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

2
Abbreviations

AIME American Invitational Mathematics Examination
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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

2.15. ASU 1975 ............................................................................................................ 72
2.16. ASU 1976 ............................................................................................................ 74
2.17. ASU 1977 ............................................................................................................ 76
2.18. ASU 1978 ............................................................................................................ 78
2.19. ASU 1979 ............................................................................................................ 80
2.20. ASU 1980 ............................................................................................................ 82
2.21. ASU 1981 ............................................................................................................ 84
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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

3.31. BMO 1995.......................................................................................................... 157
3.32. BMO 1996.......................................................................................................... 158
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3.33. BMO 1997.......................................................................................................... 159
3.34. BMO 1998.......................................................................................................... 160
3.35. BMO 1999.......................................................................................................... 161
3.36. BMO 2000.......................................................................................................... 162
3.37. BMO 2001.......................................................................................................... 163
3.38. BMO 2002.......................................................................................................... 164
3.39. BMO 2003.......................................................................................................... 165
3.40. BMO 2004.......................................................................................................... 166
4. Brasil (1979 – 2003) .................................................................................................... 167
4.1. Brasil 1979 ........................................................................................................... 168
4.2. Brasil 1980 ........................................................................................................... 169
4.3. Brasil 1981 ........................................................................................................... 170
4.4. Brasil 1982 ........................................................................................................... 171
4.5. Brasil 1983 ........................................................................................................... 172
4.6. Brasil 1984 ........................................................................................................... 173
4.7. Brasil 1985 ........................................................................................................... 174
4.8. Brasil 1986 ........................................................................................................... 175
4.9. Brasil 1987 ........................................................................................................... 176
4.10. Brasil 1988 ......................................................................................................... 177
4.11. Brasil 1989 ......................................................................................................... 178
4.12. Brasil 1990 ......................................................................................................... 179
4.13. Brasil 1991 ......................................................................................................... 180
4.14. Brasil 1992 ......................................................................................................... 181
4.15. Brasil 1993 ......................................................................................................... 182
4.16. Brasil 1994 ......................................................................................................... 183

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

6.38. 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

6.95. 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

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

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

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
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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
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17. Austrian – Polish (1978 – 2003) ................................................................................ 502
17.1. Austrian – Polish 1978 ....................................................................................... 503
17.2. Austrian – Polish 1979 ....................................................................................... 504

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
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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

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
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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

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
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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

PART I. National Olympiads

AIME (1983 – 2004)
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1st AIME 1983

1. x, y, z are real numbers greater than 1 and w is a positive real number. If log
x
w = 24, log
y
w
= 40 and log
xyz

83
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 (9x
2
sin
2
x + 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.
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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 QP
2

2
+ a
4
+ a
6
+ ... + a
98
.
2. Find the smallest positive integer n such that every digit of 15n is 0 or 8.
3. P is a point inside the triangle ABC. Lines are drawn through P parallel to the sides of the
triangle. The areas of the three resulting triangles with a vertex at P have areas 4, 9 and 49.
What is the area of ABC?

4. A sequence of positive integers includes the number 68 and has arithmetic mean 56. When
68 is removed the arithmetic mean of the remaining numbers is 55. What is the largest
number than can occur in the sequence?
5. The reals x and y satisfy log
8
x + log
4
(y
2
) = 5 and log
8
y + log
4
(x
2
) = 7. Find xy.
6. Three circles radius 3 have centers at P (14, 92), Q (17, 76) and R (19, 84). The line L

-1
3 + cot
-1
7 + cot
-1
13 + cot
-1
21).
14. What is the largest even integer that cannot be written as the sum of two odd composite
positive integers?
15. The real numbers x, y, z, w satisfy: x
2
/(n
2
- 1
2
) + y
2
/(n
2
- 3
2
) + z
2
/(n
2
- 5
2
) + w
2

2
, x
4
= 4/x
3
, ... , x
8
= 8/x
7
. Find x
1
x
2
... x
8
.
2. The triangle ABC has angle B = 90
o
. When it is rotated about AB it gives a cone volume
800π. When it is rotated about BC it gives a cone volume 1920π. Find the length AC.
3. m and n are positive integers such that N = (m + ni)
3
- 107i is a positive integer. Find N.
4. ABCD is a square side 1. Points A', B', C', D' are taken on the sides AB, BC, CD, DA
respectively so that AA'/AB = BB'/BC = CC'/CD = DD'/DA = 1/n. The strip bounded by the
lines AC' and A'C meets the strip bounded by the lines BD' and B'D in a square area 1/1985.
Find n.

5. The integer sequence a
1

center (where x + y < 180
o
). Find cos x.
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10. How many of 1, 2, 3, ... , 1000 can be expressed in the form [2x] + [4x] + [6x] + [8x], for
some real number x?
11. The foci of an ellipse are at (9, 20) and (49, 55), and it touches the x-axis. What is the
length of its major axis?
12. A bug crawls along the edges of a regular tetrahedron ABCD with edges length 1. It starts
at A and at each vertex chooses its next edge at random (so it has a 1/3 chance of going back
along the edge it came on, and a 1/3 chance of going along each of the other two). Find the
probability that after it has crawled a distance 7 it is again at A is p.
13. Let f(n) be the greatest common divisor of 100 + n
2
and 100 + (n+1)
2
for n = 1, 2, 3, ... .
What is the maximum value of f(n)?
14. In a tournament each two players played each other once. Each player got 1 for a win, 1/2
for a draw, and 0 for a loss. Let S be the set of the 10 lowest-scoring players. It is found that
every player got exactly half his total score playing against players in S. How many players
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?


= 6
x
1
+ 2x
2
+ x
3
+ x
4
+ x
5
= 12
x
1
+ x
2
+ 2x
3
+ x
4
+ x
5
= 24
x
1
+ x
2
+ x
3
+ 2x

1000000 except 1000000 itself.
9. A triangle has sides 425, 450, 510. Lines are drawn through an interior point parallel to the
sides, the intersections of these lines with the interior of the triangle have the same length.
What is it?
10. abc is a three digit number. If acb + bca + bac + cab + cba = 3194, find abc.
11. The polynomial 1 - x + x
2
- x
3
+ ... - x
15
+ x
16
- x
17
can be written as a polynomial in y = x
+ 1. Find the coefficient of y
2
.
12. Let X be a subset of {1, 2, 3, ... , 15} such that no two subsets of X have the same sum.
What is the largest possible sum for X?
13. A sequence has 15 terms, each H or T. There are 14 pairs of adjacent terms. 2 are HH, 3
are HT, 4 are TH, 5 are TT. How many sequences meet these criteria?
14. A rectangular box has 12 edges. A long diagonal intersects 6 of them. The shortest
distance of the other 6 from the long diagonal are 2√5 (twice), 30/√13 (twice), 15/√10 (twice).
Find the volume of the box.
15. The triangle ABC has medians AD, BE, CF. AD lies along the line y = x + 3, BE lies
along the line y = 2x + 4, AB has length 60 and angle C = 90
o
. Find the area of ABC.

2
.
6. ABCD is a rectangle. The points P, Q lie inside it with PQ parallel to AB. Points X, Y lie
on AB (in the order A, X, Y, B) and W, Z on CD (in the order D, W, Z, C). The four parts
AXPWD, XPQY, BYQZC, WPQZ have equal area. BC = 19, PQ = 87, XY = YB + BC + CZ
= WZ = WD + DA + AX. Find AB. 7. How many ordered triples (a, b, c) are there, such that lcm(a, b) = 1000, lcm(b, c) = 2000,
lcm(c, a) = 2000?
8. Find the largest positive integer n for which there is a unique integer k such that 8/15 <
n/(n+k) < 7/13.
9. P lies inside the triangle ABC. Angle B = 90
o
and each side subtends an angle 120
o
at P. If
PA = 10, PB = 6, find PC.
10. A walks down an up-escalator and counts 150 steps. B walks up the same escalator and
counts 75 steps. A takes three times as many steps in a given time as B. How many steps are
visible on the escalator?
11. Find the largest k such that 3
11
is the sum of k consecutive positive integers.
12. Let m be the smallest positive integer whose cube root is n + k, where n is an integer and
0 < k < 1/1000. Find n.
13. Given distinct reals x
1
, x
2

+ 324) and n = (4
4
+ 324)(16
4

+ 324)(28
4
+ 324)(40
4
+ 324)(52
4
+ 324). Find m/n.
15. Two squares are 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.