After reading the supplement to your question, I think your purpose of learning advanced mathematics is wrong, and you want to take a shortcut by learning advanced mathematics. In fact, you can't learn anything by studying advanced mathematics for this purpose, but you will be impetuous. As soon as you do the problem, you want to work it out in a simple way.

Depend, the world which have such a simple thing? In fact, there are many experts who study advanced mathematics, and their problem-solving ability is different. Their problem-solving ability is stronger than that of high school students. They only know how to think scientifically and analyze problems, instead of changing those formulas under extremely limited and special circumstances, as in high school. Of course, you can't despise high school knowledge any more than you can despise kindergarten learning Arabic numerals. Everything is step by step. )

Therefore, learning advanced mathematics is of little significance to solving problems in high school. But what you said about awesome people is also true. Through systematic advanced mathematics, people's scientific thinking ability (logic, abstraction, analysis, etc. ) you can exercise.

After you have these abilities, it's only a matter of time before you learn anything else (certainly not too long, math graduates are very popular with enterprises in the United States)

These things are also useful for high school problems, but they are very limited. The essence of high school mathematics is the application of some formulas. Of course, there are certain requirements for analysis, but the proportion is not large.

The most important thing in learning high school mathematics is to do more questions, analyze the proposition ideas, misread the problem book, insist on independent thinking, complete the problem efficiently within a limited time, analyze and summarize after completion, analyze the reasons why you can't, the reasons for thoughtlessness, and the reasons for writing slowly, and write down your own analysis in the wrong problem book. Make sure to do it right every time, train yourself to do what you can and keep a high accuracy (don't use carelessness as an excuse, because your understanding is not deep enough). At first, you would rather write slowly than make mistakes, which are full of loopholes.

Also, if you really want to take some shortcuts, I suggest you look at the expansions of some inequalities, such as Qin Sheng inequality, multiple mean inequality and harmonic inequality. , and recursive skills of series (generally in Olympic books), but never delve into it. The college entrance examination attaches importance to the general way of thinking and should adhere to the usual thinking habits.

There are also fixed patterns in high school subjects and exams. As long as you have confidence, hard work and persistence, those patterns will be clearly presented to you and make you suddenly enlightened.

You must give me extra points for writing so much. Good luck!