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Chapter 72: The difficulty of this final mainly lies in the details?
After roughly understanding the rules, Qiao Yu directly opened the browser, entered the final address, and logged into his account.

The competition is not the college entrance examination. It doesn't matter if you are one minute early or one minute late. You can't gain any advantage because the background will automatically count down. Anyway, everyone is only given eight hours to do the questions.

And it lasted for eight hours straight.

That is to say, once the timing starts, it cannot be stopped.

Whether it is eating, drinking water or going to the toilet, all the time spent in the meantime will be counted in the answering time.

Fortunately, this is not a big deal for a group of young people.

No matter they are junior high school students or high school students, they may not run very fast, but most of them can sit very steadily.

……

Soon Qiao Yu saw the final questions, and the first question made him very happy.

To be honest, if it were Qiao Yu before he solved Professor Xue Song's question, he would probably still have a headache when faced with this kind of question.

It’s not that this type of question is difficult, but it mainly tests many concepts. And the concepts that need to be deeply understood. For example, the definition of subring, the understanding of matrix ring, the concept of lattice, the classification of module isomorphism, the understanding of finite generation, etc.

But now Qiao Yu is so powerful that it is terrifying.

For example, based on the given conditions, Qiao Yu immediately determined that the matrix shape given in the question can be written as:
Obviously, this type of matrix forms a subring with a special algebraic structure, which can be set to R.

After that, it is easy. The core of the proof is nothing more than judging how many different R-lattices there are.

With a rough idea of ​​how to solve the problem in mind, Qiao Yu did not rush to start answering the questions, but quickly scanned them. The second question was easy, and the third question was not difficult either. It was not until the fourth question that he paused for a moment.

Wow, this is a problem about finding an equation that has no integer solutions. (Today I have used up all the illustrations, so I can’t give you a free answer. If you are interested, you can go to the Easter Egg Chapter.)
To be honest, Qiao Yu thought it was probably difficult for other people. But now he found that as long as he carefully examined the questions, this kind of proof question was really not difficult. It was nothing more than introducing unit roots and polynomial expressions, then simplifying the equations and analyzing the background of algebraic number theory.

Even at this point, Qiao Yu could 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 a factorized form of a polynomial, and each factor is related to the real part of the p-order root of unity. These factors correspond to Chebyshev polynomials or symmetric polynomials related to the root of unity.

Since these polynomials usually have non-integer coefficients, it can be inferred that the roots of these polynomials will not be integers.

Of course, the specific circumstances still need to be proven.

But it will suffice to formalize it further via modular p-arithmetic.

So Qiao Yu felt that this question was not difficult.

The fifth question is a linear algebra question, which involves some concepts in topological groups. It is difficult, but it happens to be Qiao Yu's strong point. The key point is to select infinite subsequences and analyze uniform convergence.

To put it bluntly, Qiao Yu believes that the person who set this question probably wanted to test the contestants' understanding of the generation of matrix groups, the product behavior of matrix sequences, and the convergence problems under matrix multiplication.

The main points of question 6 are the direct sum decomposition and tensor product operation of modules in group representation theory, as well as the isomorphism of modules and the proof of module uniqueness. The difficulty lies in how to analyze the structure of finitely generated modules under the action of p-group.

Therefore, Qiao Yu felt that as long as one understood how to establish isomorphic relationships between different modules, this problem would not be too difficult.

Question seven, oh, that’s it…only six questions.

Just these six questions were answered in eight hours, and Qiao Yu felt that this was a bit contemptuous.

Of course, it's not that I look down on him. It's mainly because I think the question setter looks down on those masters and doctors from famous universities.

After all, these people are different from Teacher Lan.

Their research direction is number theory, so it should be no problem for them to get full marks on these questions.

This made Qiao Yu a little worried. If everyone got full marks, would there be enough gold medals and bonuses prepared by the organizing committee?

So this year's finals will mainly focus on details, right?

Qiao Yu had already imagined a scene in his mind where a bunch of big-name professors were staring at the problem-solving process and picking out mistakes, then he calmed himself down and carefully clicked on the answering interface.

Once you realize that details are important, this situation will not be difficult.

……

After the contestants entered the examination room, Ranger was led by the staff to a special rest area.

The competition lasts eight hours.

Of course, this does not mean that the contestants must stay in the prepared examination room all the time.

You can eat or go to the toilet during the trip, but the time wasted will not be compensated.

This is actually the same as taking the exam at home.

Within 24 hours, choose to answer questions for eight consecutive hours. The timer starts when you log in to the test interface, and you can no longer modify your answers after the time is up. So being able to endure more than other contestants is also an advantage.

Of course, the teachers or guardians who came with the team don’t need to endure it together.

Same as breakfast.

Some people gathered together to chat, while others played with their phones.

Ranjie sat on a sofa by a window, opened his mobile phone and looked through this year's preliminary exam questions. He didn't feel bored at all.

Mathematics is like this. When you encounter something that you just can't understand, you will feel sleepy after just a few glances. But when you encounter an interesting problem that you can solve, the more you solve it, the more interesting it becomes, and you may even forget to eat and sleep.

In fact, many times the interest of junior and senior high school students in learning is cultivated slowly in this way.

At least at this moment, Ranger was focusing on a question about combination and arrangement in the preliminary round, thinking about how to make some changes, and then directly use it in training students for the Olympiad.

It is not particularly beyond the syllabus. The current high school learning content already includes permutations, combinations, the binomial theorem and part of the pigeonhole principle.

At twelve o'clock noon, the staff came to the lounge on time to pick everyone up for lunch.

It’s still the same restaurant where we had breakfast, and it’s still a buffet.

Ranger took his plate and went to the hot food area. When he was thinking about what to eat, he suddenly heard someone calling him from behind.

"Teacher Lan, Teacher Lan."

When Langjie turned his head, he saw Xiao Guo, who was responsible for receiving him yesterday, walking towards him breathlessly from behind.

"Hey, Xiao Guo, what's the matter?"

"Oh, it's like this. The expert group of the competition wants to invite you to dinner, so let me call you." Guo Songyuan explained in a low voice.

He was specially asked to come because he was responsible for receiving Lan Jie and Qiao Yu yesterday, so he could find the person directly without causing too much noise. From the content of this sentence to the tone and volume of the young man's voice, Lan Jie could roughly judge that Qiao Yu was probably sure to win the competition this time.

After all, he has no connection with the expert group, and it is impossible for others to look up to him for no reason.

You can't just call up and humiliate a student who made it to the finals from a middle school like Xingtie No. 1 Middle School, which is not ranked high, out of curiosity, right?
So he couldn't help but smile, put down the plate he had just picked up, and said calmly: "Thank you for your help, Xiao Guo."

"You're welcome. I'll take you there."

……

"This is Mr. Lan who came with Qiao Yu, Lan Jie."

"Teacher Lan, take a seat. This is the seat reserved for you."

After Guo Songyuan brought Lan Jie into the box, someone at the table immediately greeted him.

Ranger nodded calmly to the seven people at the table, then sat down in the seat reserved for him.

"Okay, we can serve the food now."

"Okay. Please wait a moment, teachers." After saying that, Guo Songyuan walked out of the box and closed the door.

The man who had just invited Lan Jie to take a seat also began to introduce himself enthusiastically: "Teacher Lan, let me introduce myself first. I am the director of the board of directors of Xiaohu University. My last name is You, and my name is Zhao Bin. I am in charge of all the logistics work for this competition."

"Director You, nice to meet you." Lan Jie nodded slightly.

If he had been faced with this scene before coming to Xiaohu University, he might have been a little nervous, but after this trip with Qiao Yu, he completely relaxed.

What do you say?
It's probably because there's a subconscious mentality of "My students are so awesome, who are you guys?"

Although it seems a bit like the fox taking advantage of the power of the tiger, he really does not show any fear in front of the expert group.

Not to mention that in terms of division of labor, it is actually these university professors who are asking them to do so. Without these high school teachers laying a good foundation for the children, the universities would not be able to recruit suitable students, right?

……

"Let me introduce you to Professor Lan. This is Professor Liu Yidong, the deputy leader of the expert group responsible for the independent marking phase of the competition and the doctoral supervisor of Dongshan University."

"Hello, Professor Liu. I've heard a lot about you. Nice to meet you."

"Teacher Lan, you are too polite. The child you brought up this time is called Qiao Yu, right? He is really amazing."

As expected, he was able to eat here thanks to Qiao Yu.

Soon, Director You of Xiaohu University introduced everyone. However, when he introduced Professor Xue Song of Yujiang University, Lan Jie almost couldn't hold back.

It seems that the equation that Qiao Yu had solved before was uploaded to the mathematics forum by this professor.

Although he used a nickname on the forum, he had directly stated his identity in the private message to Qiao Yu. Ranjie really didn't expect that he would meet this professor in person today after having just made contact with him yesterday.

"Teacher Lan, Qiao Yu is really amazing. Before we set out here, he had already solved four questions in the algebra and number theory track, and his answers were quite perfect. It is really precious that he has such ability at the age of fifteen!
I happen to be in charge of algebra and number theory, so I am particularly curious about the child's teacher, so I proposed to invite you to have a meal together. I apologize for being a little intrusive. "

Xue Song, who was sitting opposite, said with a smile.

After hearing this, Langjie smiled and suddenly had an urge to see what expression the professor would have if he knew that the question he asked in the forum was also answered by Qiao Yu.

Fortunately, I still held back.

After all, this is Qiao Yu's private matter, and it is up to the child to decide whether to make it public.

"Ah, being curious about me is useless. Qiao Yu is a special child. It's really not my teaching. At most, I just discovered him. In terms of mathematics, he can reach his current level mainly by self-study."

In front of these experts, Ranger chose to tell the truth.

It's okay to show off, but it's not necessary. Especially in front of these professionals, it's too easy to be exposed.

The track chosen by Qiao Yu was algebra and number theory.

As he said before, he really couldn't do this. What if these big guys got interested and threw out a difficult problem for discussion? What would he say?
However, his words also surprised the professors at the table.

"All by self-study?" Xue Song was stunned and subconsciously stared at Lang Jie for a moment.

Ranger acted very calmly and didn't seem to be lying.

Of course, the experts in charge of independent marking this time are not limited to the few people in the box. After all, there are more than 700 answers to mark, and even with AI assistance, the workload is very high. Not to mention the cross-marking process to prevent cheating.

It’s just that the professors in the expert group are very busy, and since it is a private online competition, the marking can also be done online.

So today only one track invited an expert representative. Xue Song was able to come 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 he just came to take a look.

It was something unintentional in the first place, and he never expected to meet such a genius.

It would be fine for other directions, but Xue Song knew too well how big a difference it made in algebra and number theory whether or not there was a good teacher.

Let's put it this way, when most 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. Most importantly, self-study will directly affect the learning effect in terms of resource selection and depth of understanding.

It’s different if you have a famous teacher to guide you.

A good tutor can directly provide students with a systematic knowledge framework, analyze various difficult points in a targeted manner, and recommend different learning and practice resources at any time based on the students' learning progress.

At least Xue Song knew very well that if he had not met several good mentors in Princeton, he would not have been able to persevere even with his talent.

Now someone told him that a fifteen-year-old student could reach this level entirely through self-study?
Thinking about how he felt in the morning when he accidentally discovered in the background that someone was finishing the questions very quickly and the answers to each question uploaded were almost flawless, he really doubted whether what Ranjie said was true.

……

Seeing that the atmosphere at the dinner table had become cold because of Lan Jie's words, Old Xue stared at Teacher Lan without saying a word, and the other professors were looking at each other. Director You quickly spoke to break the awkwardness in the box.

"Well, Mr. Lan is really modest. By the way, I forgot to ask where did you graduate from?"

"Oh, I studied both my undergraduate and master's degrees at Shuangdan University." Langjie answered with a smile. This was the only thing he could feel proud of. After all, the mathematics major at Shuangdan University was firmly ranked among the top three in China.

However, after introducing himself, Lang Jie did not forget to explain: "But I really didn't give Qiao Yu too much guidance. In fact, I don't know how to guide a student like Qiao Yu. Professor Xue just said that Qiao Yu chose the track of algebra and number theory.

Actually, I am not good at this track either. My major in master's degree is combinatorial mathematics. And as you know, Qiao Yu is a student of Xingtie No. 1 Middle School. I think you haven't heard much about this middle school, because it is not a key middle school, and the teaching staff cannot compare with those famous schools.

So if Qiao Yu can get good grades in the finals, it really has nothing to do with me or the school. It all depends on his own efforts. "

　Thanks to book friend Tianhai Xiangyun for the reward and encouragement!
　　Thanks to my book friends Stormwind Warriors, feiruyun, and xcat for their rewards and encouragement!

　　PS: The 1th day of updating words has been completed!
　　
　
(End of this chapter)