Le Hai Chau
Le Hai Khoi
Mathematical
Olympiad
Series
Vol.5
Selected Problems of the
Vietnamese Mathematical Olympiad
(1962-2009)
World Scientific
Selected Problems of the
Vietnamese Mathematical
Olympiad
(1962–2009)
7514 tp.indd 1 8/3/10 9:49 AM
Vol. 5
Mathematical
Olympiad
Series
Selected Problems of the
Vietnamese Mathematical
Olympiad
(1962–2009)
World Scientic
Le Hai Chau
Ministry of Education and Training, Vietnam
Le Hai Khoi
Nanyang Technological University, Singapore
7514 tp.indd 2 8/3/10 9:49 AM
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

Zheng Zhongyi (High School Attached to Fudan University, China)
translated by Liu Ruifang, Zhai Mingqing & Lin Yuanqing
(East China Normal University, China)
Vol. 5 Selected Problems of the Vietnamese Olympiad (1962–2009)
by Le Hai Chau (Ministry of Education and Training, Vietnam)
& Le Hai Khoi (Nanyang Technology University, Singapore)
Vol. 6 Lecture Notes on Mathematical Olympiad Courses:
For Junior Section (In 2 Volumes)
by Jiagu Xu
LaiFun - Selected Problems of the Vietnamese.pmd 8/23/2010, 3:16 PM2
Foreword
The International Mathematical Olympiad (IMO) - an annual international
mathematical competition primarily for high school students - has a his-
tory of more than half a century and is the oldest of all international science
Olympiads. Having attracted the participation of more than 100 countries
and territories, not only has the IMO been instrumental in promoting inter-
est in mathematics among high school students, it has also been successful
in the identification of mathematical talent. For example, since 1990, at
least one of the Fields Medalists in every batch had participated in an IMO
earlier and won a medal.
Vietnam began participating in the IMO in 1974 and has consistently
done very well. Up to 2009, the Vietnamese team had already won 44 gold,
82 silver and 57 bronze medals at the IMO - an impressive performance
that places it among the top ten countries in the cumulative medal tally.
This is probably related to the fact that there is a well-established tradition
in mathematical competitions in Vietnam - the Vietnamese Mathematical
Olympiad (VMO) started in 1962. The VMO and the Vietnamese IMO
teams have also helped to identify many outstanding mathematical talents
from Vietnam, including Ngo Bao Chau, whose proof of the Fundamental
Lemma in Langland’s program made it to the list of Top Ten Scientific

In 1962, the first Vietnamese Mathematical Olympiad (VMO) was held in
Hanoi. Since then the Vietnam Ministry of Education has, jointly with the
Vietnamese Mathematical Society (VMS), organized annually (except in
1973) this competition. The best winners of VMO then participated in the
Selection Test to form a team to represent Vietnam at the International
Mathematical Olympiad (IMO), in which Vietnam took part for the first
time in 1974. After 33 participations (except in 1977 and 1981) Vietnamese
students have won almost 200 medals, among them over 40 gold.
This books contains about 230 selected problems from more than 45
competitions. These problems are divided into five sections following the
classification of the IMO: Algebra, Analysis, Number Theory, Combina-
torics, and Geometry.
It should be noted that the problems presented in this book are of
average level of difficulty. In the future we hope to prepare another book
containing more difficult problems of the VMO, as well as some problems
of the Selection Tests for forming the Vietnamese teams for the IMO.
We also note that from 1990 the VMO has been divided into two eche-
lons. The first echelon is for students of the big cities and provinces, while
the second echelon is for students of the smaller cities and highland regions.
Problems for the second echelon are denoted with the letter B.
We would like to thank the World Scientific Publishing Co. for publish-
ing this book. Special thanks go to Prof. Lee Soo Ying, former Dean of
the College of Science, Prof. Ling San, Chair of the School of Physical and
Mathematical Sciences, and Prof. Chee Yeow Meng, Head of the Division
of Mathematical Sciences, Nanyang Technological University, Singapore,
for stimulating encouragement during the preparation of this book. We are
grateful to David Adams, Chan Song Heng, Chua Chek Beng, Anders Gus-
tavsson, Andrew Kricker, Sinai Robins and Zhao Liangyi from the School
of Physical Mathematical Sciences, and students Lor Choon Yee and Ong
Soon Sheng, for reading different parts of the book and for their valuable

2.3.1 Prime Numbers 21
2.3.2 Modulo operation 23
2.3.3 Fermat and Euler theorems 23
2.3.4 Numeral systems 24
2.4 Combinatorics 24
2.4.1 Counting 24
2.4.2 Newton binomial formula 25
2.4.3 Dirichlet (or Pigeonhole) principle 25
2.4.4 Graph 26
2.5 Geometry 27
2.5.1 Trigonometric relationship in a triangle and a
circle 27
ix
x CONTENTS
2.5.2 Trigonometric formulas 28
2.5.3 Some important theorems 29
2.5.4 Dihedral and trihedral angles 30
2.5.5 Tetrah e dra 31
2.5.6 Prism, parallelepiped, pyramid 31
2.5.7 Cones 31
3Problems 33
3.1 Algebra 33
3.1.1 (1962) . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 (1964) . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 (1966) . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.4 (1968) . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.5 (1969) . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.6 (1970) . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.7 (1972) . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.8 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.38 (1991 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.39 (1992 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.40 (1992 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.41 (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.42 (1994 B) . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.43 (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.44 (1995) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.45 (1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.46 (1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.47 (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.48 (1998 B) . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.49 (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.50 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.51 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.52 (2001 B) . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.53 (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.54 (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.55 (2004 B) . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.56 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.57 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.58 (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.59 (2006 B) . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.60 (2006 B) . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.61 (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.62 (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.63 (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Analysis 47
3.2.1 (1965) . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.3 (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.33 (1997 B) . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.34 (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.35 (1998 B) . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.36 (1998 B) . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.37 (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.38 (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.39 (1999 B) . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.40 (1999 B) . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.41 (1999 B) . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.42 (2000 B) . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.43 (2000) . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.44 (2000) . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.45 (2001 B) . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.46 (2001) . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.47 (2001) . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.48 (2002 B) . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.49 (2002 B) . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.50 (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.51 (2003 B) . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.52 (2003 B) . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.53 (2003 B) . . . . . . . . . . . . . . . . . . . . . . . . . 58
CONTENTS xiii
3.2.54 (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.55 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.56 (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.57 (2006 B) . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.58 (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.59 (2007) . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.60 (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.61 (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.29 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.30 (2001) . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.31 (2002 B) . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.32 (2002 B) . . . . . . . . . . . . . . . . . . . . . . . . . 67
xiv CONTENTS
3.3.33 (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.34 (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.35 (2004 B) . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.36 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.37 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.38 (2005 B) . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.39 (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.40 (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.41 (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.42 (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Combinatorics 69
3.4.1 (1969) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.2 (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.3 (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.4 (1990) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.5 (1991) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.6 (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.7 (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.8 (1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.9 (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.10 (2001) . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.11 (2004 B) . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.12 (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.13 (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.14 (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5.28 (1972) . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.29 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.30 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.31 (1978) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.32 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.33 (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.34 (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.35 (1990) . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.36 (1990 B) . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.37 (1991) . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.38 (1991 B) . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.39 (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.40 (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.41 (1995 B) . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.42 (1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.43 (1996 B) . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.44 (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.45 (1998 B) . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.46 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.47 (2000 B) . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.48 (2000) . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Solutions 85
4.1 Algebra 85
4.1.1 85
4.1.2 85
4.1.3 86
4.1.4 87
4.1.5 87
4.1.6 88
xvi CONTENTS

4.1.37 110
4.1.38 111
4.1.39 111
4.1.40 112
4.1.41 112
4.1.42 113
4.1.43 114
4.1.44 115
4.1.45 115
4.1.46 116
4.1.47 118
4.1.48 118
CONTENTS xvii
4.1.49 119
4.1.50 121
4.1.51 121
4.1.52 122
4.1.53 123
4.1.54 124
4.1.55 125
4.1.56 126
4.1.57 126
4.1.58 128
4.1.59 128
4.1.60 129
4.1.61 129
4.1.62 131
4.1.63 131
4.2 Analysis 132
4.2.1 132

4.2.31 159
4.2.32 160
4.2.33 162
4.2.34 163
4.2.35 163
4.2.36 164
4.2.37 165
4.2.38 167
4.2.39 167
4.2.40 168
4.2.41 169
4.2.42 170
4.2.43 171
4.2.44 173
4.2.45 173
4.2.46 174
4.2.47 175
4.2.48 176
4.2.49 177
4.2.50 178
4.2.51 179
4.2.52 180
4.2.53 182
4.2.54 183
4.2.55 184
4.2.56 185
4.2.57 186
4.2.58 186
4.2.59 187
4.2.60 188

4.3.27 208
4.3.28 209
4.3.29 210
4.3.30 211
4.3.31 212
4.3.32 213
4.3.33 213
4.3.34 214
4.3.35 215
4.3.36 216
4.3.37 217
4.3.38 218
4.3.39 219
4.3.40 221
4.3.41 222
4.3.42 222
4.4 Combinatorics 224
4.4.1 224
4.4.2 224
4.4.3 225
4.4.4 225
xx CONTENTS
4.4.5 226
4.4.6 227
4.4.7 227
4.4.8 228
4.4.9 229
4.4.10 230
4.4.11 231
4.4.12 232

4.5.27 271
4.5.28 273
4.5.29 274
4.5.30 276
CONTENTS xxi
4.5.31 277
4.5.32 278
4.5.33 280
4.5.34 281
4.5.35 282
4.5.36 284
4.5.37 286
4.5.38 287
4.5.39 289
4.5.40 290
4.5.41 291
4.5.42 292
4.5.43 293
4.5.44 294
4.5.45 296
4.5.46 297
4.5.47 298
4.5.48 300
5 Olympiad 2009 301
Chapter 1
The Gifted Students
On the first school opening day of the Democratic Republic of Vietnam
in September 1946, President Ho Chi Minh sent a letter to all students,
stating: “Whether or not Vietnam becomes glorious and the Vietnamese
nation becomes gloriously paired with other wealthy nations over five con-

learning hours), following the program and materials provided by local
(provincial) Departments of Education.
Stage 3. Gifted students are selected from city/province level to partic-
ipate in a mathematical competition (for year-end students of each level).
This competition is organized completely by the local city/province (set-up
questions, script marking and rewards).
Stage 4. The National Mathematical Olympiad for students of the
final grades of Secondary and High Schools is organized by the Ministry of
Education. The national jury is formed for this to be in charge of posing
questions, marking papers and suggesting prizes. The olympiad is held over
two days. Each day students solve three problems in three hours. There
are 2 types of awards: Individual prize and Team prize, each consists of
First, Second, Third and Honorable prizes.
2. During the first few years, the Ministry of Education assigned the first-
named author, Ministry’s Inspector for Mathematics, to take charge in
organizing the Olympiad, from setting the questions to marking the papers.
When the Vietnamese Mathematical Society was established (Jan 1964),
the Ministry invited the VMS to join in. Professor Le Van Thiem, the first
Director of Vietnam Institute of Mathematics, was nominated as a chair
of the jury. Since then, the VMO is organized annually by the Ministry of
Education, even during years of fierce war.
For the reader to imagine the content of the national competition, the
full questions of the first 1962 and the latest 2009 Olympiad are presented
here.
1.1. THE VIETNAMESE MATHEMATICAL OLYMPIAD 3
The first Mathematical Olympiad, 1962
Problem 1.Provethat
1
1
a


,B

the orthogonal pro-
jections of A, B on the opposite faces, respectively. Prove that AA

and
BB

intersect each other if and only if AB ⊥ CD.
Do AA

and BB

intersect each other if AC = AD = BC = BD?
Problem 4. Given a pyramid SABCD such that the base ABCD is
a square with the center O,andSO ⊥ ABCD. The height SO is h and
the angle between SAB and ABCD is α. The plane passing through the
edge AB is perpendicular to the opposite face SCD. Find the volume of
the prescribed pyramid. Investigate the obtained formula.
Problem 5. Solve the equation
sin
6
x +cos
6
x =
1
4
.
The Mathematical Olympiad, 2009

2
,x
n
=

x
2
n−1
+4x
n−1
+ x
n−1
2
,n≥ 2.
Prove that a sequence (y
n
) defined by y
n
=
n

i=1
1
x
2
i
converges and find
its limit.