A genius? I just love studying.

Chapter 71 Attacking the Earth Internet Cafe
The third question:
For real numbers T > 0, a subset Γ of the Euclidean plane R^2 is called T-dense if for any v∈R^2, there exists w∈Γ such that ||vw||≤T. Let A 2nd-order integral square matrix A∈M2 satisfy det(A)≠0.
(1) Assume tr(A) = 0. Prove that there exists C > 0 such that for any positive integer n, the set A^nZ^2:={A^nv:v∈Z^2}
(2) Assume that the characteristic polynomial of A is irreducible over the field of rational numbers, and prove the same conclusion as (1).

"really!"

After reading the questions, Chen Hui felt relieved and finally stopped thinking that he had taken a fake competition.

If the first two questions are free points so that the contestants will not get 0 points, then the third question is a bit interesting.

This is a linear algebra and advanced algebra problem.

This question is essentially asking, when a linear mapping A is repeatedly applied to these integer points, the grid formed by these integer points will continue to change. Please ask what the magnitude of the change is, from a point on the plane to the closest point in the grid.

What the question requires to prove is that when you iterate repeatedly n times, the magnitude is approximately 2 to the power of the determinant of A.

There is no doubt that this question is difficult.

But as long as the contestants have a clear understanding of the geometric structure of linear mapping, and can skillfully use the zero polynomial theorem, and can use the form of Pei Shu's theorem in number theory, pA^(n+1)+qA^n to express the closest distance to the grid point, and then link it with the determinant of A to the power of n, this problem can be solved.

It took Chen Hui fifteen minutes to complete the proof of the two questions.

Finally, the questions are no longer the ones whose answers can be seen at a glance!
With a light breath, Chen Hui looked at the fourth question.

At this time, the door curtain of Dadi Internet Cafe was lifted, and An Chengzhang walked in from outside with his head peeked out.

Although he is already 46 years old this year, this is his first time to enter an Internet cafe. People will inevitably feel a little uncomfortable when entering an unfamiliar place.

The girl at the front desk glanced at him and then looked away, continuing to make her own funny noises.

Fortunately, Chen Hui and the others were sitting not too far from the door, so An Chengzhang found the two of them with just a quick glance.

The moment he saw the two of them, he was so angry that he wanted to teach Li Hai a lesson.

What a good kid Chen Hui is!
He walked towards the two of them quickly with overwhelming anger.

After winning a game, and winning with VN as the top laner, Li Hai was in a very good mood. He stopped at the settlement screen and turned his head to look at Chen Hui's screen next to him.

"?"

After just one glance, Li Hai decided to stay away from this dirty thing.

God knows what that scribble all over the screen has to do with mathematics.

But he knew it definitely had nothing to do with him!

However, just as he turned his head, Li Hai's neck suddenly stiffened, "An... Teacher An..."

An Chengzhang glanced at Li Hai and naturally saw the settlement screen of League of Legends. His face was as gloomy as water, but he did not get angry immediately. Then, he looked at Chen Hui's screen.

"Ok?"

The next moment, An Chengzhang suddenly realized.

He then remembered that Chen Hui participated in the Barbaria Mathematical Competition, and today should have been the day the preliminaries started. He didn't expect that Chen Hui actually came to participate in the competition.

Because Zhang Anguo was in charge, he did pay less attention to Chen Hui these days.

But he never expected that Chen Hui would come to the Internet cafe to take the exam. This was too ridiculous!
What does Zhang Anguo do for a living?
But now is not the time to worry about so many things. Chen Hui is concentrating on doing the questions and it is not good to disturb him.

Question 4?
Seeing the question number, An Chengzhang frowned.

He looked at the time on his wrist. It was only about half past eight. He knew that the Barbaria Mathematics Competition started at eight o'clock, which meant that the exam had only started for more than half an hour.

As a graduate student in the Department of Mathematics, he had participated in the Barbaria Mathematical Competition and had some understanding of the difficulty of the questions.

It's quite possible to finish the first two questions in half an hour, but the third question is not so simple. It will require at least several hours of bold assumptions and careful verification before it can be solved. Therefore, Chen Hui is now looking at the fourth question. The best result is that if he has already finished the first two questions and cannot solve the third question, he can jump directly to the fourth question.

However, the difficulty of the questions in the Barbaria Mathematical Competition gradually increases. If you can't even do the third question, it will probably be difficult to do the fourth question.

If you only get two questions right, there is no chance of entering the finals.

Sighed.

He knew very well how difficult the Barbaria Mathematical Competition was for a high school student.

But it’s okay, it’s a good thing to let this kid Chen Hui suffer some setbacks.

The anger in his heart had disappeared when he saw this scene. He stood behind Chen Hui's seat with his hands behind his back and began to read the questions.

Li Hai, who had been prepared to be taught a lesson, suddenly found that Teacher An didn't even look at him. He felt very uncomfortable for a moment, not knowing whether to be happy or sad.

But he still secretly closed the League of Legends and walked out of the Internet cafe stealthily, thinking of buying some melon seeds, drinks, mineral water, etc. to show his value, so that it would be justifiable for him to follow Chen Hui to the Internet cafe.

It was not known whether An Chengzhang was completely immersed in the questions or he really ignored Li Hai, but Li Hai easily left his seat and came to the door of the Internet cafe with a smug smile on his face.

However, the next moment, his smile froze on his face.

A group of people came to the entrance of Dadi Internet Cafe. The one leading the group was Wu Huachun, the principal of Rongcheng No. 2 Middle School, followed closely by Vice Principal Zhao Defeng, and three fully armed school security guards behind them.

"this?"

Li Hai felt like his heart was about to burst out, "Isn't this a huge scene?"

"Is this Rongcheng No. 2 Middle School leading troops to attack the Dadi Internet Cafe?"

"Die!"

At this time, Wu Huachun and others were already looking at him.

Li Hai felt his mouth dry and he wanted to run away immediately, but it was as if his feet had taken root, making it difficult to even move. He could only watch as the principal and his troops approached him... Then, they passed him by...

"?"

The people walking by brought a gust of wind, which made Li Hai feel a little cool.

He then realized that his back was soaked with sweat.

It was not until Wu Huachun and his group had gone far away that he suddenly realized that he was too nervous because he felt guilty. He certainly knew the two principals, but the two principals obviously did not know him at all!
"Yes, it's Chen Hui again!"

Looking at the direction the group was heading, Li Hai figured out the answer without much effort.

Seeing this, the receptionist even raised her eyes from her phone screen. "Are they here to mess things up?"

She was startled and called her boss.

On the other side, Xia Mi, with her hair dyed icy blue, quickly nudged Lin Xiaotang's elbow and whispered stealthily, "Xiaotang, the principal is here!"

"Are they here to arrest us?"

After Lin Xiaotang looked back at this side, his expression did not change at all and he returned to his game. He did not care whether it was the principal or the county magistrate who came next.

Xia Mi blinked, and when she saw that Lin Xiaotang didn't care, she also put on her headphones, looked straight ahead, and pretended not to see the principal.

(End of this chapter)