Iranian Mathematics Competitions 1973–2007

297 Pages·2010·22.473 MB·English

by  Bamdad R. Yahaghi (auth.)

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
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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