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:
Vinh Long, April 2006
Cao Minh Quang

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

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

2.40. 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

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

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

6.60. Eötvös Competition 1960 245
6.61. Eötvös Competition 1961 246
6.62. Eötvös Competition 1962 246
6.63. Eötvös Competition 1963 246
6.64. Eötvös Competition 1964 247
6.65. Eötvös Competition 1965 247
6.66. Eötvös Competition 1966 247
6.67. Eötvös Competition 1967 248
6.68. Eötvös Competition 1968 248
6.69. Eötvös Competition 1969 248
6.70. Eötvös Competition 1970 249
6.71. Eötvös Competition 1971 249
6.72. Eötvös Competition 1972 249
6.73. Eötvös Competition 1973 250
6.74. Eötvös Competition 1974 250
6.75. Eötvös Competition 1975 250
6.76. Eötvös Competition 1976 251
6.77. Eötvös Competition 1977 251
6.78. Eötvös Competition 1978 251
6.79. Eötvös Competition 1979 252
6.80. Eötvös Competition 1980 252
6.81. Eötvös Competition 1981 252
6.82. Eötvös Competition 1982 253
6.83. Eötvös Competition 1983 253
6.84. Eötvös Competition 1984 253
6.85. Eötvös Competition 1985 254
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8.1. Irish 1988 272
8.2. Irish 1989 273
8.3. Irish 1990 274
8.4. Irish 1991 275
8.5. Irish 1992 276
8.6. Irish 1993 277
8.7. Irish 1994 278
8.8. Irish 1995 279
8.9. Irish 1996 280
8.10. Irish 1997 281
8.11. Irish 1998 282
8.12. Irish 1999 283
8.13. Irish 2000 284
8.14. Irish 2001 285
8.15. Irish 2002 286
8.16. Irish 2003 287
9. Mexican (1987 – 2003) 288
9.1. Mexican 1987 289
9.2. Mexican 1988 290
9.3. Mexican 1989 291
9.4. Mexican 1990 292
9.5. Mexican 1991 293
9.6. Mexican 1992 294
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9.7. Mexican 1993 295
9.8. Mexican 1994 296

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
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12.4. Swedish 1964 347
12.5. Swedish 1965 348
12.6. Swedish 1966 349
12.7. Swedish 1967 350

12.38. Swedish 1998 381
12.39. Swedish 1999 382
12.40. Swedish 2000 383
12.41. Swedish 2001 384
12.42. Swedish 2002 385
12.43. Swedish 2003 386
13. USAMO (1972 – 2003) 387
13.1. USAMO 1972 388
13.2. USAMO 1973 389
13.3. USAMO 1974 390
13.4. USAMO 1975 391
13.5. USAMO 1976 392
13.6. USAMO 1977 393
13.7. USAMO 1978 394
13.8. USAMO 1979 395
13.9. USAMO 1980 396
13.10. USAMO 1981 397
13.11. USAMO 1982 398
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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

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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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

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
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

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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4
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

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
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

23.25. PUTNAM 1965 682
23.26. PUTNAM 1966 683
23.27. PUTNAM 1967 684
23.28. PUTNAM 1968 685
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6
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

7
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

83
+ 8
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.

98
satisfies a
n+1
= a
n
+ 1 for n = 1, 2, , 97 and has sum 137. Find
a
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

N. Find N.
11. Three red counters, four green counters and five blue counters are placed in a row in
random order. Find the probability that no two blue counters are adjacent.
12. Let R be the reals. The function f : R → R satisfies f(0) = 0 and f(2 + x) = f(2 - x) and f(7
+ x) = f(7 - x) for all x. What is the smallest possible number of values x such that |x| ≤ 1000
and f(x) = 0?
13. Find 10 cot( cot
-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


1. Let x
1
= 97, x
2
= 2/x
1
, x
3
= 3/x
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.

4
, C
3
= D
2
and C = A + 19. Find D - B.
8. Approximate each of the numbers 2.56, 2.61, 2.65, 2.71, 2.79, 2.82, 2.86 by integers, so
that the 7 integers have the same sum and the maximum absolute error E is as small as
possible. What is 100E?
9. Three parallel chords of a circle have lengths 2, 3, 4 and subtend angles x, y, x + y at the
center (where x + y < 180
o
). Find cos x.
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2

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

1/4
= 12/(7 - x
1/4
).
2. Find (√5 + √6 + √7)(√5 + √6 - √7)(√5 - √6 + √7)(-√5 + √6 + √7).
3. Find tan(x+y) where tan x + tan y = 25 and cot x + cot y = 30.
4. 2x
1
+ x
2
+ x
3
+ x
4
+ x
5
= 6
x
1
+ 2x
2
+ x
3
+ x
4
+ x
5
= 12
x
1

= 96
Find 3x
4
+ 2x
5
.
5. Find the largest integer n such that n + 10 divides n
3
+ 100.
6. For some n, we have (1 + 2 + + n) + k = 1986, where k is one of the numbers 1, 2, , n.
Find k.
7. The sequence 1, 3, 4, 9, 10, 12, 13, 27, includes all numbers which are a sum of one or
more distinct powers of 3. What is the 100th term?
8. Find the integral part of ∑ log
10
k, where the sum is taken over all positive divisors of
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


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, -10, 5) radius 19, and the
sphere center (12, 8, -16) radius 87?
3. A nice number equals the product of its proper divisors (positive divisors excluding 1 and
the number itself). Find the sum of the first 10 nice numbers.
4. Find the area enclosed by the graph of |x - 60| + |y| = |x/4|.
5. m, n are integers such that m
2
+ 3m
2
n
2
= 30n
2
+ 517. Find 3m
2
n
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.

1
. Then we compare the second and third terms of the resulting sequence and
swap them iff the later term is smaller, and so on, until finally we compare the 39th and 40th
terms of the resulting sequence and swap them iff the last is smaller. If the sequence is
initially in random order, find the probability that x
20
ends up in the 30th place. [The original
question asked for m+n if the prob is m/n in lowest terms.]
14. Let m = (10
4
+ 324)(22
4
+ 324)(34
4
+ 324)(46
4
+ 324)(58
4
+ 324) and n = (4
4
+ 324)(16
4

+ 324)(28
4
+ 324)(40
4
+ 324)(52
4
+ 324). Find m/n.

2
x), find (log
2
x)
2
.
4. x
i
are reals such that -1 < x
i
< 1 and |x
1
| + |x
2
| + + |x
n
| = 19 + |x
1
+ + x
n
|. What is the
smallest possible value of n?
5. Find the probability that a randomly chosen positive divisor of 10
99
is divisible by 10
88
.
[The original question asked for m+n, where the prob is m/n in lowest terms.]
6. The vacant squares in the grid below are filled with positive integers so that there is an
arithmetic progression in each row and each column. What number is placed in the square

n
- z
n
) = 0. There is a unique mean line for
the points 32 + 170i, -7 + 64i, -9 + 200i, 1 + 27i, -14 + 43i which passes through the point 3i.
Find its slope.
12. P is a point inside the triangle ABC. The line PA meets BC at D. Similarly, PB meets CA
at E, and PC meets AB at F. If PD = PE = PF = 3 and PA + PB + PC = 43, find PA·PB·PC.
13. x
2
- x - 1 is a factor of a x
17
+ b x
16
+ 1 for some integers a, b. Find a.
14. The graph xy = 1 is reflected in y = 2x to give the graph 12x
2
+ rxy + sy
2
+ t = 0. Find rs.
15. The boss places letter numbers 1, 2, , 9 into the typing tray one at a time during the day
in that order. Each letter is placed on top of the pile. Every now and then the secretary takes
the top letter from the pile and types it. She leaves for lunch remarking that letter 8 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.]