Tuyển tập các đề thi Toán của các nước trên Thế Giới phần 1

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Tuyển tập các đề thi Toán của các nước trên Thế Giới phần 1

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Kì thi học sinh giỏi là kì thi quan trọng đối với mỗi học sinh. Dưới đây là tuyển tập các đề thi Toán của các nước trên Thế giới phần 1 giúp các em kiểm tra lại đánh giá kiến thức của mình và có thêm thời gian chuẩn bị ôn tập cho kì thi sắp tới được tốt hơn.

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Nội dung Text: Tuyển tập các đề thi Toán của các nước trên Thế Giới phần 1

  1. TRƯỜNG ................ KHOA……… …………..o0o………….. Tuyển tập các đề thi của các nước trên thế giới P1 - Cao Minh Quang
  2. 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 10A2 in the school year 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: kt13quang@yahoo.com Vinh Long, April 2006 Cao Minh Quang
  3. ☺ 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 2
  4. ☺ 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 3
  5. ☺ 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 4
  6. ☺ The best problems from around the world Cao Minh Quang 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 5
  7. ☺ 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 6
  8. ☺ 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 7
  9. ☺ 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 8
  10. ☺ The best problems from around the world Cao Minh Quang 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 9
  11. ☺ 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 10
  12. ☺ The best problems from around the world Cao Minh Quang 13.12. USAMO 1983................................................................................................... 399 13.13. USAMO 1984................................................................................................... 400 13.14. USAMO 1985................................................................................................... 401 13.15. USAMO 1986................................................................................................... 402 13.16. USAMO 1987................................................................................................... 403 13.17. USAMO 1988................................................................................................... 404 13.18. USAMO 1989................................................................................................... 405 13.19. USAMO 1990................................................................................................... 406 13.20. USAMO 1991................................................................................................... 407 13.21. USAMO 1992................................................................................................... 408 13.22. USAMO 1993................................................................................................... 409 13.23. USAMO 1994................................................................................................... 410 13.24. USAMO 1995................................................................................................... 411 13.25. USAMO 1996................................................................................................... 412 13.26. USAMO 1997................................................................................................... 413 13.27. USAMO 1998................................................................................................... 414 13.28. USAMO 1999................................................................................................... 415 13.29. USAMO 2000................................................................................................... 416 13.30. USAMO 2001................................................................................................... 417 13.31. USAMO 2002................................................................................................... 418 13.32. USAMO 2003................................................................................................... 419 14. Vietnam (1962 – 2003) .............................................................................................. 420 14.1. Vietnam 1962 ..................................................................................................... 421 14.2. Vietnam 1963 ..................................................................................................... 422 14.3. Vietnam 1964 ..................................................................................................... 423 14.4. Vietnam 1965 ..................................................................................................... 424 14.5. Vietnam 1966 ..................................................................................................... 425 14.6. Vietnam 1967 ..................................................................................................... 426 14.7. Vietnam 1968 ..................................................................................................... 427 14.8. Vietnam 1969 ..................................................................................................... 428 14.9. Vietnam 1970 ..................................................................................................... 429 14.10. Vietnam 1971 ................................................................................................... 430 14.11. Vietnam 1972 ................................................................................................... 431 14.12. Vietnam 1974 ................................................................................................... 432 14.13. Vietnam 1975 ................................................................................................... 433 14.14. Vietnam 1976 ................................................................................................... 434 14.15. Vietnam 1977 ................................................................................................... 435 14.16. Vietnam 1978 ................................................................................................... 436 14.17. Vietnam 1979 ................................................................................................... 437 14.18. Vietnam 1980 ................................................................................................... 438 14.19. Vietnam 1981 ................................................................................................... 439 14.20. Vietnam 1982 ................................................................................................... 440 14.21. Vietnam 1983 ................................................................................................... 441 14.22. Vietnam 1984 ................................................................................................... 442 14.23. Vietnam 1985 ................................................................................................... 443 14.24. Vietnam 1986 ................................................................................................... 444 14.25. Vietnam 1987 ................................................................................................... 445 14.26. Vietnam 1988 ................................................................................................... 446 14.27. Vietnam 1989 ................................................................................................... 447 14.28. Vietnam 1990 ................................................................................................... 448 14.29. Vietnam 1991 ................................................................................................... 449 14.30. Vietnam 1992 ................................................................................................... 450 11
  13. ☺ The best problems from around the world Cao Minh Quang 14.31. Vietnam 1993 ................................................................................................... 451 14.32. Vietnam 1994 ................................................................................................... 452 14.33. Vietnam 1995 ................................................................................................... 453 14.34. Vietnam 1996 ................................................................................................... 454 14.35. Vietnam 1997 ................................................................................................... 455 14.36. Vietnam 1998 ................................................................................................... 456 14.37. Vietnam 1999 ................................................................................................... 457 14.38. Vietnam 2000 ................................................................................................... 458 14.39. Vietnam 2001 ................................................................................................... 459 14.40. Vietnam 2002 ................................................................................................... 460 14.41. Vietnam 2003 ................................................................................................... 461 PART II. International/Regional Olympiad problems ................................................ 462 15. Iberoamerican (1985 – 2003) .................................................................................... 462 15.1. Iberoamerican 1985 ............................................................................................ 463 15.2. Iberoamerican 1987 ............................................................................................ 464 15.3. Iberoamerican 1988 ............................................................................................ 465 15.4. Iberoamerican 1989 ............................................................................................ 466 15.5. Iberoamerican 1990 ............................................................................................ 467 15.6. Iberoamerican 1991 ............................................................................................ 468 15.7. Iberoamerican 1992 ............................................................................................ 469 15.8. Iberoamerican 1993 ............................................................................................ 470 15.9. Iberoamerican 1994 ............................................................................................ 471 15.10. Iberoamerican 1995 .......................................................................................... 472 15.11. Iberoamerican 1996 .......................................................................................... 473 15.12. Iberoamerican 1997 .......................................................................................... 474 15.13. Iberoamerican 1998 .......................................................................................... 475 15.14. Iberoamerican 1999 .......................................................................................... 466 15.15. Iberoamerican 2000 .......................................................................................... 477 15.16. Iberoamerican 2001 .......................................................................................... 478 15.17. Iberoamerican 2002 .......................................................................................... 479 15.18. Iberoamerican 2003 .......................................................................................... 480 16. Balkan (1984 – 2003) ................................................................................................ 481 16.1. Balkan 1984........................................................................................................ 482 16.2. Balkan 1985........................................................................................................ 483 16.3. Balkan 1986........................................................................................................ 484 16.4. Balkan 1987........................................................................................................ 485 16.5. Balkan 1988........................................................................................................ 486 16.6. Balkan 1989........................................................................................................ 487 16.7. Balkan 1990........................................................................................................ 488 16.8. Balkan 1991........................................................................................................ 489 16.9. Balkan 1992........................................................................................................ 490 16.10. Balkan 1993...................................................................................................... 491 16.11. Balkan 1994...................................................................................................... 492 16.12. Balkan 1995...................................................................................................... 493 16.13. Balkan 1996...................................................................................................... 494 16.14. Balkan 1997...................................................................................................... 495 16.15. Balkan 1998...................................................................................................... 496 16.16. Balkan 1999...................................................................................................... 497 16.17. Balkan 2000...................................................................................................... 498 16.18. Balkan 2001...................................................................................................... 499 16.19. Balkan 2002...................................................................................................... 500 16.20. Balkan 2003...................................................................................................... 501 12
  14. ☺ The best problems from around the world Cao Minh Quang 17. Austrian – Polish (1978 – 2003) ................................................................................ 502 17.1. Austrian – Polish 1978 ....................................................................................... 503 17.2. Austrian – Polish 1979 ....................................................................................... 504 17.3. Austrian – Polish 1980 ....................................................................................... 505 17.4. Austrian – Polish 1981 ....................................................................................... 506 17.5. Austrian – Polish 1982 ....................................................................................... 507 17.6. Austrian – Polish 1983 ....................................................................................... 508 17.7. Austrian – Polish 1984 ....................................................................................... 509 17.8. Austrian – Polish 1985 ....................................................................................... 510 17.9. Austrian – Polish 1986 ....................................................................................... 511 17.10. Austrian – Polish 1987 ..................................................................................... 512 17.11. Austrian – Polish 1988 ..................................................................................... 513 17.12. Austrian – Polish 1989 ..................................................................................... 514 17.13. Austrian – Polish 1990 ..................................................................................... 515 17.14. Austrian – Polish 1991 ..................................................................................... 516 17.15. Austrian – Polish 1992 ..................................................................................... 517 17.16. Austrian – Polish 1993 ..................................................................................... 518 17.17. Austrian – Polish 1994 ..................................................................................... 519 17.18. Austrian – Polish 1995 ..................................................................................... 520 17.19. Austrian – Polish 1996 ..................................................................................... 521 17.20. Austrian – Polish 1997 ..................................................................................... 522 17.21. Austrian – Polish 1998 ..................................................................................... 523 17.22. Austrian – Polish 1999 ..................................................................................... 524 17.23. Austrian – Polish 2000 ..................................................................................... 525 17.24. Austrian – Polish 2001 ..................................................................................... 526 17.25. Austrian – Polish 2002 ..................................................................................... 527 17.26. Austrian – Polish 2003 ..................................................................................... 528 18. APMO (1989 – 2004) ................................................................................................. 529 18.1. APMO 1989 ....................................................................................................... 530 18.2. APMO 1990 ....................................................................................................... 531 18.3. APMO 1991 ....................................................................................................... 532 18.4. APMO 1992 ....................................................................................................... 533 18.5. APMO 1993 ....................................................................................................... 534 18.6. APMO 1994 ....................................................................................................... 535 18.7. APMO 1995 ....................................................................................................... 536 18.8. APMO 1996 ....................................................................................................... 537 18.9. APMO 1997 ....................................................................................................... 538 18.10. APMO 1998 ..................................................................................................... 539 18.11. APMO 1999 ..................................................................................................... 540 18.12. APMO 2000 ..................................................................................................... 541 18.13. APMO 2001 ..................................................................................................... 542 18.14. APMO 2002 ..................................................................................................... 543 18.15. APMO 2003 ..................................................................................................... 544 18.16. APMO 2004 ..................................................................................................... 545 19. IMO (1959 – 2003) .................................................................................................... 546 19.1. IMO 1959 ........................................................................................................... 547 19.2. IMO 1960 ........................................................................................................... 548 19.3. IMO 1961 ........................................................................................................... 549 19.4. IMO 1962 ........................................................................................................... 550 19.5. IMO 1963 ........................................................................................................... 551 19.6. IMO 1964 ........................................................................................................... 552 19.7. IMO 1965 ........................................................................................................... 553 13
  15. ☺ The best problems from around the world Cao Minh Quang 19.8. IMO 1966 ........................................................................................................... 554 19.9. IMO 1967 ........................................................................................................... 555 19.10. IMO 1968 ......................................................................................................... 556 19.11. IMO 1969 ......................................................................................................... 557 19.12. IMO 1970 ......................................................................................................... 558 19.13. IMO 1971 ......................................................................................................... 559 19.14. IMO 1972 ......................................................................................................... 560 19.15. IMO 1973 ......................................................................................................... 561 19.16. IMO 1974 ......................................................................................................... 562 19.17. IMO 1975 ......................................................................................................... 563 19.18. IMO 1976 ......................................................................................................... 564 19.19. IMO 1977 ......................................................................................................... 565 19.20. IMO 1978 ......................................................................................................... 566 19.21. IMO 1979 ......................................................................................................... 567 19.22. IMO 1981 ......................................................................................................... 568 19.23. IMO 1982 ......................................................................................................... 569 19.24. IMO 1983 ......................................................................................................... 570 19.25. IMO 1984 ......................................................................................................... 571 19.26. IMO 1985 ......................................................................................................... 572 19.27. IMO 1986 ......................................................................................................... 573 19.28. IMO 1987 ......................................................................................................... 574 19.29. IMO 1988 ......................................................................................................... 575 19.30. IMO 1989 ......................................................................................................... 576 19.31. IMO 1990 ......................................................................................................... 577 19.32. IMO 1991 ......................................................................................................... 578 19.33. IMO 1992 ......................................................................................................... 579 19.34. IMO 1993 ......................................................................................................... 580 19.35. IMO 1994 ......................................................................................................... 581 19.36. IMO 1995 ......................................................................................................... 582 19.37. IMO 1996 ......................................................................................................... 583 19.38. IMO 1997 ......................................................................................................... 584 19.39. IMO 1998 ......................................................................................................... 585 19.40. IMO 1999 ......................................................................................................... 586 19.41. IMO 2000 ......................................................................................................... 587 19.42. IMO 2001 ......................................................................................................... 588 19.43. IMO 2002 ......................................................................................................... 589 19.44. IMO 2003 ......................................................................................................... 590 20. Junior Balkan (1997 – 2003) .................................................................................... 591 20.1. Junior Balkan 1997............................................................................................. 592 20.2. Junior Balkan 1998............................................................................................. 593 20.3. Junior Balkan 1999............................................................................................. 594 20.4. Junior Balkan 2000............................................................................................. 595 20.5. Junior Balkan 2001............................................................................................. 596 20.6. Junior Balkan 2002............................................................................................. 597 20.7. Junior Balkan 2003............................................................................................. 598 21. Shortlist IMO (1959 – 2002) ..................................................................................... 599 21.1. Shortlist IMO 1959 – 1967 ................................................................................ 600 21.2. Shortlist IMO 1981............................................................................................. 602 21.3. Shortlist IMO 1982............................................................................................. 603 21.4. Shortlist IMO 1983............................................................................................. 604 21.5. Shortlist IMO 1984............................................................................................. 606 21.6. Shortlist IMO 1985............................................................................................. 608 14
  16. ☺ The best problems from around the world Cao Minh Quang 21.7. Shortlist IMO 1986............................................................................................. 610 21.8. Shortlist IMO 1987............................................................................................. 612 21.9. Shortlist IMO 1988............................................................................................. 614 21.10. Shortlist IMO 1989........................................................................................... 616 21.11. Shortlist IMO 1990........................................................................................... 618 21.12. Shortlist IMO 1991........................................................................................... 620 21.13. Shortlist IMO 1992........................................................................................... 623 21.14. Shortlist IMO 1993........................................................................................... 624 21.15. Shortlist IMO 1994........................................................................................... 626 21.16. Shortlist IMO 1995........................................................................................... 628 21.17. Shortlist IMO 1996........................................................................................... 630 21.18. Shortlist IMO 1997........................................................................................... 632 21.19. Shortlist IMO 1998........................................................................................... 634 21.20. Shortlist IMO 1999........................................................................................... 636 21.21. Shortlist IMO 2000........................................................................................... 638 21.22. Shortlist IMO 2001........................................................................................... 641 21.22. Shortlist IMO 2002........................................................................................... 643 22. OMCC (1999 – 2003) ................................................................................................. 645 22.1. OMCC 1999 ....................................................................................................... 646 22.2. OMCC 2000 ....................................................................................................... 647 22.3. OMCC 2001 ....................................................................................................... 648 22.4. OMCC 2002 ....................................................................................................... 649 22.5. OMCC 2003 ....................................................................................................... 650 23. PUTNAM (1938 – 2003) ............................................................................................ 651 23.1. PUTNAM 1938 .................................................................................................. 652 23.2. PUTNAM 1939 .................................................................................................. 654 23.3. PUTNAM 1940 .................................................................................................. 656 23.4. PUTNAM 1941 .................................................................................................. 657 23.5. PUTNAM 1942 .................................................................................................. 659 23.6. PUTNAM 1946 .................................................................................................. 660 23.7. PUTNAM 1947 .................................................................................................. 661 23.8. PUTNAM 1948 .................................................................................................. 662 23.9. PUTNAM 1949 .................................................................................................. 663 23.10. PUTNAM 1950 ................................................................................................ 664 23.11. PUTNAM 1951 ................................................................................................ 666 23.12. PUTNAM 1952 ................................................................................................ 667 23.13. PUTNAM 1953 ................................................................................................ 668 23.14. PUTNAM 1954 ................................................................................................ 669 23.15. PUTNAM 1955 ................................................................................................ 670 23.16. PUTNAM 1956 ................................................................................................ 671 23.17. PUTNAM 1957 ................................................................................................ 672 23.18. PUTNAM 1958 ................................................................................................ 673 23.19. PUTNAM 1959 ................................................................................................ 675 23.20. PUTNAM 1960 ................................................................................................ 677 23.21. PUTNAM 1961 ................................................................................................ 678 23.22. PUTNAM 1962 ................................................................................................ 679 23.23. PUTNAM 1963 ................................................................................................ 680 23.24. PUTNAM 1964 ................................................................................................ 681 23.25. PUTNAM 1965 ................................................................................................ 682 23.26. PUTNAM 1966 ................................................................................................ 683 23.27. PUTNAM 1967 ................................................................................................ 684 23.28. PUTNAM 1968 ................................................................................................ 685 15
  17. ☺ The best problems from around the world Cao Minh Quang 23.29. PUTNAM 1969 ................................................................................................ 686 23.30. PUTNAM 1970 ................................................................................................ 687 23.31. PUTNAM 1971 ................................................................................................ 688 23.32. PUTNAM 1972 ................................................................................................ 689 23.33. PUTNAM 1973 ................................................................................................ 690 23.34. PUTNAM 1974 ................................................................................................ 691 23.35. PUTNAM 1975 ................................................................................................ 692 23.36. PUTNAM 1976 ................................................................................................ 693 23.37. PUTNAM 1977 ................................................................................................ 694 23.38. PUTNAM 1978 ................................................................................................ 695 23.39. PUTNAM 1979 ................................................................................................ 696 23.40. PUTNAM 1980 ................................................................................................ 697 23.41. PUTNAM 1981 ................................................................................................ 698 23.42. PUTNAM 1982 ................................................................................................ 699 23.43. PUTNAM 1983 ................................................................................................ 700 23.44. PUTNAM 1984 ................................................................................................ 701 23.45. PUTNAM 1985 ................................................................................................ 702 23.46. PUTNAM 1986 ................................................................................................ 703 23.47. PUTNAM 1987 ................................................................................................ 704 23.48. PUTNAM 1988 ................................................................................................ 705 23.49. PUTNAM 1989 ................................................................................................ 706 23.50. PUTNAM 1990 ................................................................................................ 707 23.51. PUTNAM 1991 ................................................................................................ 708 23.52. PUTNAM 1992 ................................................................................................ 709 23.53. PUTNAM 1993 ................................................................................................ 710 23.54. PUTNAM 1994 ................................................................................................ 711 23.55. PUTNAM 1995 ................................................................................................ 712 23.56. PUTNAM 1996 ................................................................................................ 713 23.57. PUTNAM 1997 ................................................................................................ 714 23.58. PUTNAM 1998 ................................................................................................ 715 23.59. PUTNAM 1999 ................................................................................................ 716 23.60. PUTNAM 2000 ................................................................................................ 717 23.61. PUTNAM 2001 ................................................................................................ 718 23.62. PUTNAM 2002 ................................................................................................ 719 23.63. PUTNAM 2003 ................................................................................................ 720 24. Seminar (1 – 109) ...................................................................................................... 721 References ......................................................................................................................... 729 16
  18. ☺ The best problems from around the world Cao Minh Quang PART I. National Olympiads AIME (1983 – 2004) 17
  19. ☺ The best problems from around the world Cao Minh Quang 1st AIME 1983 1. x, y, z are real numbers greater than 1 and w is a positive real number. If logxw = 24, logyw = 40 and logxyzw = 12, find logzw. 2. Find the minimum value of |x - p| + |x - 15| + |x - p - 15| for x in the range p ≤ x ≤ 15, where 0 < p < 15. 3. Find the product of the real roots of the equation x2 + 18x + 30 = 2 √(x2 + 18x + 45). 4. A and C lie on a circle center O with radius √50. The point B inside the circle is such that ∠ ABC = 90o, AB = 6, BC = 2. Find OB. 5. w and z are complex numbers such that w2 + z2 = 7, w3 + z3 = 10. What is the largest possible real value of w + z? 6. What is the remainder on dividing 683 + 883 by 49? 7. 25 knights are seated at a round table and 3 are chosen at random. Find the probability that at least two of the chosen 3 are sitting next to each other. 8. What is the largest 2-digit prime factor of the binomial coefficient 200C100? 9. Find the minimum value of (9x2sin2x + 4)/(x sin x) for 0 < x < π. 10. How many 4 digit numbers with first digit 1 have exactly two identical digits (like 1447, 1005 or 1231)? 11. ABCD is a square side 6√2. EF is parallel to the square and has length 12√2. The faces BCF and ADE are equilateral. What is the volume of the solid ABCDEF? 12. The chord CD is perpendicular to the diameter AB and meets it at H. The distances AB and CD are integral. The distance AB has 2 digits and the distance CD is obtained by reversing the digits of AB. The distance OH is a non-zero rational. Find AB. 18
  20. ☺ The best problems from around the world Cao Minh Quang 13. For each non-empty subset of {1, 2, 3, 4, 5, 6, 7} arrange the members in decreasing order with alternating signs and take the sum. For example, for the subset {5} we get 5. For {6, 3, 1} we get 6 - 3 + 1 = 4. Find the sum of all the resulting numbers. 14. The distance AB is 12. The circle center A radius 8 and the circle center B radius 6 meet at P (and another point). A line through P meets the circles again at Q and R (with Q on the larger circle), so that QP = PR. Find QP2. 15. BC is a chord length 6 of a circle center O radius 5. A is a point on the circle closer to B than C such that there is just one chord AD which is bisected by BC. Find sin AOB. 19
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