The function is in the basic part.
Functions mainly play the role of foreshadowing, and the topics in this part are generally not difficult, mainly basic algebraic deformation and discussion. This part of the entry book is almost the same, refer to the "Olympic Mathematics Course" one volume higher. The difficulty of function part is function equation and Gaussian function.
This part of the problem of functional equation often appears in competitions, and Cauchy method is the most common and important method to solve this kind of problem. At the same time, we should pay attention to zero point, fixed point, special value and common substitution. In the process of learning functional equations, we can refer to the solution of differential equations appropriately. For some problems that are difficult to see the original function, we can usually assume that the function is differentiable, solve the original function with differential equations, and then give the proof of the elementary method according to the characteristics of the original function. Reference function equations and classical algebraic volumes.
Gaussian function is an important number theory function, which has many uses in number theory. It is of great help to simplify the problem-solving process to grasp its deformation skills quantitatively. At the same time, pay attention to the substitution skills when dealing with Gaussian function. Referring to the Gaussian function part of the mathematics competition research course, the series of questions in the 2005 national team trials are the key parts of high school learning, and their topics can involve any part of algebra, so they are favored by proposers. This part of learning needs to master the general term solution of various common sequences and the related theory of fixed points, and pay attention to the cultivation of computing ability. Refer to the series of "A Course of Olympiad Mathematics" and "A Course of Mathematical Competition Research".
plural
The complex part mainly focuses on the combination of numbers and shapes, the thinking of complex geometric problems and algebraic problems. Pay attention to the thinking of classic topics. This part of the topic involves many important methods in mathematics, and simple topics should be carefully studied. Refer to the complex part in the research process of mathematics competition.
inequality
Inequality is an essential question type in mathematics competition, and few people can solve it every time a new question type appears. There are many methods and strong algebraic skills in this part, but most of them are only variants of A-G inequality and Cauchy inequality. Therefore, when solving problems, we must think clearly, don't get lost in the ocean of formulas, and don't use higher inequalities indiscriminately. Of course, we should also understand the inequalities in higher mathematics, such as Zhan Sen inequality. When solving problems, we should make full use of the conditions of taking the equal sign to seek clues to solve problems, and mainly write out the conditions of taking the equal sign when writing. Refer to the inequality part in the research process of mathematics competition, the title of the algebra volume of the textbook, and the titles of previous competitions.
multinomial
Polynomial is the part in which the way of thinking is biased towards higher mathematics in mathematics competition. When solving a problem, we mainly investigate two expressions of a formula, that is, we should pay attention to the investigation of special values. Note that this is generally a complex number, so it will involve the processing skills of complex numbers, especially Chebyshev polynomials. At the same time, master Lagrange and Newton interpolation formulas skillfully. Refer to the polynomial part, inscription and algebra volume, inequality and Cauchy inequality in the Olympic number series, and the topics of previous competitions in the course of Olympic number research.
geometry
High school geometry includes plane geometry, analytic geometry and solid geometry. Generally speaking, the latter two will only appear in one exam, and it is not difficult, mainly to examine the mastery of basic knowledge points and the proficiency of calculation; Plane geometry is one of the necessary questions in the competition, which examines the players' grasp of graphics and the active degree of thinking. The basic knowledge of plane geometry will be mentioned in every competition book. You need to master Menelius Theorem, Cheval Theorem, siemsen Theorem, euler theorem Theorem and Ptolemy Theorem. Very familiar with the common conclusions in geometry and various geometric transformations, including translation, rotation, phase similarity, polar coordinates and inversion. There are not many knowledge points in this part, mainly because the player grasps the graphic structure. When dealing with problems, we should pay attention to choosing various methods flexibly, and don't think that properly introducing triangles, analytic geometry, vectors and complex numbers is quite beneficial to prove problems. Refer to Modern Euclidean Geometry, Hunan Geometry Volume and affiliated high school of south china normal university Geometry.
-Geometric inequality is very difficult, which is more difficult than the conventional plane problem. Players who participate in high-level competitions need to strengthen their training. Reference geometric inequality.
Analytic geometry
This part of the topic generally involves a lot of calculation, and the focus is on the training of calculation ability. Don't pursue the simplest method from the beginning, just ensure the correctness of the calculation. After reaching a certain level, thinking about the simplicity of practice will naturally appear, and attention should be paid to the naturalness of thinking and the symmetry of methods. Referring to the second-year volume of Olympic Mathematics Course, Skills of Analytic Geometry is a single work.
solid geometry
This part is the training of spatial imagination, and the general topics are very simple, so even people with weak spatial imagination can solve most problems through analytic geometry. Pay attention to the beauty of drawing and the accuracy of calculation. Refer to the solid geometry part of "Olympiad Mathematics Course" and "Mathematics Competition Research Course"
number theory
Number theory is a very beautiful part of the competition, which involves many classic skills in elementary number theory. Through this part of the study, we can master the process and method of defining a new system, so we must pay attention to this part of the content as a system and inseparable. To learn number theory, we must seriously study elementary number theory. For some parts that are not detailed, you can refer to Professor Hua's Introduction to Number Theory and master the basic ideas and methods skillfully. Many difficult problems are made up by simple topic methods. Refer to elementary number theory, introduction to number theory, the number theory part of affiliated high school of south china normal university's problem set, and the book of number theory.
-Classical indefinite equation
This part is a classic part, and its basic skills are constant modulus, factorization and algebraic deformation. The topic is generally not difficult, as long as you pay attention to special circumstances. This part-Pell equation is a hot topic in recent years, and its general solution formulas and derivation need to be mastered. To master this part of knowledge, we need to learn Legendre symbols, Gauss quadratic reciprocal law, Jacobian symbols, continued fractions, rational approximation of irrational numbers and so on.
-Exponents and primitive roots
Although this part will not be explicitly put forward in the competition, many ideas are actually used in this part of knowledge, so it is very beneficial to master it skillfully.
combine
This part is really a hodgepodge. The three aspects mentioned above will be used here, and there are also some methods of their own. Every problem will have different methods, so thinking needs to be highly divergent. Generally speaking, except for classic problems, which can be solved by some universal methods, the rest of the problems are completely a manifestation of mathematical intuition, which requires a lot of training and constant summary to correct the deviation of thinking in solving problems. Reference inscription combination volume, affiliated high school of south china normal university problem set combination part, mathematics competition research course combination part.
Training of Mathematical Competitors
The math contest is boring. If you are not interested, it is a waste of time to hold it. Therefore, for a competitor, the first is the interest in mathematics. Then there is confidence. You will encounter many difficulties at the beginning of your study. Even if your level is relatively high, you will enter a very long platform period. At these times, self-confidence is your motivation to continue learning and your weapon to break through obstacles. For players who want to participate in the competition, if they lack self-confidence, they will often appear to be lacking in confidence in the examination room, and there will be bad emotions such as anxiety when solving problems, which will seriously affect their performance. Therefore, self-confidence is a necessary condition for their success. It is the cultivation of study habits after having a good attitude. The first thing is to have a long-term and short-term plan and constantly urge yourself to complete the plan according to the plan. When studying, be practical, make clear the basic problems, and don't hide them because of embarrassment. The tedious calculation and writing must be done carefully, so as not to lose points because of the tension in the examination room. When the level reaches a new height, it is necessary to start summing up frequently, such as writing down the better questions you have done recently and the ways to answer certain questions. In this way, after a period of time, there will be a set of self-compiled learning materials, and it is best to review these materials before the big exam. At ordinary times, you should always read your own summary and thoroughly understand every question. Also pay attention to the latest information consciously. Some math enthusiasts have the latest competition questions on their websites, such as Mathlinks. For higher level players, the cultivation of thinking mode is very important. To train your first feeling, try to let yourself know the direction as soon as you see the topic, so that even if you can't, there will still be some process points. Of course, this can't be done just by talking. This requires long-term training and high mathematical talent.
My failure
We have three seed players this year-me, Ye, and. Of the three, Ye walked to Zhejiang University with the first place in the league, entered the national team and won the IMO perfect gold medal, while I entered Shanghai Jiaotong University in the college entrance examination. Three people usually study together, the level is not much different, but the results are far apart. In the days of preparing for the college entrance examination, I often think about this problem, hoping to help the college entrance examination. Although it has always been said that it is a matter of mentality, I never thought it was like this. I didn't know it was true until I took the college entrance examination. It took me two and a half years to enter the competition, and I found that I won four second prizes and had to go back to class to prepare for the college entrance examination. It may be really scary to finish all the courses in high school in six months, but I did it anyway, and I was admitted to Shanghai Jiaotong University, while many other people who usually got higher grades than me and took longer to prepare for the college entrance examination had lower grades than me. Why? Because my goal at this time is only Huazhong University of Science and Technology, I believe I can do it. I am so confident that I don't have any burden in my studies and exams. I can do well in the college entrance examination, while others may take the exam with too heavy a burden ... When I think about my own league, I really think too much before the exam, so I lack confidence. Although I feel good about myself, I actually have a poor mentality, so I make mistakes in exams again and again.
I hope that future contestants can learn this lesson, greet each game with the best attitude and achieve the best results.