After roughly understanding the rules, Qiao Yu directly opened the browser, entered the finals address, and logged in to his account.
The competition is not a college entrance examination. It doesn't matter if it's one minute early or one minute late. You can't take any advantage, because the background will automatically time it. Anyway, everyone is only given a total of eight hours to do the questions. And it is eight consecutive hours.
That is to say, once the timer starts, it cannot be stopped.
Whether you eat, drink, or go to the toilet in between, it is all counted in the answering time. Fortunately, this is not a big deal for a group of young people.
Whether they are junior high school students or high school students, they may not necessarily run very fast, but most of them can sit very stable...
Soon Qiao Yu saw the final questions, and the first question made him very happy.
To be honest, if it were Qiao Yu who solved Professor Xue Song's question, he would probably still have a headache when encountering this kind of question.
It’s not that these questions are difficult, the main reason is that they test a lot of concepts. And they require a deep understanding of the concepts. For example, the definition of rings, the understanding of phase array rings, the concept of lattice, the isomorphism classification of modules and finite
Generative understanding and so on... But the current Qiao Yu is really frighteningly strong.
For example, based on the conditions given, Qiao Yu immediately judged that the shape of the matrix given in the question could be written as: Obviously this type of matrix forms a sub-ring with a special algebraic structure, which can be set to R.
Then it becomes simple. The core of the proof is nothing more than determining how many different R-lattices there are.
He probably had some ideas for solving the problem in his mind, so Qiao Yu didn't rush to start answering the questions. Instead, he quickly scanned the direction. The second question was easy; the third question was not difficult either. It wasn't until the fourth question that he paused for a moment. Okay.
Guys, this is a problem of finding an equation that does not have an integer solution. (I have run out of illustrations today, so I can’t give you any more questions. If you are interested, you can read the easter egg chapter.)
To be honest, for other people, Qiao Yu thought it was probably quite difficult. But now he found that as long as he carefully reviewed the question, this kind of proof question was really not difficult. It was nothing more than introducing unit roots and polynomial expressions, and then simplifying the equations
,Analytical algebraic number theory background
Even at this point, Qiao Yu could already see that the roots of this equation had no integer solutions.
Because in the step of simplifying the equation, the left side of the equation can be regarded as the factorization form of a certain polynomial, and each factor is related to the real part of the p·th unit root. These factors correspond to Chebyshev polynomials or unit roots.
Related symmetric polynomials.
Such polynomials usually have non-integer coefficients, so it can basically be inferred that the roots of these polynomials will not be integers. Of course, the specific situation still needs to be proven.
But it is enough to further formalize it through modulo p arithmetic, so Qiao Yu thinks this question is not difficult.
The fifth question, a linear algebra question, involves some concepts in topological groups. It is difficult, but it happens to be Qiao Yu’s strength. The focus is nothing more than selecting infinite subsequences and analyzing uniform convergence. To put it bluntly,
Qiao Yu believes that the person who asked this question probably wanted to test the contestants’ understanding of the generation of matrix groups, the product behavior of matrix sequences, and the convergence issues under matrix multiplication.
The sixth question, the main test points are probably the direct sum decomposition of modules in group representation theory, tensor product operation, and the proof of isomorphism and uniqueness of modules. The difficulty lies in how to analyze the finite generation of modules under the action of p·group Structure. So Qiao Yu feels that as long as you understand how to establish an isomorphic relationship between different modules, this question is not too difficult.
Question 7, oh, no more... there are only six questions.
Just six questions, and a full eight hours were given. Qiao Yu thought about how this felt a bit contemptuous.
Of course, it’s not that I look down on him, it’s mainly because I think that people in the field look down on those with master’s degrees and Ph.D.s from prestigious universities. After all, these people are different from Teacher Lan.
The direction of their research is number theory, so it's probably no problem to get full marks on these questions.
This makes Qiao Yu a little worried. If everyone gets full marks in the exam, will the gold medal and bonus prepared by the organizing committee be enough? So this year's finals are mainly about card details, right?
Qiao Yu had already pictured a bunch of big-name professors staring at the problem-solving process and picking up mistakes, and then calmed down and cautiously clicked on the answer interface. Once you realize that details are important, this kind of game will not be difficult.
...
After the contestants entered the examination room, Lang Jie was led by the staff to a special rest area. The competition lasted eight hours.
Of course, this does not mean that the contestants must stay in the prepared examination room all the time.
If you need to eat, go to the toilet, etc. in the middle, you can do it. It’s just that the lost time will not be made up. This is actually the same as taking an exam at home.
Within twenty-four hours, choose to answer the questions for eight consecutive hours. Log in to the exam interface to start the timer. After the time is up, you cannot modify the answers. So being able to survive longer than other contestants is also an advantage. Of course, these teachers who come with the team Or the guardian does not need to stay with you.
Same as at breakfast.
Some people gathered together to chat, while others were minding their own business playing with their mobile phones.
Lan Jie was sitting on the sofa by the window, turning on his mobile phone and flipping through this year's preliminary test questions. He didn't feel too bored at all.
Mathematics is like this. When you encounter something that you can't understand, you really want to sleep after just a few glances. But when you encounter a topic that is interesting and just right, the more you solve it, the more interesting it becomes, and you may even forget to eat and sleep because of it. In fact, there are many This is how the interest in learning among middle school and high school children was slowly developed.
At least at this time, Lang Jie was looking at a question about combination arrangement in the preliminaries and was immersed in it, thinking about how to make some changes, and then directly use it in Olympiad training for students. It is not too over the top. Current high school learning content This book contains parts of permutation, combination, binomial theorem and pigeonhole principle.
At 12 o'clock at noon, the staff came to the lounge on time to pick everyone up for lunch. It was still the same restaurant where breakfast was served, and it was still a buffet style.
Lan Jie took the dinner plate and came to the hot dining area. When he was thinking about what to eat, he suddenly heard someone calling him from behind. "Teacher Lan, Teacher Lan."
Lan Jie turned around and saw Xiao Guo, who was in charge of receiving him yesterday, panting and walking towards him from behind. "Hey, Xiao Guo, what's the matter?"
"Oh, it's like this. The competition expert team wants to invite you to have dinner with us. Let me call you." Guo Songyuan explained in a low voice.
He was specifically asked to come here because he could directly find Lan Jie and Qiao Yu, who were in charge of reception yesterday, without causing too much noise.
From the content of this sentence to the tone and volume of the young man speaking in front of him, Lan Jie could roughly judge that Qiao Yu was probably safe in this competition. After all, he had no interaction with the expert group, and neither could anyone else.
She glanced at him inexplicably.
You can't just call him up and humiliate him just because a middle school like Curiosity Xingtie No. 1 Middle School, which is not ranked well, suddenly showed up with a student who can reach the finals, right? So he couldn't help but smile and put the plate he just took
He put it down casually and said calmly: "Sorry to trouble you, Xiao Guo."
"You're welcome, I'll take you there."
"This is Teacher Lan, Lan Jie, who came with Qiao Yu." "Teacher Lan, sit down, there is a seat specially reserved for you."
After Guo Songyuan brought Lan Jie into the box, someone at the table immediately greeted him.
Lan Jie calmly nodded to the seven people at the table, and then sat at the seat reserved for him. "Okay, we can arrange for the food to be served."
"Okay. Teachers, wait a moment." After saying that, Guo Songyuan walked out of the box and closed the box door.
The person who just invited Lan Jie to take a seat also enthusiastically began to introduce: "Mr. Lan, let me introduce myself first. I am the director of our Xiaohu University Board of Directors. My surname is You and my name is Zhaobin. I am responsible for handling all aspects of this competition.
Logistics work." "Director You, nice to meet you, nice to meet you." Lan Jie nodded slightly.
Before coming to Xiaohu University, he might have been a little nervous when suddenly faced with this scene, but after this trip with Qiao Yu, he was completely relaxed. How should I say?
Probably because of the subconscious mentality of "My students are so awesome, who are you all?" Although it seems a bit pretentious, but in front of the expert group, I can really show no timidity at all.
Not to mention that in terms of division of labor, it is actually these university professors who require them. Without high school teachers like them to lay a good foundation for children, universities would not be able to recruit suitable students, right?...
"Let me introduce to Mr. Lan. This is Professor Liu Yidong, the doctoral supervisor of Dongshan University and the deputy leader of the expert group responsible for the independent marking phase of the competition." "Hello, Professor Liu, I have known you for a long time. Nice to meet you."
"Teacher Lan is so angry. The child you brought out this time is named Qiao Yu, right? That's really amazing." As expected, he was able to come here to eat because of Qiao Yu's favor.
Soon, Director You from Xiaohu University introduced everyone. But when he introduced Professor Xue Song from Yujiang University, Lan Jie almost stopped. The equation that Qiao Yu made before seemed to be
This professor posted it on the mathematics forum.
Although he used the nickname on the forum, he directly stated his identity in the private message to Qiao Yu. Lan Jie really didn't expect that he just met this professor yesterday and met him directly today.
"Teacher Lan, this kid Qiao Yu is really awesome. Before we set off here, he had already solved four questions on the algebra and number theory track, and his answers were quite perfect. He has this ability at the age of fifteen. Ability is really valuable! I happen to be in charge of algebra and number theory, so I am particularly curious about my child’s teacher, so I made an impromptu proposal to invite you over to have a meal with me. I’m a little presumptuous, so please forgive me.”
Xue Song, who was sitting opposite, smiled and said again.
After hearing this, Lan Jie smiled and suddenly had the urge to see what the professor would look like if he knew that the question he asked on the forum was also answered by Qiao Yu. Fortunately, it was still I endured it.
After all, this is Qiao Yu's private matter, and it is up to the children to decide whether to make it public or not.
"Hey, it's useless to be curious about me. Qiao Yu is a special child. I really didn't teach him. At most, I just discovered him. In terms of mathematics, he can achieve his current level mainly through self-study. ." In front of these experts, Lang Jie chose to tell the truth.
It's not impossible to show off, but it's not necessary. Not to mention showing off in front of these professionals, it's too easy to reveal your secrets. Qiao Yu's chosen track is algebra and number theory.
As he said before, this is really out of his reach. What if these big guys get interested and raise a difficult problem for discussion, what should he say? But his words also made everyone at the table The professors were quite surprised.
"It's all self-study?"
Of course, the experts at the end of the package are not the only ones who are responsible for independent examination this time. After all, the answers to more than 700 points need to be marked. Even with the help of A, the workload is very high. Not to mention that there is a cross-marking link to prevent cheating. .It’s just that because the professors in the expert group are very busy, and this is a private online competition, the marking can also be done online.
So today is just an expert representative invited by the track. Xue Song can be here because he is a professor at Yujiang University, which is in the same city as Xiaohu University. It is very convenient to come here, so I just came to have a look. It was originally unintentional. He really didn't expect to meet such a genius.
That's all in other directions. Xue Song knows very well the difference between having a good teacher in algebra and number theory.
Let’s put it this way, when ordinary people learn number theory, difficulties appear one after another. If there is no timely feedback and guidance, it is enough to make mathematics enthusiasts collapse. The most important thing is that self-study has a lot to do with the selection of resources and the depth of understanding. It will also directly affect the learning effect.
It’s different when you have guidance from a famous teacher
A good tutor can directly provide students with a systematic knowledge framework, analyze various difficulties in a targeted manner, and can recommend different learning resources and practice resources at any time based on the student's learning progress. At least Xue Song knows very well, If he hadn't met several good mentors at Princeton, he wouldn't have been able to persist even with his talent.
Now someone told him that a fifteen-year-old student could reach this level through self-study?
Thinking of how he felt when he accidentally discovered someone backstage in the morning who was doing the questions very quickly and uploading answers to each question that were almost impeccable, he really doubted whether Lan Jie was telling the truth. aBB
Seeing that the atmosphere at the dinner table had cooled down because of Lan Jie's words, Lao Xue stared at Teacher Lan without saying a word, and other professors also looked at each other. Director You quickly spoke to break the awkwardness in the box. "Well, Teacher Lan is really really cool. I'm so humble. By the way, I forgot to ask where you graduated from?"
"Oh, I studied at Shuangdan University for both my undergraduate and master's degrees." Lan Jie replied with a smile. Speaking of which, this is the only thing he can feel proud of. After all, Shuangdan University's mathematics major is firmly among the best in China. Can be ranked in the top three.
After finishing this, Lan still didn't forget to explain: "But I really don't have many good intentions from Qiao Di. In fact, I don't know how to lead such a student. I deleted what he said, and what I can't do is say It's a matter of discussion.
In fact, I am not long in this field. My main field of study during my master's degree was combinatorial mathematics. And as you know, Qiao Yu is a student of Xingtie No. 1 Middle School. I think you have never heard of this middle school, because it is not a middle school in the first place. The teaching staff of any key middle school cannot compare with those of famous schools.
Chapter completed!