Covers All 27 Chapters
,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
,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.
, 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
,This page intentionally left blank
, Contents
Editorial Board v
Preface vii
1. Linear Equations with Absolute Values 1
2. Linear Inequalities with Absolute Values 7
3. Polynomial Factorization (I) 13
4. Polynomial Factorization (II) 23
5. Calculation of Rational Fractions 33
6. Partial Fractions 39
7. Polynomial Equations and Fractional Equations
with Unknown Constants 45
8. Real Numbers 53
9. Quadratic Radicals 61
10. Evaluating Algebraic Expressions 71
11. Symmetric Polynomials 79
xi
,xii Problems and Solutions in Mathematical Olympiad (Secondary 2)
12. Proof of Identities 87
13. Linear Functions 95
14. Inversely Proportional Functions 111
15. Statistics 127
16. The Sides and Angles of a Triangle 137
17. Congruent Triangles 143
18. Isosceles Triangles 155
19. Right Triangles 165
20. Parallelograms 177
21. Trapezoids 191
22. The Angles and Diagonals of a Polygon 203
23. Proportion of Segments 213
24. Similar Triangles 225
25. The Midsegment 235
26. Translation and Symmetry 245
27. The Area 257
Contents xiii
Solutions
1. Linear Equations with Absolute Values 271
2. Linear Inequalities with Absolute Values 273
3. Polynomial Factorization (I) 275
, 4. Polynomial Factorization (II) 279
5. Calculation of Rational Fractions 283
6. Partial Fractions 285
7. Polynomial Equations and Fractional Equations
with Unknown Constants 289
8. Real Numbers 293
9. Quadratic Radicals 297
10. Evaluating Algebraic Expressions 301
11. Symmetric Polynomials 305
12. Proving Identities 309
13. Linear Functions 315
14. Inversely Proportional Functions 321
15. Statistics 327
16. The Sides and Angles of a Triangle 331
17. Congruent Triangles 335
18. Isosceles Triangles 339
19. Right Triangles 343
20. Parallelograms 347
21. Trapezoids 351
22. The Angles and Diagonals of a Polygon 357
23. Proportion of Segments 361
,24. Similar Triangles 365
25. The Midsegment 369
26. Translation and Symmetry 375
27. The Area 381
, Chapter 1
Linear Equations with
Absolute Values
We know that the absolute value of a positive number is itself, and the absolute
value of a negative number is its opposite, while the absolute value of 0 is
0. In other words,
x, x ≥ 0,
|x| =
−x, x < 0.
Consequently,
|x| = |−x|, |xy| = |x||y|;
If |x + y| = a, then a ≥ 0;
If |x + y| = |x − y|, then x = 0 or y = 0.
On the number axis, if a point A represents the number a, then |a| is the
distance from A to the origin, and |a − b| is the distance between the points
representing the numbers a and b. For example, |9+ 7| = |9 − (−7)|, which is
the distance between the points representing 9 and −7.
Equations that have unknowns inside the absolute value symbol are
called equations with absolute values.
To solve an equation with absolute values, the key is to remove the
absolute value symbol based on its definition. In order to remove the abso-
lute value symbol, we sometimes need to divide the range of a variable into
parts for further analysis.
Example 1. The sum of all roots of the equation |2x − 4| = 5 is ( ).
(A) −0.5 (B) 4.5 (C) 5 (D) 4
1
,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
,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.
, 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
,This page intentionally left blank
, Contents
Editorial Board v
Preface vii
1. Linear Equations with Absolute Values 1
2. Linear Inequalities with Absolute Values 7
3. Polynomial Factorization (I) 13
4. Polynomial Factorization (II) 23
5. Calculation of Rational Fractions 33
6. Partial Fractions 39
7. Polynomial Equations and Fractional Equations
with Unknown Constants 45
8. Real Numbers 53
9. Quadratic Radicals 61
10. Evaluating Algebraic Expressions 71
11. Symmetric Polynomials 79
xi
,xii Problems and Solutions in Mathematical Olympiad (Secondary 2)
12. Proof of Identities 87
13. Linear Functions 95
14. Inversely Proportional Functions 111
15. Statistics 127
16. The Sides and Angles of a Triangle 137
17. Congruent Triangles 143
18. Isosceles Triangles 155
19. Right Triangles 165
20. Parallelograms 177
21. Trapezoids 191
22. The Angles and Diagonals of a Polygon 203
23. Proportion of Segments 213
24. Similar Triangles 225
25. The Midsegment 235
26. Translation and Symmetry 245
27. The Area 257
Contents xiii
Solutions
1. Linear Equations with Absolute Values 271
2. Linear Inequalities with Absolute Values 273
3. Polynomial Factorization (I) 275
, 4. Polynomial Factorization (II) 279
5. Calculation of Rational Fractions 283
6. Partial Fractions 285
7. Polynomial Equations and Fractional Equations
with Unknown Constants 289
8. Real Numbers 293
9. Quadratic Radicals 297
10. Evaluating Algebraic Expressions 301
11. Symmetric Polynomials 305
12. Proving Identities 309
13. Linear Functions 315
14. Inversely Proportional Functions 321
15. Statistics 327
16. The Sides and Angles of a Triangle 331
17. Congruent Triangles 335
18. Isosceles Triangles 339
19. Right Triangles 343
20. Parallelograms 347
21. Trapezoids 351
22. The Angles and Diagonals of a Polygon 357
23. Proportion of Segments 361
,24. Similar Triangles 365
25. The Midsegment 369
26. Translation and Symmetry 375
27. The Area 381
, Chapter 1
Linear Equations with
Absolute Values
We know that the absolute value of a positive number is itself, and the absolute
value of a negative number is its opposite, while the absolute value of 0 is
0. In other words,
x, x ≥ 0,
|x| =
−x, x < 0.
Consequently,
|x| = |−x|, |xy| = |x||y|;
If |x + y| = a, then a ≥ 0;
If |x + y| = |x − y|, then x = 0 or y = 0.
On the number axis, if a point A represents the number a, then |a| is the
distance from A to the origin, and |a − b| is the distance between the points
representing the numbers a and b. For example, |9+ 7| = |9 − (−7)|, which is
the distance between the points representing 9 and −7.
Equations that have unknowns inside the absolute value symbol are
called equations with absolute values.
To solve an equation with absolute values, the key is to remove the
absolute value symbol based on its definition. In order to remove the abso-
lute value symbol, we sometimes need to divide the range of a variable into
parts for further analysis.
Example 1. The sum of all roots of the equation |2x − 4| = 5 is ( ).
(A) −0.5 (B) 4.5 (C) 5 (D) 4
1
