Competition, IASBS of Zanjan, May 2002 43 1.27. Twenty-Seventh Competition, Bu-Ali Sina (Avecina) University of Hamedan, May 2003 44 1.28. Twenty-Eighth Competition, Sharif University of Technology, May 2004 46 1.29. Twenty-Ninth Competition, University of Mazandaran in Babolsar, May 2005 48 1.30. Thirtieth Competition, Tafresh University, May 2006 50 1.31. Thirty-First Competition, Ferdowsi University of Mashhad, May 2007 51 Part 2. Solutions 55 2.1. First Competition 56 2.2. Second Competition 61 2.3. Third Competition 70 2.4. Fourth Competition 76 2.5. Fifth Competition 80 2.6. Sixth Competition 90 2.7. Seventh Competition 94 2.8. Eighth Competition 99 2.9. Ninth Competition 105 2.10. Tenth Competition 118 2.11. Eleventh Competition 124 2.12. Twelfth Competition 128 2.13. Thirteenth Competition 135 2.14. Fourteenth Competition 144 2.15. Fifteenth Competition 153 2.16. Sixteenth Competition 159 2.17. Seventeenth Competition 163 2.18. Eighteenth Competition 166 2.19. Nineteenth Competition 171 2.20. Twentieth Competition 177 2.21. Twenty-First Competition 184 2.22. Twenty-Second Competition 188 2.23. Twenty-Third Competition 192 2.24. Twenty-Fourth Competition 197 2.25. Twenty-Fifth Competition 209 2.26. Twenty-Sixth Competition 215 2.27. Twenty-Seventh Competition 218 2.28. Twenty-Eighth Competition 224 CONTENTS ix 2.29. Twenty-Ninth Competition 230 2.30. Thirtieth Competition 235 2.31. Thirty-First Competition 241 Part 3. Problem Index 249 First Competition 250 Second Competition 250 Third Competition 251 Fourth Competition 251 Fifth Competition 251 Sixth Competition 252 Seventh Competition 252 Eighth Competition 252 Ninth Competition 253 Tenth Competition 253 Eleventh Competition 254 Twelfth Competition 254 Thirteenth Competition 254 Fourteenth Competition 255 Fifteenth Competition 255 Sixteenth Competition 256 Seventeenth Competition 256 Eighteenth Competition 256 Nineteenth Competition 257 Twentieth Competition 257 Twenty-First Competition 257 Twenty-Second Competition 257 Twenty-Third Competition 258 Twenty-Fourth Competition 258 Twenty-Fifth Competition 258 Twenty-Sixth Competition 259 Twenty-Seventh Competition 259 Twenty-Eighth Competition 259 Twenty-Ninth Competition 260 Thirtieth Competition 260 Thirty-First Competition 260 Appendix A. Historical Introduction And The Winners List 263 A.1. Iranian University Students Mathematics Competitions - a historical introduction 263 A.2. Winners of the Competitions 267 Index 279 Foreword This volume offers a co11ection of problems with solutions in different branches of mathematics given in the annual student competitions of the Ira nian Mathematical Society (IMS) during the past 31 years. After the establishment of numerous new 1ranian universities and mathe matics departments in the 1960's, there was a growing need for co11aboration among these departments. Thus the first national conference was held in Shiraz in 1970. The Iranian Mathematical Society was born just after this meeting, and was registered in 1971. One of the important activities of the 1MS is holding university student competitions which were inaugurated in 1972 by our Society's General Assem bly. The first such competition was held in 1973 and the 31st held in 2007 at the Ferdowsi University of Mashhad. This co11ection includes a11 the problems of these contests with nice and interesting solutions presented by a former brilliant student of Sharif University of Technology and one of the winners of these competitions, Dr. Bamdad Yahaghi, who now holds a Ph.D. in Mathematics. I am sure that this exce11ent problem book, which is published simultane ously with its Pers i an version, would be a very helpful reference for a11 students of mathematics around the world. A.R. Medghalchi President of the 1MS xi