Top academic

Chapter 1 This must be a ghost!
May 2024, .

After May Day, Xingcheng entered the rainy season, and the rain has not stopped for two weeks.

However, Ranjie, the head of the mathematics teaching and research group of the high school department of Xingcheng Railway No. 1 Middle School and the coach of the Olympic Games, did not care too much about the depressing and gloomy weather. He was following two students in white school uniforms from a distance with an umbrella.

Here's the thing.

About ten minutes ago, after Ranjie finished tutoring the school's top math students for the Olympiad, he did not choose to go home for dinner because he had an evening study session for the rocket class in the second grade of high school.

In addition, the weather was too stuffy and rainy, so I didn’t want to eat in the cafeteria, so I just went to a wonton noodle restaurant outside the school and had a bowl of wontons.

After eating the wontons, as soon as I walked out the door, I saw two freshman students who had made the fastest progress in the Olympiad cram school. Lu Jia and Ma Yufei were eating buns together while sneaking into a small alley outside the school gate.

When you are growing up, you just eat two buns to kill yourself?!

Originally, Ranger wanted to stop the two boys and ask them what they were doing outside the school instead of waiting for evening self-study in school. But suddenly, he had an idea and thought that he was free anyway and might as well take a walk after dinner, so he simply followed them.

So this scene happened.

Soon, Ranger saw two guys walking straight to an Internet cafe in the alley and then went inside.

Ah……

This time we can catch him in the act.

Although the two children have made rapid progress recently, I am sorry that they have to run to surf the Internet when they only have an hour and a half to eat in the evening. I have to criticize them.

Also, this internet cafe is too much.

The banner prohibiting minors from surfing the Internet is still hanging on the door, but the boss dared to let these high school students in freely. It is obvious that he needs re-education. Lan Jie thinks it is necessary to report them to the Cultural Bureau under their real names.

So Teacher Lan took out her mobile phone without hesitation, turned on the recording mode, and followed the two people into the Internet cafe.

……

Although the computers in the Internet cafe in the alley seemed to have just been replaced with new ones, it was still as smoky and dirty as Ranger remembered.

Although China has already issued regulations prohibiting smoking in indoor public places, and these slogans are posted everywhere in Internet cafes, it is obvious that these small Internet cafes have not implemented them thoroughly.

But think about it, if they dare to allow minors to go online, why would they care whether they smoke or not?
But what surprised Ranger was that after the two children entered the Internet cafe, they did not go to the bar counter to turn on the computer directly, but went straight to the booth in the corner of the back row of the Internet cafe.

Lan Jie hid his phone and walked behind the two of them. He just happened to hear Ma Yufei's voice, who was resting his hand on the edge of the booth: "Brother Qiao, I have another request from you today."

"Old rules."

"Okay, let's charge fifty, right?"

"Ok!"

At this time, Langjie had already walked past a seat where someone was playing a game. The previous conversation aroused his curiosity, so he simply sat down in an empty seat. Then he saw Ma Yufei turn around and walk towards the bar.

Soon, Ranger heard the sound of a successful fifty-yuan recharge on a membership card from the bar.

Wow, no wonder they only had buns for dinner. It seems like they spent all their money on recharging other people’s cards!

Ranger became even angrier and couldn't help but turn his head to look inside the booth.

No, it didn't look like a big brother. Instead, it was a guy who looked younger than the two of them, who was playing games seriously on the computer inside the booth.

The profile looks very young, at most a middle school student. At this moment, the young face wearing headphones was staring at the screen intently, with a lollipop in his mouth. Lu Jia, who was sitting next to him in front of a black computer, was also looking at the screen seriously, muttering quietly: "Left, there is someone on the left... Hey, how could he be beaten to death?"

At this time, Ma Yufei had walked back and blocked Ranger's sight. Ranger only saw the guy raised his hand and made a gesture to take off his headphones.

"Brother Qiao, it's charged."

"Read it." "The question is this: suppose there are positive integers a and b, such that ab+1 can divide a+b. Prove that (a+b)/(ab+1) is the square of some integer."

Hearing this, Ranjie almost vomited blood. It turned out that he recharged the money just to ask him for help to solve the problem?
This was exactly the thinking question he assigned to everyone at the end of the Olympiad class today.

It can also be said to be a very classic Olympiad question.

It may seem simple just by listening to the questions, but once you try to do them you will find that they are extremely difficult.

This type of question has appeared in the International Mathematical Olympiad before, and only two contestants scored full marks on this question that year. Even many judges tried to solve the problem, but failed to score full marks.

Although he made some changes, it was still very difficult.

As a result, these two guys actually used this question to ask a bad boy who hangs out in an Internet cafe?
As expected by Ranger, the boy sat there lost in thought, not saying a word for a long time. After all, this problem was not that easy to solve.

After about two or three minutes, the boy suddenly said, "Show me the question again."

"Okay." Ma Yufei immediately handed over a piece of A4 paper.

Just like that, Ranger watched the boy staring at the question and thinking. After a few minutes, Ranger finally couldn't help it and was about to stand up and go over to find out. The boy suddenly spoke again: "Do you know about Veda's jump?"

"Vedic jump? The teacher taught Vieta's theorem today. Does this have anything to do with Vieta's jump?"

"So what did I say? You should learn mathematics in a practical way. Vieta's jump is to use Vieta's theorem to solve problems by infinite descent.

First, use the proof by contradiction, assuming that (a+b)/(ab+1) is not the square of an integer, and then assume that c and d are a set of arrays that meet the conditions, and c+d is the smallest of all arrays that meet the conditions.

Let c be greater than or equal to d, then c squared plus d squared divided by cd plus 1 equals k, and factoring yields c squared minus kcd plus d squared minus k equals zero..."

"Wait, Brother Qiao, please speak slower, I can't keep up with you..."

Ranger, who was so stunned that he didn't even bother to hide himself, stood up and walked towards the booth.

Fortunately, at this time Ma Yufei and Lu Jia both took out paper and pen to take notes seriously, and did not notice the movement around them at all.

At this time, the young guy with unknown origins had already taken his eyes off the topic and started playing the game on the screen again. It seemed to be a gunfight game...

"Okay, Brother Qiao, go on."

"Let the other solution of the corresponding two-variable equation be e, and use Vieta's theorem directly. e plus c equals kd..."

Langjie listened blankly to the guy sitting in front of the computer, who was playing the game while explaining the complete solution to this problem. It sounded like there were no mistakes or omissions, but when this guy set the unknown number, he seemed to like to use letters randomly.

But the idea is definitely correct!
Vieta's leap plus proof by contradiction!

The two children who scored full marks that year both used this solution.

But it’s not over yet…

Because that guy is still chattering...

"In fact, there is a geometric method. To put it simply, this problem is about the distribution of points on a geometric figure. Construct points and slopes on a plane, find the slope and the midpoint, and use distance and ratio to prove it. But drawing a picture is too troublesome."

"It's okay, Brother Qiao, you can tell me about it. I'll write it down and think about the geometric method later."

"I told you to draw a picture! And the geometric method is too complicated to explain. Since your teacher taught Vieta's theorem today, you must use Vieta's theorem to solve the problem. But if you recharge me for another fifty memberships..."

Before Qiao Yu finished speaking, he heard two guys next to him who were one grade higher than him shouting in fear almost at the same time: "Teacher Lan..." Then a strong force came from the back of his neck, and his clothes made his neck hurt a little, so he could only stand up...

Then a voice exploded in my ears: "Who are you?"

　New person, new book, asking for everything.

　　
　
(End of this chapter)