The author of this article is Tang Jingkai, a student from Chengdu No. 7 Middle School. He started studying mathematics competitions in his first year of high school. In his third year of high school, he was selected for the national training team with a good score of tenth in the country and was recommended to Yao class.
Compared to many national gold medalists, Tang Jingkai is relatively late to start, but since the beginning of the introduction, he has invested in mathematics with great enthusiasm and confidence. The previous two failures did not let him down, but instead made him go all out in the last year without regrets.
After the finals, Sister Zhongxin approached classmate Tang Jingkai, hoping that he would share his learning experience with the classmates, so the eloquent 5,000-character long essay was presented in front of everyone, from the competitive journey to the learning method to the recommendation of the book list. This is really a very careful and dry experience post, I hope it can be helpful to the students in the competition.
Hello everyone, I am Tang Jingkai from Chengdu No. 7 Middle School. In 2019, Peking University Golden Autumn Camp won the first place in the first prize. In 2020, the National High School Mathematics League entered the Sichuan Provincial Team with 272 points. In the final, he entered the National Training Team with a national score of 10th and was recommended to Tsinghua Yao Class.
In addition, in the formal (on-site) examinations of all training institutions, the fifth, fourth, and third place were ranked once, the second place was twice, and the rest were first place. It seems to go well, but in fact, my racing career has had ups and downs.
Road to competition
Compared with my classmates around, I actually studied math competition later, and I started to learn math competition formally a month before the first year of high school. At the beginning, I didn't know anything about mathematics competitions, and the choice actually came from the confidence and preference of mathematics. But to put it bluntly, I was selected into the math competition small class and started my road of competition.
At the end of July, we started to listen to the lectures of the seniors. At that time, it was in the clouds, the quadratic residue, inversion, Sondat, cyclotomic polynomials, Lucas, Kummer, primitive roots... everything. The senior's handout is called "High Mountain Yangzhi", which I think is quite appropriate.
This was followed by more than ten consecutive days of simulation of the league. My first attempt was almost the worst in the class. Twenty to thirty points is the norm, and the worst one got ten points. I did not understand the lectures of the seniors as well as I expected. Algebra and number theory are still not as good as the topics, obviously. In the second test, you can almost only do geometry problems each time, and you can often score 20 points on the combination of students who hit zero. This gave me a lot of confidence, and convinced me that I was suitable for learning math and competition.
I didn't have much hope in the first league, after all, I only studied for a month, thinking about trying the water, so I went to take the exam. I was so excited when I tried it, and I thought it was much easier than usual (although it is). After reviewing the questions after the second test, you can see that the first question can be reduced to n=2, and the third question uses the drawer principle. Helpless foundation is too bad. The first question n=2 was written on the answer sheet for half an hour and overturned the water glass and had to rewrite it. The next time was stuck in the geometry problem that I thought I was best at at the time. There was no progress in 2,3,4.
As expected, the league scored 84+30 and only won the second prize. Nearly half of the classmates went to the Qingbei Golden Autumn Camp, and I could only study the content of the college entrance examination at school. That was the first time I felt the shame of not being strong enough. I am determined to become stronger.
There was no arrangement for suspension of classes in the first year of high school, so I suspended classes myself. I don't listen to the usual comprehensive courses very much, and I rely on copying my homework (you don't want to learn from me), and I can do my own competitions when I have time. At that time, there seemed to be a kind of enthusiasm driving me, and I can still pull out my biology books, English workbooks or chemistry papers full of drafts.
The first thing I did was "Olympic Mathematics Tutorial". When I finished the second third and second third of the high school in November, I felt the enlightenment, as if I had some integration. The first test went up, the second test went up, and the weekly league simulation results also went up. I have at least three questions almost every time, and often four. At that time, I had a hearty thrill in the simulation of the league. It seemed that every question was just on the boundary of my ability, and I could just do it every time, and it was often a simple method different from the standard answer. Taking advantage of this momentum, I finished the small blueprint "Number Theory" in four days, and it took another three months to finish the "Research Tutorial".
Throughout the whole winter vacation, my racing life can be described as a spring breeze. The test scores during the training have been mentioned earlier. I have always been the first in the same level when the seniors in the school asked me. I was the first in the whole level except for one. In June, July and August, the simulation minimum of two hundred points is also the lowest, and early submission of papers is the norm. At that time, I was immersed in a kind of fluttering confidence, as if I was the chosen son of heaven. Needless to say, the provincial team, the CMO should be at least the gold medal, how can you go to the elite class. At that time, I met a close classmate and said, "Called the senior."
However, life will never be as smooth as you imagined, and you will always hit you in unexpected places. With the defeat of Beixia, I only took the second-class appointment, and I ushered in my second league.
On September 8, 2019, that was my second league. I tried a simple question wrong and I had a good mentality. The first two questions in the second test were quickly completed, but unfortunately the process was too lengthy to write. It was almost 11 o'clock after finishing writing. The third question was the number theory that I was best at at the time, and I happened to have done a similar number theory before the exam. I tried the previous method first, but it didn't work. I started to think normally again, and found that if the conclusion is correct, then the condition that an is an integer can be removed, repeated several times, and it remains the same. Turning to look at the fourth question, not only did not find that 908 is a special data, but also misunderstood the question, and did not make progress for 20 minutes.
At this moment, I feel my face is hot. I went out to wash my face with cold water, and went back to continue working on the question. I found that I had misunderstood the fourth question, so the previous ideas were useless. I decided to die on the number theory that I was best at at the time. However, the previous discovery allowed me to directly remove the condition that an is an integer (in order to remove interference), and did not make it in the end, so I had to write some points for each of the three or four questions.
After the exam, I went back to school in the afternoon. Almost everyone made questions 1 and 2. About half of the people who are good in number theory made the third question, and those who were bad in number theory made the third question. So although the senior teachers comforted me that I still had a chance, I knew that I could no longer enter the provincial team, and I knew that I had lived up to the expectations of those who cared for me. I cried a lot on the way home. Fortunately, being accompanied by my dear friends and family members made me feel a little better.
The results were achieved, as expected, 97+100, and did not enter the provincial team. More than half of the classmates in the class have suspended classes to prepare for the winter camp, and I can only go back to the class for synthesis. I can imagine them frolicking or studying in the competition classroom, but I can only silently move my comprehensive textbook from the competition building back to the teaching building, listening to classes, taking exams, and doing homework. I have imagined that I will show off my talents in CMO countless times, but the reality is that I can only applaud and applaud my classmates.
One year's time, one year of hard work, it seems that something has changed, and nothing seems to have changed. The policy is getting tighter and tighter. If you do not have a provincial team in your third year, you will be extremely passive. At the beginning, I like to say that I would retire if I didn't enter the provincial team in my second year of high school, but I didn't expect it to be true. Is my road of competition really coming to an end like this?
Perhaps because of unwillingness, perhaps because of pride, or perhaps because of love, I said to myself: "No, even if I didn't join the provincial team, I will continue to move forward like them."
I arranged my schedule to be the same as that of the provincial teammates. Everything was so similar, I started to suspend classes myself again. I don't listen to the usual comprehensive courses very much, and I rely on copying my homework (and I have to persuade everyone not to learn from me). At that time, I was tossed around the country, and I also vented and brushed up questions. When I finished the IMO SL, ELMO, USA TST, USATSTST in recent years, and when I brushed the "Towards IMO" of the past ten years, the epidemic came to an end.
I started back to school in April, and I started preparing for the league when I returned to school. I reflect on my last defeat in the league. In addition to playing this metaphysical reason, the main reason is that the trial is not strong enough, the mentality is not good enough, the process is written too long, and the strength is not strong enough.
I did all the test questions sent by the teacher, the simulations I bought myself, or the set of test questions one by one. In order to prevent impetuous mentality, unless I am in a hurry, I will not submit papers in advance. The gaps in the preparation for the league are also for CMO and above in order to maintain the state. In this way, I finally ushered in my third league match in the various exams, courses and self-study.
In fact, for the third time in the league, I did not get rid of the problem of writing too finely as soon as I got to the formal exam process (and later CMO also), but fortunately, when the strength was reached, it was a matter of course. The first attempt is smooth and steady, and the paper is finished ahead of schedule. In the second test, the third question was made between the issuance of the paper and the opening of the test. Then, I did questions 1, 2, and 4 without any twists and turns. Finally, I finished the paper ahead of schedule. Finally entered the provincial team smoothly.
I don’t want to say much about the winter camp in the last year. It seems to have done well in the exam, but I still want to be better. I only hope that my road of competition will not end here, and that I can give a satisfactory explanation for my three years in high school.
I want to thank my parents and coach (Mr Jiang Haibing). Without them, I would not be where I am today. I also want to thank my seniors and classmates for their encouragement and inspiration. Finally, I also want to thank Sister Zhongxin for giving me so many opportunities to talk about it here. thank you all!
study method
The most important thing is to say first: There must be a heart to be the first. Don't indulge in discussions with classmates. When the strength and thinking are not in place, it is often decadent and waste of time.
When reading the book, you must do it one by one, and don't skip the sample questions. Often the most essential part of a book is its sample questions.
You must think more about the questions yourself, don’t just look at the answers if you can’t do it in an hour or two (except for beginners, at that time you should give priority to building the foundation). Merely remembering the answer is no use, only thinking and understanding can improve the strength. It is recommended that you prepare a notebook to record the questions and methods you have done in case you forget it in the future.
Going out for training means going out to do problems, and a good coach is a coach who can find good problems. Good questions are the most fundamental thing. You must have the ability to reject them. If you think that the question is unlikely to help you, you should skip it (similarly, you must also have the ability to refuse to go out for training). AOPS (search AOPS directly) is a very good website where you can find almost all the questions you want.
Must learn to organize. Organizing notes directly in class is actually not very effective, in fact, it is dictation or dictation. After class, it is recommended that you copy topics that you think are inspiring in your notebook, and do it again with memory and understanding. The wrong questions in the exam should also be redone, so that you will understand more deeply. It is recommended to use a notebook for organizing, which feels smoother, and a loose-leaf notebook for wrong questions, which is easier to organize.
Finally, before dividing into sections, let’s talk about a few familiar books:
"Olympic Mathematics Tutorial" is personally recommended for beginners. It is recommended to look backwards from the third year of high school. It is enough to see the analytical geometry of the second year. If you understand it, it will be very helpful to your strength. You can see if you want to.
"Research Tutorial" is actually relatively simple in number theory and geometry. Algebra is a simple method. Personally, I like it. Its essence lies in combination, especially graph theory. I understand that most of the graph theories in the competition are no problem. Up.
If you want to watch the small blue book series, it is recommended to watch 4, 5, 6, 7, 10, 13, but the overall difficulty is not very large (the seniors evaluate the difficulty of CMO), if you have certain strength and hope to improve, you don’t need to watch it. Up.
The series of "Lectures by the Propositioner" is mainly to expand horizons. I highly recommend "Elementary Number Theory". This book is basically no problem after finishing the number theory. In addition, if you want to read it, I recommend "Analytic Geometry" and "Sets and Correspondence". The book raises the level of thinking.
Personally do not recommend the "Olympic Classics" series. The above questions are mixed. The biggest advantage after doing it is that you can often see the questions you have done and become a Chinese question bank in the eyes of students. If you want to do it, it is recommended to do it as a problem set when there is no problem.
The two PFTBs (problems from the book and proofs from the book) are fine and you can watch and play.
In fact, it is not necessary to do a series of questions to do the questions. You can do the real questions when you think you have a certain level. At the same time, you can read some college books after one round of study. You will find that many topics have advanced backgrounds. Give it a try
There is nothing to say in the first try. Practice with the school coach, sort out the wrong questions, adjust the status, and you can take the test in one try. Algebra I believe everyone must learn inequalities first. I think that learning the mean and Cauchy in inequality first is very unfriendly to beginners (of course, the mean Cauchy must know it). At that time, I didn’t understand why the mean Cauchy had to assign coefficients in this way, so I was confused. Water, I feel that my inequality is simply hopeless.
Personally, I think the most important thing for beginners to master is the violent method—that is, deployment and adjustment (the application of adjustment is wider). First, the machine is simple and can be described as a general method. Most inequalities can be done in this way, which can increase confidence. heart. The second is to be familiar with algebraic structure and improve algebraic literacy. After several months of expansion and adjustment, your algebra level has also risen. At this time, if you look at the mean Cauchy, you can probably understand a little bit of it yourself, and you can use it yourself. You can read "The Secret of Inequalities", which has many ideas.
In fact, the problem of the sequence of numbers is not essentially a problem of the sequence of numbers, but just a layer of sequence. It is natural to master other aspects well.
Polynomials do not need to be deliberately practiced. It is a very comprehensive content. When your level improves, the polynomial level will also increase.
Geometry is actually the least talented plate among several plates. It can be slowly raised through accumulation and practice. When you are a beginner, just find this textbook to familiarize yourself with the basic theorems and then do the questions yourself. Don’t indulge in doing geometry problems when you are up to the level of geometry, but you should also keep your hand feeling. Just one a day. If you don’t do geometry for a long time, it will lead to a cliff-like decline in geometry level (a painful lesson).
Normally, when doing questions, you should think about geometric methods to cultivate geometric intuition, but do not despise and must master the calculation methods (plural numbers, triangles, center of gravity coordinates, etc.), as long as you can solve the problem method is a good way. The trigonometric method is covered everywhere, but I haven't seen any books on complex number calculations in detail. If you want to know the complex number method, ask your senior.
The center of gravity coordinate has miraculous effects in some problems. For example, the "root axis IG" problem of pure geometry can be solved quickly with the center of gravity coordinate. If you want to learn, "Secondary Mathematics" in 2017, the second issue of Li Zhaohui's "Discussion on the Proof of Plane Geometry Based on the Center of Gravity Coordinate System" is very good. ("Intermediate Mathematics" can be ordered, watch and play when you are fine.)
If you have no questions, you can go to pure geometry. Many of the above questions have TST difficulty. To lay the foundation for number theory, just do the books that I told you before. Number theory must learn "black technology", don't be afraid to use advanced theorems, do what you can do, and use what you can. When you have more energy, you can check out Pan Chengdong, Pan Chengbiao's "Elementary Number Theory" and PFTB. Combinations do not actually need to be practiced deliberately, because combination ideas are more or less used in the problems of each section. Mathematics induction method must be mastered proficiently, after mastering a large part of the problem (not limited to combination) can be done.
In fact, there are almost general methods for combinatorial identities in the competition. You only need to multiply the variables on both sides of the equation and then sum them diagonally (as shown below). In fact, this is the summation of power series. It is recommended to master the Taylor expansion of simple functions (if you have more power, it is also recommended to master the Fourier expansion, which has amazing effects in some problems).
Combinatorial geometry only needs to be proficient in the Heley theorem and basic combination skills geometry knowledge to be able to do most of the combination geometry problems in the competition. In fact, most of the graph theory in the competition can use mathematical induction. If you want to improve your level of graph theory, it is recommended to read "Introduction to Graph Theory" (the key lemma of the fourth question in the 2019 League is one of the exercises in one of the chapters), instead of looking at the "Graph Theory" of "Proposition Man Lecture", above The content of is basically covered in "Introduction to Graph Theory". Details can be viewed on the Centroid Physics Network .
I have said so much without knowing it, I hope it will be helpful to everyone.