Best problems from around the world - mathematical olympiads

729 Pages·2006·2.85 MB·English

by  Cao Minh Quang.

PREFACE

Collecting the Mathematics tests from the contests choosing the best students is not
only my favorite interest but also many different people’s. This selected book is an adequate
collection of the Math tests in the Mathematical Olympiads tests from 14 countries, from
different regions and from the International Mathematical Olympiads tests as well.

I had a lot of effort to finish this book. Besides, I’m also grateful to all students
who gave me much support in my collection. They include students in class 11 of specialized
Chemistry – Biologry, class 10 specialized Mathematics and class 10A in the school year
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2003 – 2004, Nguyen Binh Khiem specialized High School in Vinh Long town.

This book may be lack of some Mathematical Olympiads tests from different
countries. Therefore, I would like to receive both your supplement and your supplementary
ideas. Please write or mail to me.

• Address: Cao Minh Quang, Mathematic teacher, Nguyen Binh Khiem specialized
High School, Vinh Long town.
• Email: [e-mail]

Vinh Long, April 2006
Cao Minh Quang
☺ The best problems from around the world Cao Minh Quang
Abbreviations

AIME American Invitational Mathematics Examination
ASU All Soviet Union Math Competitions
BMO British Mathematical Olympiads
CanMO Canadian Mathematical Olympiads
INMO Indian National Mathematical Olympiads
USAMO United States Mathematical Olympiads
APMO Asian Pacific Mathematical Olympiads
IMO International Mathematical Olympiads

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☺ The best problems from around the world Cao Minh Quang
CONTENTS

Page
Preface.................................................................................................................................1
Abbreviations......................................................................................................................2
Contents...............................................................................................................................3
PART I. National Olympiads..........................................................................................17
1. AIME (1983 – 2004) .....................................................................................................17
1.1. AIME 1983............................................................................................................18
1.2. AIME 1984............................................................................................................20
1.3. AIME 1985............................................................................................................21
1.4. AIME 1986............................................................................................................23
1.5. AIME 1987............................................................................................................24
1.6. AIME 1988............................................................................................................25
1.7. AIME 1989............................................................................................................26
1.8. AIME 1990............................................................................................................27
1.9. AIME 1991............................................................................................................28
1.10. AIME 1992..........................................................................................................29
1.11. AIME 1993..........................................................................................................30
1.12. AIME 1994..........................................................................................................32
1.13. AIME 1995..........................................................................................................33
1.14. AIME 1996..........................................................................................................35
1.15. AIME 1997..........................................................................................................36
1.16. AIME 1998..........................................................................................................37
1.17. AIME 1999..........................................................................................................39
1.18. AIME 2000..........................................................................................................40
1.19. AIME 2001..........................................................................................................42
1.20. AIME 2002..........................................................................................................45
1.21. AIME 2003..........................................................................................................48
1.22. AIME 2004..........................................................................................................50
2. ASU (1961 – 2002).........................................................................................................51
2.1. ASU 1961..............................................................................................................52
2.2. ASU 1962..............................................................................................................54
2.3. ASU 1963..............................................................................................................55
2.4. ASU 1964..............................................................................................................56
2.5. ASU 1965..............................................................................................................57
2.6. ASU 1966..............................................................................................................59
2.7. ASU 1967..............................................................................................................60
2.8. ASU 1968..............................................................................................................61
2.9. ASU 1969..............................................................................................................63
2.10. ASU 1970............................................................................................................64
2.11. ASU 1971............................................................................................................65
2.12. ASU 1972............................................................................................................67
2.13. ASU 1973............................................................................................................68
2.14. ASU 1974............................................................................................................70
2.15. ASU 1975............................................................................................................72
2.16. ASU 1976............................................................................................................74
2.17. ASU 1977............................................................................................................76
2.18. ASU 1978............................................................................................................78
2.19. ASU 1979............................................................................................................80
2.20. ASU 1980............................................................................................................82
2.21. ASU 1981............................................................................................................84

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☺ The best problems from around the world Cao Minh Quang
2.22. ASU 1982............................................................................................................86
2.23. ASU 1983............................................................................................................88
2.24. ASU 1984............................................................................................................90
2.25. ASU 1985............................................................................................................92
2.26. ASU 1986............................................................................................................94
2.27. ASU 1987............................................................................................................96
2.28. ASU 1988............................................................................................................98
2.29. ASU 1989...........................................................................................................100
2.30. ASU 1990...........................................................................................................102
2.31. ASU 1991...........................................................................................................104
2.32. CIS 1992.............................................................................................................106
2.33. Russian 1995 .....................................................................................................108
2.34. Russian 1996......................................................................................................110
2.35. Russian 1997......................................................................................................112
2.36. Russian 1998......................................................................................................114
2.37. Russian 1999......................................................................................................116
2.38. Russian 2000......................................................................................................118
2.39. Russian 2001......................................................................................................121
2.40. Russian 2002......................................................................................................123
3. BMO (1965 – 2004) .....................................................................................................125
3.1. BMO 1965............................................................................................................126
3.2. BMO 1966............................................................................................................127
3.3. BMO 1967............................................................................................................128
3.4. BMO 1968............................................................................................................129
3.5. BMO 1969............................................................................................................130
3.6. BMO 1970............................................................................................................131
3.7. BMO 1971............................................................................................................132
3.8. BMO 1972............................................................................................................133
3.9. BMO 1973............................................................................................................134
3.10. BMO 1974..........................................................................................................136
3.11. BMO 1975..........................................................................................................137
3.12. BMO 1976..........................................................................................................138
3.13. BMO 1977..........................................................................................................139
3.14. BMO 1978..........................................................................................................140
3.15. BMO 1979..........................................................................................................141
3.16. BMO 1980..........................................................................................................142
3.17. BMO 1981..........................................................................................................143
3.18. BMO 1982..........................................................................................................144
3.19. BMO 1983..........................................................................................................145
3.20. BMO 1984..........................................................................................................146
3.21. BMO 1985..........................................................................................................147
3.22. BMO 1986..........................................................................................................148
3.23. BMO 1987..........................................................................................................149
3.24. BMO 1988..........................................................................................................150
3.25. BMO 1989..........................................................................................................151
3.26. BMO 1990..........................................................................................................152
3.27. BMO 1991..........................................................................................................153
3.28. BMO 1992..........................................................................................................154
3.29. BMO 1993..........................................................................................................155
3.30. BMO 1994..........................................................................................................156
3.31. BMO 1995..........................................................................................................157
3.32. BMO 1996..........................................................................................................158

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3.33. BMO 1997..........................................................................................................159
3.34. BMO 1998..........................................................................................................160
3.35. BMO 1999..........................................................................................................161
3.36. BMO 2000..........................................................................................................162
3.37. BMO 2001..........................................................................................................163
3.38. BMO 2002..........................................................................................................164
3.39. BMO 2003..........................................................................................................165
3.40. BMO 2004..........................................................................................................166
4. Brasil (1979 – 2003) ....................................................................................................167
4.1. Brasil 1979...........................................................................................................168
4.2. Brasil 1980...........................................................................................................169
4.3. Brasil 1981...........................................................................................................170
4.4. Brasil 1982...........................................................................................................171
4.5. Brasil 1983...........................................................................................................172
4.6. Brasil 1984...........................................................................................................173
4.7. Brasil 1985...........................................................................................................174
4.8. Brasil 1986...........................................................................................................175
4.9. Brasil 1987...........................................................................................................176
4.10. Brasil 1988.........................................................................................................177
4.11. Brasil 1989.........................................................................................................178
4.12. Brasil 1990.........................................................................................................179
4.13. Brasil 1991.........................................................................................................180
4.14. Brasil 1992.........................................................................................................181
4.15. Brasil 1993.........................................................................................................182
4.16. Brasil 1994.........................................................................................................183
4.17. Brasil 1995.........................................................................................................184
4.18. Brasil 1996.........................................................................................................185
4.19. Brasil 1997.........................................................................................................186
4.20. Brasil 1998.........................................................................................................187
4.21. Brasil 1999.........................................................................................................188
4.22. Brasil 2000.........................................................................................................189
4.23. Brasil 2001.........................................................................................................190
4.24. Brasil 2002.........................................................................................................191
4.25. Brasil 2003.........................................................................................................192
5. CanMO (1969 – 2003) .................................................................................................193
5.1. CanMO 1969........................................................................................................194
5.2. CanMO 1970........................................................................................................195
5.3. CanMO 1971........................................................................................................196
5.4. CanMO 1972........................................................................................................197
5.5. CanMO 1973........................................................................................................198
5.6. CanMO 1974........................................................................................................199
5.7. CanMO 1975........................................................................................................200
5.8. CanMO 1976........................................................................................................201
5.9. CanMO 1977........................................................................................................202
5.10. CanMO 1978......................................................................................................203
5.11. CanMO 1979......................................................................................................204
5.12. CanMO 1980......................................................................................................205
5.13. CanMO 1981......................................................................................................206
5.14. CanMO 1982......................................................................................................207
5.15. CanMO 1983......................................................................................................208
5.16. CanMO 1984......................................................................................................209
5.17. CanMO 1985......................................................................................................210

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☺ The best problems from around the world Cao Minh Quang
5.18. CanMO 1986......................................................................................................211
5.19. CanMO 1987......................................................................................................212
5.20. CanMO 1988......................................................................................................213
5.21. CanMO 1989......................................................................................................214
5.22. CanMO 1990......................................................................................................215
5.23. CanMO 1991......................................................................................................216
5.24. CanMO 1992......................................................................................................217
5.25. CanMO 1993......................................................................................................218
5.26. CanMO 1994......................................................................................................219
5.27. CanMO 1995......................................................................................................220
5.28. CanMO 1996......................................................................................................221
5.29. CanMO 1997......................................................................................................222
5.30. CanMO 1998......................................................................................................223
5.31. CanMO 1999......................................................................................................224
5.32. CanMO 2000......................................................................................................225
5.33. CanMO 2001......................................................................................................226
5.34. CanMO 2002......................................................................................................227
5.35. CanMO 2003......................................................................................................228
6. Eötvös Competition (1894 – 2004) ..............................................................................229
6.1. Eötvös Competition 1894.....................................................................................230
6.2. Eötvös Competition 1895.....................................................................................230
6.3. Eötvös Competition 1896.....................................................................................230
6.4. Eötvös Competition 1897.....................................................................................230
6.5. Eötvös Competition 1898.....................................................................................231
6.6. Eötvös Competition 1899.....................................................................................231
6.7. Eötvös Competition 1900.....................................................................................231
6.8. Eötvös Competition 1901.....................................................................................231
6.9. Eötvös Competition 1902.....................................................................................232
6.10. Eötvös Competition 1903...................................................................................232
6.11. Eötvös Competition 1904...................................................................................232
6.12. Eötvös Competition 1905...................................................................................232
6.13. Eötvös Competition 1906...................................................................................233
6.14. Eötvös Competition 1907...................................................................................233
6.15. Eötvös Competition 1908...................................................................................233
6.16. Eötvös Competition 1909...................................................................................233
6.17. Eötvös Competition 1910...................................................................................234
6.18. Eötvös Competition 1911...................................................................................234
6.19. Eötvös Competition 1912...................................................................................234
6.20. Eötvös Competition 1913...................................................................................234
6.21. Eötvös Competition 1914...................................................................................235
6.22. Eötvös Competition 1915...................................................................................235
6.23. Eötvös Competition 1916...................................................................................235
6.24. Eötvös Competition 1917...................................................................................235
6.25. Eötvös Competition 1918...................................................................................236
6.26. Eötvös Competition 1922...................................................................................236
6.27. Eötvös Competition 1923...................................................................................236
6.28. Eötvös Competition 1924...................................................................................236
6.29. Eötvös Competition 1925...................................................................................237
6.30. Eötvös Competition 1926...................................................................................237
6.31. Eötvös Competition 1927...................................................................................237
6.32. Eötvös Competition 1928...................................................................................237
6.33. Eötvös Competition 1929...................................................................................238

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☺ The best problems from around the world Cao Minh Quang
6.34. Eötvös Competition 1930...................................................................................238
6.35. Eötvös Competition 1931...................................................................................238
6.36. Eötvös Competition 1932...................................................................................238
6.37. Eötvös Competition 1933...................................................................................239
6.38. Eötvös Competition 1934...................................................................................239
6.39. Eötvös Competition 1935...................................................................................239
6.40. Eötvös Competition 1936...................................................................................240
6.41. Eötvös Competition 1937...................................................................................240
6.42. Eötvös Competition 1938...................................................................................240
6.43. Eötvös Competition 1939...................................................................................240
6.44. Eötvös Competition 1940...................................................................................241
6.45. Eötvös Competition 1941...................................................................................241
6.46. Eötvös Competition 1942...................................................................................241
6.47. Eötvös Competition 1943...................................................................................242
6.48. Eötvös Competition 1947...................................................................................242
6.49. Eötvös Competition 1948...................................................................................242
6.50. Eötvös Competition 1949...................................................................................242
6.51. Eötvös Competition 1950...................................................................................243
6.52. Eötvös Competition 1951...................................................................................243
6.53. Eötvös Competition 1952...................................................................................243
6.54. Eötvös Competition 1953...................................................................................244
6.55. Eötvös Competition 1954...................................................................................244
6.56. Eötvös Competition 1955...................................................................................244
6.57. Eötvös Competition 1957...................................................................................244
6.58. Eötvös Competition 1958...................................................................................245
6.59. Eötvös Competition 1959...................................................................................245
6.60. Eötvös Competition 1960...................................................................................245
6.61. Eötvös Competition 1961...................................................................................246
6.62. Eötvös Competition 1962...................................................................................246
6.63. Eötvös Competition 1963...................................................................................246
6.64. Eötvös Competition 1964...................................................................................247
6.65. Eötvös Competition 1965...................................................................................247
6.66. Eötvös Competition 1966...................................................................................247
6.67. Eötvös Competition 1967...................................................................................248
6.68. Eötvös Competition 1968...................................................................................248
6.69. Eötvös Competition 1969...................................................................................248
6.70. Eötvös Competition 1970...................................................................................249
6.71. Eötvös Competition 1971...................................................................................249
6.72. Eötvös Competition 1972...................................................................................249
6.73. Eötvös Competition 1973...................................................................................250
6.74. Eötvös Competition 1974...................................................................................250
6.75. Eötvös Competition 1975...................................................................................250
6.76. Eötvös Competition 1976...................................................................................251
6.77. Eötvös Competition 1977...................................................................................251
6.78. Eötvös Competition 1978...................................................................................251
6.79. Eötvös Competition 1979...................................................................................252
6.80. Eötvös Competition 1980...................................................................................252
6.81. Eötvös Competition 1981...................................................................................252
6.82. Eötvös Competition 1982...................................................................................253
6.83. Eötvös Competition 1983...................................................................................253
6.84. Eötvös Competition 1984...................................................................................253
6.85. Eötvös Competition 1985...................................................................................254

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☺ The best problems from around the world Cao Minh Quang
6.86. Eötvös Competition 1986...................................................................................254
6.87. Eötvös Competition 1987...................................................................................254
6.88. Eötvös Competition 1988...................................................................................255
6.89. Eötvös Competition 1989...................................................................................255
6.90. Eötvös Competition 1990...................................................................................255
6.91. Eötvös Competition 1991...................................................................................256
6.92. Eötvös Competition 1992...................................................................................256
6.93. Eötvös Competition 1993...................................................................................256
6.94. Eötvös Competition 1994...................................................................................257
6.95. Eötvös Competition 1995...................................................................................257
6.96. Eötvös Competition 1996...................................................................................257
6.97. Eötvös Competition 1997...................................................................................258
6.98. Eötvös Competition 1998...................................................................................258
6.99. Eötvös Competition 1999...................................................................................258
6.100. Eötvös Competition 2000.................................................................................258
6.101. Eötvös Competition 2001.................................................................................259
6.102. Eötvös Competition 2002.................................................................................259
7. INMO (1995 – 2004) ...................................................................................................260
7.1. INMO 1995..........................................................................................................261
7.2. INMO 1996..........................................................................................................262
7.3. INMO 1997..........................................................................................................263
7.4. INMO 1998..........................................................................................................264
7.5. INMO 1999..........................................................................................................265
7.6. INMO 2000..........................................................................................................266
7.7. INMO 2001..........................................................................................................267
7.8. INMO 2002..........................................................................................................268
7.9. INMO 2003..........................................................................................................269
7.10. INMO 2004........................................................................................................270
8. Irish (1988 – 2003) ......................................................................................................271
8.1. Irish 1988..............................................................................................................272
8.2. Irish 1989..............................................................................................................273
8.3. Irish 1990..............................................................................................................274
8.4. Irish 1991..............................................................................................................275
8.5. Irish 1992..............................................................................................................276
8.6. Irish 1993..............................................................................................................277
8.7. Irish 1994..............................................................................................................278
8.8. Irish 1995..............................................................................................................279
8.9. Irish 1996..............................................................................................................280
8.10. Irish 1997............................................................................................................281
8.11. Irish 1998............................................................................................................282
8.12. Irish 1999............................................................................................................283
8.13. Irish 2000............................................................................................................284
8.14. Irish 2001............................................................................................................285
8.15. Irish 2002............................................................................................................286
8.16. Irish 2003............................................................................................................287
9. Mexican (1987 – 2003) ................................................................................................288
9.1. Mexican 1987.......................................................................................................289
9.2. Mexican 1988.......................................................................................................290
9.3. Mexican 1989.......................................................................................................291
9.4. Mexican 1990.......................................................................................................292
9.5. Mexican 1991.......................................................................................................293
9.6. Mexican 1992.......................................................................................................294

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9.7. Mexican 1993.......................................................................................................295
9.8. Mexican 1994.......................................................................................................296
9.9. Mexican 1995.......................................................................................................297
9.10. Mexican 1996.....................................................................................................298
9.11. Mexican 1997.....................................................................................................299
9.12. Mexican 1998.....................................................................................................300
9.13. Mexican 1999.....................................................................................................301
9.14. Mexican 2000.....................................................................................................302
9.15. Mexican 2001.....................................................................................................303
9.16. Mexican 2003.....................................................................................................304
9.17. Mexican 2004.....................................................................................................305
10. Polish (1983 – 2003) ..................................................................................................306
10.1. Polish 1983.........................................................................................................307
10.2. Polish 1984.........................................................................................................308
10.3. Polish 1985.........................................................................................................309
10.4. Polish 1986.........................................................................................................310
10.5. Polish 1987.........................................................................................................311
10.6. Polish 1988.........................................................................................................312
10.7. Polish 1989.........................................................................................................313
10.8. Polish 1990.........................................................................................................314
10.9. Polish 1991.........................................................................................................315
10.10. Polish 1992.......................................................................................................316
10.11. Polish 1993.......................................................................................................317
10.12. Polish 1994.......................................................................................................318
10.13. Polish 1995.......................................................................................................319
10.14. Polish 1996.......................................................................................................320
10.15. Polish 1997.......................................................................................................321
10.16. Polish 1998.......................................................................................................322
10.17. Polish 1999.......................................................................................................323
10.18. Polish 2000.......................................................................................................324
10.19. Polish 2001.......................................................................................................325
10.20. Polish 2002.......................................................................................................326
10.21. Polish 2003.......................................................................................................327
11. Spanish (1990 – 2003) ...............................................................................................328
11.1. Spanish 1990......................................................................................................329
11.2. Spanish 1991......................................................................................................330
11.3. Spanish 1992......................................................................................................331
11.4. Spanish 1993......................................................................................................332
11.5. Spanish 1994......................................................................................................333
11.6. Spanish 1995......................................................................................................334
11.7. Spanish 1996......................................................................................................335
11.8. Spanish 1997......................................................................................................336
11.9. Spanish 1998......................................................................................................337
11.10. Spanish 1999....................................................................................................338
11.11. Spanish 2000....................................................................................................339
11.12. Spanish 2001....................................................................................................340
11.13. Spanish 2002....................................................................................................341
11.14. Spanish 2003....................................................................................................342
12. Swedish (1961 – 2003) ...............................................................................................343
12.1. Swedish 1961.....................................................................................................344
12.2. Swedish 1962.....................................................................................................345
12.3. Swedish 1963.....................................................................................................346

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☺ The best problems from around the world Cao Minh Quang
12.4. Swedish 1964.....................................................................................................347
12.5. Swedish 1965.....................................................................................................348
12.6. Swedish 1966.....................................................................................................349
12.7. Swedish 1967.....................................................................................................350
12.8. Swedish 1968.....................................................................................................351
12.9. Swedish 1969.....................................................................................................352
12.10. Swedish 1970....................................................................................................353
12.11. Swedish 1971....................................................................................................354
12.12. Swedish 1972....................................................................................................355
12.13. Swedish 1973....................................................................................................356
12.14. Swedish 1974....................................................................................................357
12.15. Swedish 1975....................................................................................................358
12.16. Swedish 1976....................................................................................................359
12.17. Swedish 1977....................................................................................................360
12.18. Swedish 1978....................................................................................................361
12.19. Swedish 1979....................................................................................................362
12.20. Swedish 980.....................................................................................................363
12.21. Swedish 1981....................................................................................................364
12.22. Swedish 1982....................................................................................................365
12.23. Swedish 1983....................................................................................................366
12.24. Swedish 1984....................................................................................................367
12.25. Swedish 1985....................................................................................................368
12.26. Swedish 1986....................................................................................................369
12.27. Swedish 1987....................................................................................................370
12.28. Swedish 1988....................................................................................................371
12.29. Swedish 1989....................................................................................................372
12.30. Swedish 1990....................................................................................................373
12.31. Swedish 1991....................................................................................................374
12.32. Swedish 1992....................................................................................................375
12.33. Swedish 1993....................................................................................................376
12.34. Swedish 1994....................................................................................................377
12.35. Swedish 1995....................................................................................................378
12.36. Swedish 1996....................................................................................................379
12.37. Swedish 1997....................................................................................................380
12.38. Swedish 1998....................................................................................................381
12.39. Swedish 1999....................................................................................................382
12.40. Swedish 2000....................................................................................................383
12.41. Swedish 2001....................................................................................................384
12.42. Swedish 2002....................................................................................................385
12.43. Swedish 2003....................................................................................................386
13. USAMO (1972 – 2003) ..............................................................................................387
13.1. USAMO 1972.....................................................................................................388
13.2. USAMO 1973.....................................................................................................389
13.3. USAMO 1974.....................................................................................................390
13.4. USAMO 1975.....................................................................................................391
13.5. USAMO 1976.....................................................................................................392
13.6. USAMO 1977.....................................................................................................393
13.7. USAMO 1978.....................................................................................................394
13.8. USAMO 1979.....................................................................................................395
13.9. USAMO 1980.....................................................................................................396
13.10. USAMO 1981...................................................................................................397
13.11. USAMO 1982...................................................................................................398

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