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PREFACE Collecting the Mathematics tests from the contests choosing the best students is not only my favorite interest but also many different people’s. This selected book is an adequate collection of the Math tests in the Mathematical Olympiads tests from 14 countries, from different regions and from the International Mathematical Olympiads tests as well. I had a lot of effort to finish this book. Besides, I’m also grateful to all students who gave me much support in my collection. They include students in class 11 of specialized Chemistry – Biologry, class 10 specialized Mathematics and class 10A 2 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 ☺ The best problems from around the world Cao Minh Quan g 2 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 ☺ The best problems from around the world Cao Minh Quan g 3 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 ☺ The best problems from around the world Cao Minh Quan g 4 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 ☺ The best problems from around the world Cao Minh Quan g 5 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 ☺ The best problems from around the world Cao Minh Quan g 6 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 ☺ The best problems from around the world Cao Minh Quan g 7 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 ☺ The best problems from around the world Cao Minh Quan g 8 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 ☺ The best problems from around the world Cao Minh Quan g 9 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 34 5 12.3. Swedish 1963 346 ☺ The best problems from around the world Cao Minh Quan g 10 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 [...]... 729 16 ☺ The best problems from around the world Cao Minh Quang PART I National Olympiads AIME (1983 – 2004) 17 ☺ 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 ≤... 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 ☺ The best problems from around the world Cao Minh Quang 13 For each non-empty... were in the tournament? 15 A 12 x 12 square is divided into two pieces by joining to adjacent side midpoints Copies of the triangular piece are placed on alternate edges of a regular hexagon and copies of the other piece are placed on the other edges The resulting figure is then folded to give a polyhedron with 7 faces What is the volume of the polyhedron? 22 ☺ The best problems from around the world. .. along the line y = 2x + 4, AB has length 60 and angle C = 90o Find the area of ABC 23 ☺ The best problems from around the world Cao Minh Quang 5th AIME 1987 1 How many pairs of non-negative integers (m, n) each sum to 1492 without any carries? 2 What is the greatest distance between the sphere center (-2 , -1 0, 5) radius 19, and the sphere center (12, 8, -1 6) radius 87? 3 A nice number equals the product... elements of the subset have difference 4 or 7 14 Any number of the form M + Ni with M and N integers may be written in the complex base (i - n) as am(i - n)m + am-1(i - n)m-1 + + a1(i - n) + a0 for some m >= 0, where the digits ak lie in the range 0, 1, 2, , n2 Find the sum of all ordinary integers which can be written to base i - 3 as 4-digit numbers 15 In the triangle ABC, the segments have the lengths... inscribed in a right-angled triangle as shown The first has area 441 and the second area 440 Find the sum of the two shorter sides of the triangle 24 ☺ The best problems from around the world Cao Minh Quang 6th AIME 1988 1 A lock has 10 buttons A combination is any subset of 5 buttons It can be opened by pressing the buttons in the combination in any order How many combinations are there? Suppose it... The best problems from around the world Cao Minh Quang 8th AIME 1990 1 The sequence 2, 3, 5, 6, 7, 10, 11, consists of all positive integers that are not a square or a cube Find the 500th term 2 Find (52 + 6√43)3/2 - (52 - 6√43)3/2 3 Each angle of a regular r-gon is 59/58 times larger than each angle of a regular s-gon What is the largest possible value of s? 4 Find the positive solution to 1/(x 2-. .. before they can see each other again 30 ☺ The best problems from around the world Cao Minh Quang 14 R is a 6 x 8 rectangle R' is another rectangle with one vertex on each side of R R' can be rotated slightly and still remain within R Find the smallest perimeter that R' can have 15 The triangle ABC has AB = 1995, BC = 1993, CA = 1994 CX is an altitude Find the distance between the points at which the incircles... ☺ The best problems from around the world Cao Minh Quang 12th AIME 1994 1 The sequence 3, 15, 24, 48, is those multiples of 3 which are one less than a square Find the remainder when the 1994th term is divided by 1000 2 The large circle has diameter 40 and the small circle diameter 10 They touch at P PQ is a diameter of the small circle ABCD is a square touching the small circle at Q Find AB 3 The. .. has already been typed How many possible orders there are for the typing of the remaining letters [For example, letters 1, 7 and 8 might already have been typed, and the remaining letters might be typed in the order 6, 5, 9, 4, 3, 2 So the sequence 6, 5, 9, 4, 3, 2 is one possibility The empty sequence is another.] 25 ☺ The best problems from around the world Cao Minh Quang 7th AIME 1989 1 Find sqrt(1 . digits of AB. The distance OH is a non-zero rational. Find AB. ☺ The best problems from around the world Cao Minh Quan g 19 13. For each non-empty subset. The best problems from around the world Cao Minh Quan g 1 7 PART I. National Olympiads AIME (1983 – 2004) ☺ The best