Chapter 72: The difficulty of this final is mainly about card details?
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 the college entrance examination. It doesn’t matter if you are one minute early or one minute late. You can't take any advantage, because the background will automatically time the test. Anyway, everyone is only given a total of eight hours to do the questions.
And it was eight hours straight.
That is to say, once the timer starts, it cannot be stopped.
Whether you eat, drink, or go to the toilet in between, all are 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 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 who answered 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, it’s mainly that they test a lot of concepts. And it requires a deep understanding of the concepts. For example, the definition of subrings, the understanding of matrix rings, the concept of lattice, the isomorphism classification of modules, the understanding of limited generativity, etc...
But now Qiao Yu is really terrifyingly powerful. .
For example, according to the given conditions, Qiao Yu immediately judged that the shape of the matrix given in the question can be written as:
Obviously, this type of matrix forms a sub-ring with a special algebraic structure. Can be set to R.
Then it’s simple. The core of the proof is nothing more than determining how many different R-lattice there are.
Qiao Yu probably had some ideas for solving the problem in his mind, so Qiao Yu didn’t rush to answer the questions. Instead, he quickly scanned the question. The second question was easy; the third question was not difficult either. It wasn’t until the fourth question that I paused for a moment.
Hey guys, this is a problem of finding an equation that does not have an integer solution. (I have used up the number of illustrations today, so I can’t give you any questions. If you are interested, you can read the easter egg chapter.)
To be honest, for other people, Qiao Yu thinks it is probably quite difficult. But now he found that as long as he carefully reviewed the questions, this kind of proof question was really not difficult. It is nothing more than introducing unit roots and polynomial expressions, then simplifying equations, and analyzing the background of algebraic number theory.
Even at this point, Qiao Yu could already see that the roots of this equation had no integer solutions.
Because in the equation simplification step, the left side of the equation can be regarded as the factorized 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 symmetric polynomials related to unit roots.
This type of polynomial usually has non-integer coefficients, so it can basically be inferred that the roots of these polynomials will not be integers.
Of course the specific circumstances still need to be proven.
But it is enough to further formalize it by modulo p arithmetic.
So Qiao Yu thinks this question is not difficult.
The fifth question, a linear algebra question, only 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 thinks 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.
For the sixth question, the main test points are probably the direct sum decomposition and tensor product operation of modules in group representation theory, as well as the proof of module isomorphism and uniqueness. The difficulty lies in how to analyze the structure of finitely generated modules under the action of p-group.
So Qiao Yu feels that as long as you understand how to establish an isomorphic relationship between different models, this question is not too difficult.
The seventh question, 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 that I think people look down on those master’s and doctoral students from prestigious universities.
After all, these people are different from Teacher Lan.
The direction of their research is number theory. It is 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 making 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 lasts eight hours.
Of course, it 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. during the trip, you can do it. The time lost will not be replenished.
This is actually the same as taking an exam at home.
Within twenty-four hours, choose to answer questions for eight consecutive hours. Log in to the exam interface to start timing, and you cannot modify your answers after the time is up. So being able to survive longer than other players can be considered an advantage.
Of course, teachers or guardians like them who come with the team don’t need to stay with them.
Same as breakfast.
Some people gathered together to chat, and some were minding their own business playing with their mobile phones.
Ran Jie was sitting on a sofa by a window, turning on his mobile phone and flipping through this year's preliminary test questions. He didn't feel bored at all.
Mathematics is like this. When you encounter something that you can’t understand no matter how hard you learn it, you really feel like falling asleep after just a few glances at it. But when you encounter a question 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 over it.
In fact, in many cases, the interest in learning among middle school and high school children is slowly developed in this way.
At least at this time, Lang Jie looked at a question about combination arrangement in the preliminaries and got involved, thinking about how to make some changes, and then directly use it to train students for the Olympiad.
It’s not particularly out of line. Nowadays, high school learning content includes permutation, combination, binomial theorem and pigeonhole principle.
At 12 noon, the staff came to the lounge on time to pick everyone up for lunch.
It’s still the same restaurant where you have breakfast, it’s still a buffet style.
Lang 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."
As soon as Lan Jie turned his head, he 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, that's it. The competition expert team wants to invite you to dinner together, let me call you. "Guo Songyuan explained in a low voice.
He was specifically asked to come here because he could directly find the people 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's voice, Lan Jie could probably tell that Qiao Yu was probably safe in this competition.
After all, he has no interaction with other expert groups, and it is impossible for others to look up to him for no reason.
You can’t just call him up and humiliate him just because a student from a low-ranking middle school like Curiosity Star No. 1 Middle School suddenly pops up who can reach the finals, right?
So he couldn't help but smile, put down the plate he just picked up, and said calmly: "Sorry to trouble you, Xiao Guo."
"You're welcome, I'll take you there."
......
"This is Teacher Lan who came with Qiao Yu, Lan Jie."
"Teacher Lan , take a seat, there is a seat specially reserved for you."
After Guo Songyuan brought Lan Jie into the box, someone at the table greeted him immediately.
Lang Jie calmly nodded to the seven people on the table, and then sat in 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 door.
The person who just invited Lan Jie to sit down also enthusiastically began to introduce: "Mr. Lan, let me introduce myself first. I am the director of the board of directors of Xiaohu University. My surname is You and my name is Zhaobin. I am responsible for all Handling some logistical work for this competition."
"Director 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.
What do you say?
Probably there is a subconscious mentality of "my students are so awesome, how old are you all?"
Although it seems a bit pretentious, in front of the expert group, he can really show no timidity at all.
Not to mention that in terms of division of labor, it is actually these university professors who ask 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 Teacher Lan. This is the deputy leader of the expert group responsible for the independent marking stage of the competition, Professor Liu Yidong, the doctoral supervisor of Dongshan University."
"Liu Professor, hello, Jiu Thank you for your honor.”
“Teacher Lan, you are so polite. The child you brought out this time is called Qiao Yu, right?”
As expected, he can. Coming here to eat is to take advantage of Qiao Yu.
Soon, Director You of Xiaohu University introduced everyone. But when he was introduced to Professor Xue Song from Yujiang University, Lan Jie almost lost his temper.
The equation that Qiao Yu made before seems to have been posted to the mathematics forum by this professor.
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 he answered them perfectly. He was only ten years old. It is so valuable to have this ability at the age of five!
I happen to be in charge of algebra and number theory, so I am particularly curious about your child’s teacher, so I temporarily proposed to invite you over for a meal. It’s 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 do if he knew that the questions he asked on the forum were also answered by Qiao Yu. What expression.
Fortunately, I 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.
"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 mathematics, he can reach his current level. The main thing is self-study.”
In front of these experts, Langer 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 tracks are algebra and number theory.
As he said before, this is really out of his reach. If these big guys get interested and raise a difficult problem for discussion, what should they say?
However, his words also surprised the professors at the table.
"It's all self-study?" Xue Song was stunned and subconsciously stared at Lan Jie for a moment.
Ranjie acted very calmly, not like he was lying.
Of course, the experts in the box are not the only ones responsible for independent marking this time. After all, more than 700 answers need to be marked. Even with AI assistance, the workload is still very high. Not to mention there is a cross-marking process to prevent cheating.
But 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 was able to attend because he was a professor at Yujiang University, which is in the same city as Xiaohu University. It was very convenient to come here, so he decided to come and have a look.
It was an unintentional thing, and he really didn't expect to meet such a wizard.
It’s just another direction. 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 math enthusiasts collapse. The most important thing is that self-study will also directly affect the learning effect in terms of resource selection and depth of understanding.
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 recommend different learning resources and practice resources at any time based on the student's learning progress.
At least Xue Song knew very well that 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 the answers uploaded to each question were almost flawless, he really doubted whether Lan Jie was telling the truth.
......
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. The other professors also looked at each other, and Director You quickly spoke. Breaking the awkwardness in the box.
"Well, Teacher Lan is so humble. By the way, I forgot to ask where you graduated from?"
"Oh, I studied at Shuangdan University for my undergraduate and master's degrees." Lan Jie replied with a smile. After all, Shuangdan University's mathematics major is firmly ranked among the top three in China.
However, after introducing himself, Lan Jie did not forget to explain: "But I really didn't give Qiao Yu much tutoring. In fact, I don't know how to tutor a student like Qiao Yu. Professor Xue As I said just now, Qiao Yu chose the track of algebra and number theory.
Actually, I am not good at this track either. My main focus during my master's degree was combinatorial mathematics. . And you also know that Qiao Yu is a student of Xingtie No. 1 Middle School. You have never heard of this middle school, because it is not a key middle school, and the teaching staff is definitely not comparable to those of 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 the book friend Tianhai Xiangyun for the reward and encouragement!
Thanks to the book friends Stormwind City soldiers, feiruyun, and x cat for their rewards and encouragement!
PS: The first day of the 10,000-word update has been completed!
(End of this chapter)