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Bất Đẳng Thức Luôn Có Một Sức Cuốn Hút Kinh Khủng, Một Số tài Liệu và Sách Bổ ích Cho Việc Học Tập.
1. An Introduction to Inequalities [E. Beckenbach, R. Bellman]
Book title: An Introduction to Inequalities.
Authors: Edwin Beckenbach, Richard Bellman.
Publisher: The L. W. Singer Company.
ISBN: 61-6228.
Preface
Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement (appropriately by a philosopher) is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art. Secondly, the fundamental results of mathematics are often inequalities rather than equalities.
In the pages that follow, we have presented three aspects of the theory of inequalities. First, in Chapters 1, 2, and 3, we have the axiomatic aspect. Secondly, in Chapter 4, we use the products of the preceding chapters to derive the basic inequalities of analysis, results that are used over and over by the practicing mathematician. In Chapter 5, we show how to use these results to derive a number of interesting and important maximum and minimum properties of the elementary symmetric figures of geometry: the square, cube, equilateral triangle, and so on. Finally, in Chapter 6, some properties of distance are studied and some unusual distance functions are exhibited.
There is thus something for many tastes, material that may be read consecutively or separately. Some readers will want to understand the axiomatic approach that is basic to higher mathematics. They will enjoy the first three chapters. In addition, in Chapter 3 there are many illuminating graphs associated with inequalities. Other readers will prefer for the moment to take these results for granted and turn immediately to the more analytic results. They will find Chapter 4 to their taste. There will be some who are interested in the many ways in which the elementary inequalities can be used to solve problems that ordinarily are treated by means of calculus. Chapter 5 is intended for these. Readers interested in generalizing notions and results will enjoy the analysis of some strange non-Euclidean distances described in Chapter 6.
Those whose appetites have been whetted by the material presented here will want to read the classic work on the subject, Inequalities, by G. H. Hardy, J. E. Littlewood, and G. P6lya, Cambridge University Press, London, 1934. A more recent work containing different types of results is Inequalities, by E. F. Beckenbach and R. Bellman, Ergebnisse der Mathematik, Julius Springer Verlag, Berlin, 1961.
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2.The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities [J. Michael Steele]
Book title: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities.
Author: J. Michael Steele.
Publisher: Cambridge University Press.
ISBN: 9780521837750.
Preface
In the fine arts, a master class is a small class where students and coaches work together to support a high level of technical and creative excellence. This book tries to capture the spirit of a master class while providing coaching for readers who want to refine their skills as solvers of problems, especially those problems dealing with mathematical inequalities.
The most important prerequisite for benefiting from this book is the desire to master the craft of discovery and proof. The formal requirements are quite modest. Anyone with a solid course in calculus is well prepared for almost everything to be found here, and perhaps half of the material does not even require calculus. Nevertheless, the book develops many results which are rarely seen, and even experienced readers are likely to find material that is challenging and informative.
With the Cauchy–Schwarz inequality as the initial guide, the reader is led through a sequence of interrelated problems whose solutions are presented as they might have been discovered – either by one of history’s famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way one finds systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and all of the so-called classical inequalities, including those of H¨older, Hilbert, and Hardy.
To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions. Still, as George Polya and others have taught us, there are principles of problem solving. With practice and good coaching we can all improve our skills. Just like singers, actors, or pianists, we have a path toward a deeper mastery of our craft.
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3. Analytic Inequalities [D. S. Mitrinovic, P. M. Vasic]
Book title: Analytic Inequalities
Authors: D. S. Mitrinovic – P. M. Vasic.
Publisher: Springer.
ISBN:
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4. Elementary Inequalities [D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok]
Book title: Elementary Inequalities.
Authors: D. S. Mitrinovic – E. S. Barnes – D. C. B. Marsh – J. R. M. Radok.
Publisher: P. Noordhoff LTD – Groningen.
ISBN:
5. Recent Advances in Geometric Inequalities [D. S. Mitrinovic, J. E. Pecaric, V. Volenec]
Book title: Recent Advances in Geometric Inequalities.
Authors: D. S. Mitrinovic – J. E. Pecaric – V. Volenec.
Publisher: Kluwer.
ISBN: 90-277-2565-9.
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Inequalities [G. H. Hardy, J. E. Littlewood, G. Polya]
Book title: Inequalities.
Authors: G. H. Hardy – J. E. Littlewood – G. Polya.
Publisher: Cambridge University Press.
ISBN:
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Inequalities – A Mathematical Olympiad Approach [R. B. Manfrino, J. A. G. Ortega, R. V. Delgado]
7. Inequalities – A Mathematical Olympiad Approach.
Author: Radmila Bulajich Manfrino – José Antonio Gómez Ortega – Rogelio Valdez Delgado.
Publisher: Birkhäuser.
ISBN: 978-3-0346-0049-1.
Preface
This book is intended for the Mathematical Olympiad students who wish to prepare for the study of inequalities, a topic now of frequent use at various levels of mathematical competitions. In this volume we present both classic inequalities and the more useful inequalities for confronting and solving optimization problems. An important part of this book deals with geometric inequalities and this fact makes a big difference with respect to most of the books that deal with this topic in the mathematical olympiad.
The book has been organized in four chapters which have each of them a different character. Chapter 1 is dedicated to present basic inequalities. Most of them are numerical inequalities generally lacking any geometric meaning. However, where it is possible to provide a geometric interpretation, we include it as we go along. We emphasize the importance of some of these inequalities, such as the inequality between the arithmetic mean and the geometric mean, the Cauchy-Schwarz inequality, the rearrangement inequality, the Jensen inequality, the Muirhead theorem, among others. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad problems. We also emphasize how the substitution strategy is used to deduce several inequalities.
The main topic in Chapter 2 is the use of geometric inequalities. There we apply basic numerical inequalities, as described in Chapter 1, to geometric problems to provide examples of how they are used. We also work out inequalities which have a strong geometric content, starting with basic facts, such as the triangle inequality and the Euler inequality. We introduce examples where the symmetrical properties of the variables help to solve some problems. Among these, we pay special attention to the Ravi transformation and the correspondence between an inequality in terms of the side lengths of a triangle $a,$ $b,$ $c$ and the inequalities that correspond to the terms $s,$ $r$ and $R,$ the semiperimeter, the inradius and the circumradius of a triangle, respectively. We also include several classic geometric problems, indicating the methods used to solve them.
In Chapter 3 we present one hundred and twenty inequality problems that have appeared in recent events, covering all levels, from the national and up to the regional and international olympiad competitions.
In Chapter 4 we provide solutions to each of the two hundred and ten exercises in Chapters 1 and 2, and to the problems presented in Chapter 3. Most of the solutions to exercises or problems that have appeared in international mathematical competitions were taken from the official solutions provided at the time of the competitions. This is why we do not give individual credits for them.
Some of the exercises and problems concerning inequalities can be solved using different techniques, therefore you will find some exercises repeated in different sections. This indicates that the technique outlined in the corresponding section can be used as a tool for solving the particular exercise.
The material presented in this book has been accumulated over the last fifteen years mainly during work sessions with the students that won the national contest of the Mexican Mathematical Olympiad. These students were developing their skills and mathematical knowledge in preparation for the international competitions in which Mexico participates.
8. Geometric Problems on Maxima and Minima – T. Andreescu, O. Mushkarov, L. Stoyanov
Book title: Geometric Problems on Maxima and Minima.
Authors: Titu Andreescu – Oleg Mushkarov – Luchezar Stoyanov.
Publisher: Birkhauser Boston.
ISBN: 978-0-8176-3517-6
Preface
Problems on maxima and minima arise naturally not only in science and engineering and their applications but also in daily life. A great variety of these have geometric nature: finding the shortest path between two objects satisfying certain conditions or a figure of minimal perimeter, area, or volume is a type of problem frequently met. Not surprisingly, people have been dealing with such problems for a very long time. Some of them, now regarded as famous, were dealt with by the ancient Greeks, whose intuition allowed them to discover the solutions of these problems even though for many of them they did not have the mathematical tools to provide rigorous proofs.
For example, one might mention here Heron’s (first century CE) discovery that the light ray in space incoming from a point $A$ and outgoing through a point $B$ after reflection at a mirror $alpha$ travels the shortest possible path from $A$ to $B$ having a common point with $alpha.$
Another famous problem, the so-called isoperimetric problem, was considered for example by Descartes (1596–1650): Of all plane figures with a given perimeter, find the one with greatest area. That the “perfect figure” solving the problem is the circle was known to Descartes (and possibly much earlier); however, a rigorous proof that this is indeed the solution was first given by Jacob Steiner in the nineteenth century.
A slightly different isoperimetric problem is attributed to Dido, the legendary queen of Carthage. She was allowed by the natives to purchase a piece of land on the coast of Africa “not larger than what an oxhide can surround.” Cutting the oxhide into narrow strips, she made a long string with which she was supposed to surround as large as possible area on the seashore. How to do this in an optimal way is a problem closely related to the previous one, and in fact a solution is easily found once one knows the maximizing property of the circle.
Another problem that is both interesting and easy to state was posed in 1775 by I. F. Fagnano: Inscribe a triangle of minimal perimeter in a given acute-angled triangle. An elegant solution to this relatively simple “network problem” was given by Hermann Schwarz (1843–1921).
Most of these classical problems are discussed in Chapter 1, which presents several different methods for solving geometric problems on maxima and minima. One of these concerns applications of geometric transformations, e.g., reflection through a line or plane, rotation. The second is about appropriate use of inequalities. Another analytic method is the application of tools from the differential calculus. The last two methods considered in Chapter 1 are more geometric in nature; these are the method of partial variation and the tangency principle. Their names speak for themselves.
Chapter 2 is devoted to several types of geometric problems on maxima and minima that are frequently met. Here for example we discuss a variety of isoperimetric problems similar in nature to the ones mentioned above. Various distinguished points in the triangle and the tetrahedron can be described as the solutions of some specific problems on maxima or minima. Section 2.2 considers examples of this kind. An interesting type of problem, called Malfatti’s problems, are contained in Section 2.3; these concern the positioning of several disks in a given figure in the plane so that the sum of the areas of the disks is maximal. Section 2.4 deals with some problems on maxima and minima arising in combinatorial geometry.
Chapter 3 collects some geometric problems on maxima and minima that could not be put into any of the first two chapters. Finally, Chapter 4 provides solutions and hints to all problems considered in the first three chapters.
Each section in the book is augmented by exercises and more solid problems for individual work. To make it easier to follow the arguments in the book a large number of figures is provided.
The present book is partly based on its Bulgarian version Extremal Problems in Geometry, written by O. Mushkarov and L. Stoyanov and published in 1989 (see [16]). This new version retains about half of the contents of the old one.
Altogether the book contains hundreds of geometric problems on maxima or minima. Despite the great variety of problems considered-from very old and classical ones like the ones mentioned above to problems discussed very recently in journal articles or used in various mathematics competitions around the world-the whole exposition of the book is kept at a sufficiently elementary level so that it can be understood by high-school students.
Apart from trying to be comprehensive in terms of types of problems and techniques for their solutions, we have also tried to offer various different levels of difficulty, thus making the book possible to use by people with different interests in mathematics, different abilities, and of different age groups. We hope we have achieved this to a reasonable extent.
The book reflects the experience of the authors as university teachers and as people who have been deeply involved in various mathematics competitions in different parts of the world for more than 25 years. The authors hope that the book will appeal to a wide audience of high-school students and mathematics teachers, graduate students, professional mathematicians, and puzzle enthusiasts. The book will be particularly useful to students involved in mathematics competitions around the world.
10. Secrets in Inequalities, volume 1, basic inequalities – P. K. Hung
Book title: Secrets in Inequalities, volume 1, basic inequlaities.
Authors: Pham Kim Hung.
Publisher: GIL publishing house.
ISBN: 978-973-9417-88-4.
Preface
You are now keeping in your hands this new book of elementary inequalities. “Yet another book of inequalities?” We hear you asking, and you may be right. Speaking with the author’s words:
“Myriads of inequalities and inequality techniques appear nowadays in books and contests. Trying to learn all of them by heart is hopeless and useless. Alternatively, this books objective is to help you understand how inequalities work and how you can set up your own techniques on the spot, not just remember the ones you already learned. To get such a pragmatic mastery of inequalities, you surely need a comprehensive knowledge of basic inequalities at first. The goal of the first part of the book (chapters 1-8) is to lay down the foundations you will need in the second part (chapter 9), where solving problems will give you some practice. It is important to try and solve the problems by yourself as hard as you can, since only practice will develop your understanding, especially the problems in the second part. On that note, this books objective is not to present beautiful solutions to the problems, but to present such a variety of problems and techniques that will give you the best kind of practice.”
It is true that there are very many books on inequalities and you have all the right to be bored and tired of them. But we tell you that this is not the case with this one. Just read the proof of Nesbitt’s Inequality in the very beginning of the material, and you will understand exactly what we mean.
Now that you read it you should trust us that you will find in this book new and beautiful proofs for old inequalities and this alone can be a good reason to read it, or even just to take a quick look at it. You will find a first chapter dedicated to the classical inequalities: from AM-GM and Cauchy-Schwartz inequalities to the use of derivatives, to Chebyshev’s and rearrangements’ inequalities, you will find here the most important and beautiful stuff related to these classical topics. And then you have spectacular topics: you have symmetric inequalities, and inequalities with convex functions and even a less known method of balancing coefficients. And the author would add
“You may think they are too simple to have a serious review. However, I emphasize that this review is essential in any inequalities book. Why? Because they make at least half of what you need to know in the realm of inequalities. Furthermore, really understanding them at a deep level is not easy at all. Again, this is the goal of the first part of the book, and it is the foremost goal of this book.”
Every topic is described through various and numerous examples taken from many sources, especially from math contests around the world, from recent contests and recent books, or from (more or less) specialized sites on the Internet, which makes the book very lively and interesting to read for those who are involved in such activities, students and teachers from all over the world.
The author seems to be very interested in creating new inequalities: this may be seen in the whole presentation of the material, but mostly in the special chapter 2 (dedicated to this topic), or, again, in the end of the book. Every step in every proof is explained in such a manner that it seems very natural to think of; this also comes from the author’s longing for. a deep understanding of inequalities, longing that he passes on to the reader. Many exercises are left for those who are interested and, as a real professional solver, the author always advises us to try to find our own solution first, and only then read his one.
We will finish this introduction with the words of the author:
“Don’t let the problems overwhelm you, though they are quite impressive problems, study applications of the first five basic inequalities mentioned above, plus the Abel formula, symmetric inequalities and the derivative method. Now relax with the AM-GM inequality – the foundational brick of inequalities.”
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Secrets inequalities volume 2 of Pham Kim Hung don’t have ebook.
11. Old and New Inequalities – T. Andreescu, V. Cirtoaje, G. Dospinescu, M. Lascu
Authors: Titu Andreescu, Vasile Cirtoaje, Gabriel Dospinescu, Mircea Lascu.
Publisher: GIL publishing house.
ISBN: 973-9417-35-3.
Preface
This work blends together classic inequality results with brand new problems, some of which devised only a few days ago. What could be special about it when so many inequality problem books have already been written? We strongly believe that even if the topic we plunge into is so general and popular our book is very different. Of course, it is quite easy to say this, so we will give some supporting arguments. This book contains a large variety of problems involving inequalities, most of them difficult, questions that became famous in competitions because of their beauty and difficulty. And, even more importantly, throughout the text we employ our own solutions and propose a large numbers of new original problems. There are memorable problems in this book and memorable solutions as well. This is why this book will clearly appeal to students who are used to use Cauchy-Schwarz as a verb and want to further improve their algebraic skills and techniques. They will find here tough problems, new results, and even problems that could lead to research. The student who is not as keen in this field will also be exposed to a wide variety of moderate and easy problems, ideas, techniques, and all the ingredients leading to a good preparation for mathematical contests. Some of the problems we chose to present are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities. Anyone will find here a challenge to prove his or her skills. If we have not convinced you, then please take a look at the last problems and hopefully you will agree with us.
12. Inequalities with Beautiful Solutions – V. Cirtoaje, V. Q. B. Can, T. Q. Anh
Authors: Vasile Cirtoaje – Võ Quốc Bá Cẩn – Trần Quốc Anh.
Publisher: GIL publishing house.
ISBN: 9786065000148.
Foreword
“Let solutions say the method!” is the way this book is written. Readers don’t find here the entire theory, strong theorems as well as detailed explanation of the methods. But you can find here a lot of beautiful problems with beautiful solutions. Most of these solutions are simple and elementary, the authors try to avoid as much as possible of using advanced methods of proving inequalities. The main weapons here are skilful technics of handling with algebraic expressions and virtuous applications of classical inequalities. It makes the book more romantic rather than academic. And even a student of 8th, 9th grade can read most of the content of this book.
Reading the book, you sometimes are surprising with the way the authors solve the problems. “How simple! Why didn’t I think about this?”, you ask. If you read this book for fun only, it is OK. But if you want to learn something from this book? Don’t be surprised only. You should ask more “Where does the solution come from? Why and how the authors think about the way?”. The answer is not simple, and you may not to find it immediately in single solution. Try to gather the answer from several solutions. If you succeed, you are going on the right way, the way that the authors of the book want you to go.
Tran Nam Dung
Ho Chi Minh city University of Science
Preface
“The only way to learn Mathematics is to do Mathematics.”
Paul Halmos
The inequalities appeared in Mathematics a long time ago, have developed and evolved stably in course of time, and even more in our days. As stated by Richard Bellman in 1978, “…there are at least three reasons for the study of inequalities: practical, theoretical, and aesthetic; …beauty is in the eyes of the inequality beholder; …it is generally agreed that certain pieces of music, art or mathematics are beautiful; there is an elegance to inequalities that makes them very attractive”. We add two new reasons to the three ones already formulated by Bellman: fascination to create a new strong and beautiful inequality, and happiness to prove such an inequality by an original and nice way. For all these reasons, the inequalities became very popular in advanced and elementary Mathematics, being very useful in level-transfer tests, in university entrance tests, and especially in national and international contests for excellent students. This explains why a large number of people are so concerned with mathematical inequalities.
Nowadays, many clever people find out a lot of new ideas and methods to deal with inequalities, and a lot of “modern style” reference books are published. In our viewpoint, the methods for solving inequalities are very important, but above all, learning how to think for creating or solving an inequality is even more important. We wrote the book “Inequalities with beautiful solutions” having in view these things, as well as our desire to make known to the inequality lovers some new inequalities of the authors.
With more than 200 problems, which are carefully and logically arranged, the book will help the readers form a general overview on the inequality field, as well as learn the secret of “finding way” to deal with inequalities and other mathematical problems. We look forward to receiving heart-felt comments from the readers to improve the book in the next republication.
Finally, we would like to thank Dr. Tran Nam Dung, Ho Chi Minh University of Nature, for his helpful suggestions, and our friends, Nguyen Van Dung, Tran Quang Hung, for their creative solutions and contribution in solving many puzzles.
The authors, October 2009
http://www.mediafire.com/?7a63xcs1la6gfas
13. Bất đẳng thức và những lời giải hay
Book title: Bất đẳng thức và những lời giải hay.
Authors: Võ Quốc Bá Cẩn – Trần Quốc Anh.
Publisher: Hanoi publishing house.
ISBN: 0101200338914.
Preface
Mình sẽ viết gì trong cuốn sách sắp tới? Câu hỏi này thực sự khiến chúng tôi trăn trở mỗi khi nghĩ về nó. Và sau nhiều đêm suy nghĩ, sau nhiều lần tổng kết từ những cuốn sách thành công, cuối cùng chúng tôi cũng tìm được cho mình câu trả lời: Thực ra một cuốn sách hay chỉ khi nó viết về những điều độc giả cần đọc. Đơn giản là vậy nhưng điều mà học sinh, sinh viên cần nhất trong khi học bất đẳng thức là gì? Đó không chỉ là kiến thức hay bài tập, mà chính là cách tư duy, cách giải quyết vấn đề không những áp dụng được trong toán học mà trong cả cuộc sống mỗi chúng ta. Muốn thành công, bạn phải vượt qua những thử thách không phải ai cũng vượt qua được. Cuộc sống luôn thay đổi, những vấn đề mới xuất hiện một cách liên tục. Nhiều khi, đi theo những lối mòn không giúp bạn giải quyết được vấn đề mà ngược lại, chỉ làm mất thời gian của bạn. Câu hỏi đặt ra là “khi đối mặt với chúng, bạn nên làm thế nào?” Chúng tôi sẽ chia sẻ với các bạn thế này: Trong bất đẳng thức nói riêng, có rất nhiều những bài toán đẹp không có lời giải trong một thời gian dài. Đó là do “những ý tưởng cũ không còn phù hợp với những bài toán mới”. Bạn muốn có kết quả tốt hơn, bạn phải làm khác đi, tư duy của bạn phải tuyệt vời hơn những người bình thường. Và cuốn sách này ra đời nhằm mục đích thay đổi tư duy của bạn, mang đến cho bạn cách nhìn hoàn toàn mới về những vấn đề “mở” thậm chí tới hàng chục năm. Chúng tôi tin chắc rằng khi nắm bắt được những ý tưởng ấy, cuộc sống của bạn sẽ thay đổi nhiều hơn bạn có thể tưởng tượng. Đây cũng chính là lí do khiến chúng tôi cố gắng thực hiện quyển sách này.
Mặc dù cố gắng biên soạn thật kĩ càng nhưng sai sót là không thể tránh khỏi, rất mong bạn đọc thông cảm và góp ý để giúp chúng tôi hoàn thiện cuốn sách hơn. Mọi ý kiến đóng góp, xin gửi về địa chỉ: babylearnmath@yahoo.com. Xin chân thành cảm ơn.
Old and New Inequalities 2, Vo Quoc Ba Can – Cosmin Pohoata
14. Old and New Inequalities, volume 2.
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Authors: Võ Quốc Bá Cẩn – Cosmin Pohoata.
Publisher: GIL publishing house.
ISBN: 9786065000032.
Preface
”The last thing one knows when writing a book is what to put first”
Blaise Pascal
Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art, as G. H. Hardy liked to consider it. Secondly, the fundamental results of mathematics are often inequalities rather than equalities.
In the pages that follow, we present a large variety of problems involving such inequalities, questions that became famous in (mathematical) competitions or journals because of their beauty. The most important prerequisite for benefiting from this book is the desire to master the craft of discovery and proof. The formal requirements are quite modest. Anyone who knows basic inequalities such as the ones of Cauchy-Schwarz, Holder, Schur, Chebyshev or Bernoulli is well prepared for almost everything to be found here. The student who is not that experienced will also be exposed in the first part to a wide combination of moderate and easy problems, ideas, techniques, and all the ingredients leading to a good preparation for mathematical contests. Some of the problems we chose to discuss are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities. Nevertheless, the book develops many results which are rarely seen, and even experienced readers are likely to find material that is challenging and informative.
To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions. Still, as George P´olya and the others have taught us, there are principles of problem solving. With practice and good coaching we can all improve our skills. Just like singers, actors, or pianists, we have a path toward a deeper mastery of our craft.
15. 567 Nice And Hard Inequalities(My Book)
Authors: Nguyễn Duy Tùng.
Publisher: HaNoi publishing house.
Preface
Sayan is writting.
Algebraic Inequalities Old and New Methods
Vasile Cirtoaje
Book title: Algebraic Inequalities Old and New Methods.
Authors: Vasile Cirtoaje.
Publisher: GIL publishing house.
ISBN: (10) 973-9417-66-3.
ISBN: (13) 978-973-9417-66-2.
16. Những Viên Kim Cương Trong Bất Đẳng Thức Toán Học.
Authors: Trần Phương.
Publisher: Hanoi publishing house.
ISBN: 97860222313.