After reading the recommended answer, I thought it was well written, but I didn't say it very carefully. .....................................................................................................................................................

If you want to prepare for the high school math league, you must know the main types of high school math league exams. According to my impression, every big question in the second volume of the league is tens of points, and the content of the questions is divided into: 1. Plane geometry II. Number theory 3. Function 4. Inequality 5. Combination (sometimes alone, or using analytic geometry to solve plane geometry).

As for these five types of questions, according to my personal opinion, most senior high school students who are preparing for the Olympic Mathematics don't have so much energy and ability to study and master every aspect (except for a few talented people who have been trained since childhood, I am personally ashamed to be inferior to those people), so I can only choose 3 to 4 types from these five types according to my personal preferences and abilities to study and strive to reach a certain depth.

According to my personal experience, I would like to discuss with you four categories: 1, plane geometry 2, number theory 3, function 4 and inequality. As for 5, I will skip it automatically, because I only know a little.

1. Plane geometry

There are two common methods to solve plane geometry in the alliance: pure geometry method and analytical geometry method.

Pure geometric method: the proof of this kind of method needs keen eyes to find the key to prove the problem and make corresponding useful auxiliary lines; Then the proof process is deduced through strict logical reasoning. It is suggested that before starting to learn plane geometry, it is best to read Euclid's Elements of Geometry and get familiar with it. Although the element of geometry looks simple at our high school level, I think it is more important. It is the basis for you to learn geometry. When learning, it is natural to learn how Euclid's "23 definitions and 5 postulates and 5 axioms" are clear at a glance. Moreover, when learning plane geometry, we should learn more about Euclid, use our known knowledge to find out how the theorems of plane geometry we have learned or encountered are derived, and be familiar with this process so as to use these theorems flexibly in the future. Personally, if you want to study plane geometry in depth, you should buy a book that only talks about plane geometry. )

Analytic geometry method: It is not easy to write out the whole process when solving plane geometry by analytic geometry method. Although the key is how to establish a good coordinate system and then transform the conditions and the problems to be proved into analytical expressions, the key is how to transform the equations listed according to the conditions into the equations to be proved. This process is very complicated and troublesome, and I personally feel awkward to do it. But I think to break through this difficulty, we should look for a clever simplified method according to the process of solving by pure geometry method. As for how to learn this method well, you'd better learn the function equation first.

2. Number theory

Number theory is a branch of mathematics, which can easily attract math lovers, but many problems in number theory can easily make people feel helpless. I suggest you buy some books that only talk about number theory when you study number theory. Don't dwell on the questions you can't do when you study by yourself, or look at how the answers are worked out first, but I still want to remind you of the sentences when you look at the answers (first understand the steps of the answering process, the number theory knowledge and related theorems used in the answers, and understand what role these theorems play in this question and under what conditions; Then copy the answer once or twice; Write it again by yourself, and then compare it with the answer to see what is different; Finally, find someone who is similar to this problem and practice the knowledge and theorems of number theory that you have just learned by yourself. )

3. Function

There are many and miscellaneous contents in this aspect of function, but one key point is that many of his contents in this aspect are the contents of high school courses, which are very close to the college entrance examination. Personally, I think the function problem in the league is more difficult than the college entrance examination, and sometimes it is not as difficult as the college entrance examination. I think that when learning functions, we need to do well in the college entrance examination, so that we can do well in the college entrance examination and cope with the function problems in the league.

4. Inequality

The proof of inequality can better reflect the flexibility of personal thinking. On the one hand, it not only tests the application of some important inequalities after the contraction of the inequality to be proved; On the other hand, the flexible use of functions requires you to skillfully construct a function, and then list a clever inequality through some prominent properties of the constructed function, and then prove it with this inequality. This process is a bit abstract and needs to be studied slowly at school.

It is suggested that when learning inequality, you still need a tutorial book on inequality to learn. I hope you can learn plane geometry as mentioned above, know how to prove and how to prove the important inequalities you have learned and encountered, and then imagine whether you can figure it out.

If you think what I said is very useful for you to prepare for the senior high school league, or if you feel the effect is good after trying it yourself, you may wish to support my answer.