STUDY NOTES & REFERENCE GUIDE
Covers All 27 Chapters
nolanjack - Stuvia | Page 1 of 442
, STUDY NOTES & REFERENCE GUIDE
Preface
It is said that in many countries, especially the United States, children
are afraid of mathematics and regard it as an “unpopular subject.” But in
China, the situation is very different. Many children love mathematics, and
their math scores are also very good. Indeed, mathematics is a subject that
the Chinese are good at. If you see a few Chinese students in elementary
and middle schools in the United States, then the top few in the class of
mathematics are none other than them.
At the early stage of counting numbers, Chinese children already show
their advantages.
Chinese people can express integers from 1 to 10 with one hand, whereas
those in other countries would have to use two.
The Chinese have long had the concept of digits, and they use the most
convenient decimal system (many countries still have the remnants of base
12 and base 60 systems).
Chinese characters are all single syllables, which are easy to recite. For
example, the multiplication table can be quickly mastered by students,
and even the slow learners know the concept of “three times seven equals
twenty one.” However, for foreigners, as soon as they study multiplication,
their heads get bigger. Believe it or not, you could try and memorize the
multiplication table in English and then recite it; it is actually much harder
to do so in English.
It takes the Chinese one or two minutes to memorize π = 3.14159 · · ·
to the fifth decimal place. However, in order to recite these digits, the
Russians wrote a poem. The first sentence contains three words, the second
sentence contains one, and so on. To recite π, recite poetry first. In our
opinion, as conveyed by Problems and Solutions in Mathematical Olympiad
vii
nolanjack - Stuvia | Page 2 of 442
, STUDY NOTES & REFERENCE GUIDE
viii Problems and Solutions in Mathematical Olympiad (Secondary 2)
Secondary 3, this is just simply asking for trouble, but they treat it as a
magical way of memorization.
Application problems for the four arithmetic operations and their arith-
metic solutions are also a major feature of Chinese mathematics. Since
ancient times, the Chinese have compiled a lot of application questions
which have contact or close relations with reality and daily life. Their solu-
tions are simple and elegant, as well as smart and diverse, which helps
increase students’ interest in learning and enlighten students. For exam-
ple: “There are one hundred monks and one hundred buns. One big monk
eats three buns and three little monks eat one bun. How many big monks
and how many little monks are there?”
Most foreigners can only solve equations, but Chinese have a variety of
arithmetic solutions. As an example, one can turn each big monk into 9
little monks, and 100 buns indicate that there are 300 little monks, which
contain 200 added little monks. As each big monk becomes a little monk,
8 more little monks are created, so 200/8 = 25 is the number of big monks,
and naturally, there are 75 little monks. Another way to solve the problem
is to group a big monk and three little monks together, and so each per-
son eats a bun on average, which is exactly equal to the overall average.
Thus, the big monks and the little monks are not more and less after being
organized this way; that is, the number of big monks is 100/(3 + 1) = 25.
The Chinese are good at calculating, especially mental arithmetic. In
ancient times, some people used their fingers to calculate (the so-called
“counting by pinching fingers”). At the same time, China has long had
computing devices, such as counting chips and abaci. The latter can be
said to be the prototype of computers.
In the introductory stage of mathematics – the study of arithmetic, our
country had obvious advantages, so mathematics is often the subject that
our smart children love.
Geometric reasoning was not well developed in ancient China (but there
were many books on the calculation of geometric figures in our country),
and it was slightly inferior to that of the Greeks. However, the Chinese
are good at learning from others. At present, the geometric level of middle
school students in our country is far ahead of the rest of the world. Once,
a foreign education delegation came to a junior high school class in our
country. They thought that the geometric content taught was too in-depth
for students to comprehend, but after attending the class, they had to admit
that the content was not only understood by Chinese students but also well
mastered.
nolanjack - Stuvia | Page 3 of 442
, STUDY NOTES & REFERENCE GUIDE
Preface ix
The achievements of mathematics education in our country are remark-
able. In international mathematics competitions, Chinese contestants have
won numerous medals, which is the most powerful proof. Ever since our
country officially sent a team to participate in the International Mathemat-
ical Olympiad in 1986, the Chinese team has won 14 team championships,
which can be described as quite impressive. Professor Shiing-Shen Chern,
a famous contemporary mathematician, once admired this in particular.
He said, “One thing to celebrate this year is that China won the first place
in the international math competition . . . Last year it was also the first
place.” (Shiing-Shen Chern’s speech, How to Build China into a Mathe-
matical Power, at Cheng Kung University in Taiwan in October 1990.)
Professor Chern also predicted: “China will become a mathematical
power in the 21st century.”
It is certainly not an easy task to become a mathematical power. It
cannot be achieved overnight. It requires unremitting efforts. The purpose
of this series of books is as follows: (1) to further popularize the knowledge
of mathematics, to make mathematics be loved by more young people, and
to help them achieve good results; (2) to enable students who love mathe-
matics to get better development and learn more knowledge and methods
through the series of books.
“The important things in the world must be done in detail.” We hope
and believe that the publication of this series of books will play a role in
making our country a mathematical power. This series was first published
in 2000. According to the requirements of the curriculum reform, each vol-
ume is revised to different degrees.
A well-known mathematician, academician of the Chinese Academy of
Sciences, and former chairman of the Chinese Mathematical Olympiad,
Professor Yuan Wang, served as a consultant for this series of books and
wrote inscriptions for young math enthusiasts. We express our heartfelt
thanks. We would also like to thank East China Normal University Press,
and in particular Mr. Ming Ni and Mr. Ling-zhi Kong. Without them, this
series of books would not have been possible.
Zun Shan and Bin Xiong
May 2018
nolanjack - Stuvia | Page 4 of 442
Covers All 27 Chapters
nolanjack - Stuvia | Page 1 of 442
, STUDY NOTES & REFERENCE GUIDE
Preface
It is said that in many countries, especially the United States, children
are afraid of mathematics and regard it as an “unpopular subject.” But in
China, the situation is very different. Many children love mathematics, and
their math scores are also very good. Indeed, mathematics is a subject that
the Chinese are good at. If you see a few Chinese students in elementary
and middle schools in the United States, then the top few in the class of
mathematics are none other than them.
At the early stage of counting numbers, Chinese children already show
their advantages.
Chinese people can express integers from 1 to 10 with one hand, whereas
those in other countries would have to use two.
The Chinese have long had the concept of digits, and they use the most
convenient decimal system (many countries still have the remnants of base
12 and base 60 systems).
Chinese characters are all single syllables, which are easy to recite. For
example, the multiplication table can be quickly mastered by students,
and even the slow learners know the concept of “three times seven equals
twenty one.” However, for foreigners, as soon as they study multiplication,
their heads get bigger. Believe it or not, you could try and memorize the
multiplication table in English and then recite it; it is actually much harder
to do so in English.
It takes the Chinese one or two minutes to memorize π = 3.14159 · · ·
to the fifth decimal place. However, in order to recite these digits, the
Russians wrote a poem. The first sentence contains three words, the second
sentence contains one, and so on. To recite π, recite poetry first. In our
opinion, as conveyed by Problems and Solutions in Mathematical Olympiad
vii
nolanjack - Stuvia | Page 2 of 442
, STUDY NOTES & REFERENCE GUIDE
viii Problems and Solutions in Mathematical Olympiad (Secondary 2)
Secondary 3, this is just simply asking for trouble, but they treat it as a
magical way of memorization.
Application problems for the four arithmetic operations and their arith-
metic solutions are also a major feature of Chinese mathematics. Since
ancient times, the Chinese have compiled a lot of application questions
which have contact or close relations with reality and daily life. Their solu-
tions are simple and elegant, as well as smart and diverse, which helps
increase students’ interest in learning and enlighten students. For exam-
ple: “There are one hundred monks and one hundred buns. One big monk
eats three buns and three little monks eat one bun. How many big monks
and how many little monks are there?”
Most foreigners can only solve equations, but Chinese have a variety of
arithmetic solutions. As an example, one can turn each big monk into 9
little monks, and 100 buns indicate that there are 300 little monks, which
contain 200 added little monks. As each big monk becomes a little monk,
8 more little monks are created, so 200/8 = 25 is the number of big monks,
and naturally, there are 75 little monks. Another way to solve the problem
is to group a big monk and three little monks together, and so each per-
son eats a bun on average, which is exactly equal to the overall average.
Thus, the big monks and the little monks are not more and less after being
organized this way; that is, the number of big monks is 100/(3 + 1) = 25.
The Chinese are good at calculating, especially mental arithmetic. In
ancient times, some people used their fingers to calculate (the so-called
“counting by pinching fingers”). At the same time, China has long had
computing devices, such as counting chips and abaci. The latter can be
said to be the prototype of computers.
In the introductory stage of mathematics – the study of arithmetic, our
country had obvious advantages, so mathematics is often the subject that
our smart children love.
Geometric reasoning was not well developed in ancient China (but there
were many books on the calculation of geometric figures in our country),
and it was slightly inferior to that of the Greeks. However, the Chinese
are good at learning from others. At present, the geometric level of middle
school students in our country is far ahead of the rest of the world. Once,
a foreign education delegation came to a junior high school class in our
country. They thought that the geometric content taught was too in-depth
for students to comprehend, but after attending the class, they had to admit
that the content was not only understood by Chinese students but also well
mastered.
nolanjack - Stuvia | Page 3 of 442
, STUDY NOTES & REFERENCE GUIDE
Preface ix
The achievements of mathematics education in our country are remark-
able. In international mathematics competitions, Chinese contestants have
won numerous medals, which is the most powerful proof. Ever since our
country officially sent a team to participate in the International Mathemat-
ical Olympiad in 1986, the Chinese team has won 14 team championships,
which can be described as quite impressive. Professor Shiing-Shen Chern,
a famous contemporary mathematician, once admired this in particular.
He said, “One thing to celebrate this year is that China won the first place
in the international math competition . . . Last year it was also the first
place.” (Shiing-Shen Chern’s speech, How to Build China into a Mathe-
matical Power, at Cheng Kung University in Taiwan in October 1990.)
Professor Chern also predicted: “China will become a mathematical
power in the 21st century.”
It is certainly not an easy task to become a mathematical power. It
cannot be achieved overnight. It requires unremitting efforts. The purpose
of this series of books is as follows: (1) to further popularize the knowledge
of mathematics, to make mathematics be loved by more young people, and
to help them achieve good results; (2) to enable students who love mathe-
matics to get better development and learn more knowledge and methods
through the series of books.
“The important things in the world must be done in detail.” We hope
and believe that the publication of this series of books will play a role in
making our country a mathematical power. This series was first published
in 2000. According to the requirements of the curriculum reform, each vol-
ume is revised to different degrees.
A well-known mathematician, academician of the Chinese Academy of
Sciences, and former chairman of the Chinese Mathematical Olympiad,
Professor Yuan Wang, served as a consultant for this series of books and
wrote inscriptions for young math enthusiasts. We express our heartfelt
thanks. We would also like to thank East China Normal University Press,
and in particular Mr. Ming Ni and Mr. Ling-zhi Kong. Without them, this
series of books would not have been possible.
Zun Shan and Bin Xiong
May 2018
nolanjack - Stuvia | Page 4 of 442
