Chapter 1 This must be a ghost!
2024, May 17th.
After May Day, Star City entered the rainy season. It has been raining for two consecutive weeks without stopping.
However, Lang Jie, director of the high school mathematics teaching and research group and Olympiad coach of Star City Railway No. 1 Middle School, did not care too much about the depressing gloomy weather. He was holding an umbrella and following two people in white school uniforms from a distance. students behind.
Here’s the thing.
About ten minutes ago, after Lang Jie gave an Olympiad tutoring class to the school's top math students, he did not choose to go home for dinner because he still had a night self-study class for the sophomore rocket class in the evening.
In addition, the weather was too boring on a cloudy and rainy day, so I didn’t want to eat in the cafeteria, so I simply went to a wonton noodle shop outside the school and ate a bowl of wontons.
After eating the wontons, as soon as I went out, I saw the two first-year students in the Osai cram school who have made the most progress recently, Lu Jia and Ma Yufei, gnawing on steamed buns and sneaking into the school. In a small alley outside the gate.
You just eat two steamed buns to kill yourself when you are growing taller? !
Originally, Lan Jie wanted to stop the two boys and ask them why they were not waiting at school for self-study in the evening and running outside the school, but suddenly a thought came to his mind and he thought that they were idle anyway. I just happened to finish eating and take a walk, so I simply followed him.
So there was this scene.
Soon, Lang Jie saw two guys walking straight to an Internet cafe in the alley, and then walked in.
Huh...
Now I can catch the current situation.
Although the two children have made rapid progress recently, I'm sorry, they only have an hour and a half to eat in the evening, and they still have to go online, so they must be criticized.
In addition, this Internet cafe is too much.
The banner that prohibits minors from accessing the Internet is still hanging on the door, but they dare to let these first-year high school students in casually. It is obvious that the boss lacks re-education. Lan Jie feels that it is necessary to report to the Cultural Affairs Bureau in real name.
So Teacher Lan took out her mobile phone without hesitation, turned on the video mode, and then followed the two of them into the Internet cafe.
......
Although the computers in the Internet cafe in the alley seemed to have just been replaced with new ones, they were still as smoky as Lan Jie remembered.
Although China has already issued regulations prohibiting smoking in indoor public places, and these slogans have been posted everywhere in Internet cafes, it is obvious that such small Internet cafes have not implemented them thoroughly.
But think about it, if you dare to allow minors to go online, how can you care about whether those who go online smoke or not?
But what surprised Lan Jie was that after the two children entered the Internet cafe, they did not go to the bar 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 cell phone and walked calmly from behind the two of them. He happened to hear Ma Yufei, who was resting his hand on the edge of the booth, speaking: "Brother Qiao, I'm asking you again today."
< br>“Old rules.”
"Okay, let's recharge fifty, right?"
"Yeah!"
At this time, Lan Jie had already passed by a machine where someone was playing a game. The conversation aroused his curiosity, so he simply sat on an empty seat, and then saw Ma Yufei turning towards the bar.
Soon Lan Jie heard the sound of a member's membership card successfully recharging fifty yuan from the bar.
Good guys, no wonder the two of them only ate steamed buns for dinner and spent all their love money on other people’s cards!
Lan Jie became even more angry and couldn't help but turn his head and look directly into the booth.
No, it's not like a big brother, but a guy who looks younger than the two of them. He is playing games seriously in the booth against the computer inside.
Looking from the side, his face is very green, and at most he looks like a middle school student. At this time, the young face wearing headphones was staring at the screen with a lollipop in his mouth. Lu Jia, who was sitting next to him in front of the black computer, was also looking at the screen carefully and still whispering. He muttered: "On the left, there is someone on the left... Hey, why are you beaten to death like this?"
At this time, Ma Yufei had come back and blocked Lan Jie's gaze. Lan Jie only saw that guy. He raised his hand and made a gesture of taking off his earphones.
“Brother Qiao, it’s ready.”
"Read it." "The question is as follows. Assume that positive integers a and b satisfy that a+1 can divide a+b. Prove that (a+b)/(ab+1) is the square of a certain integer."
Hearing this, Lan Jie almost vomited blood. Is he investing money in love just to ask him to help solve the problem?
This is exactly the after-school thinking question he assigned to everyone at the end of today’s Olympiad class.
It can also be said to be a very classic Olympiad question.
It may be easy to just listen to the questions, but if you try to do them, you will know that it is extremely difficult.
This question type has also appeared in the International Mathematical Olympiad before. Only two contestants got full marks for this question that year. Many judges even tried to solve the problem but failed to get full marks.
Although he has slightly transformed, the difficulty is still very high.
In the end, these two guys asked a delinquent boy who was hanging out in an Internet cafe to ask this question?
It was no accident to Lan Jie. The young man sat there deep in thought and remained silent for a long time. After all, this question is really not that easy to solve.
After about two or three minutes like this, the young man suddenly said: "Show me the question again."
"Okay." Ma Yufei immediately handed over an A4 paper.
Just like that, Lan Jie watched the young man staring at the question and thinking deeply. After a few more minutes, Lan Jie finally couldn't help it anymore and was about to stand up and go over to find out. The boy suddenly spoke again: "Do you know about Vedic jump?"
"Vedic jump? The teacher taught Vedic theorem today. Does this have anything to do with Vedic jump?"
"So what do I say? When you study mathematics, you must learn it and apply it flexibly. Vedic jump is to use Vedic theorem to perform infinite descent to solve problems.
First, use proof by contradiction, first assume 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.
Suppose c is greater than or equal to d, then c square plus d square divided by cd plus 1 is equal to k. Factoring can be obtained by c square minus k, cd plus d square minus k is equal to zero..."
"Wait, Brother Qiao, please speak slower, I can't keep up here..."
Lan Jie, who was too stunned to even bother to hide his figure, stood up in a daze and walked towards the booth.
Fortunately, both Ma Yufei and Lu Jia took out their bags at this time. I took notes carefully with pen and paper, not even noticing the movement next to me.
At this time, the young guy with unknown origins had already retracted his gaze from the question and started playing the game on the screen again. It seemed to be a gun battle game...
"Okay Okay, Brother Qiao, continue.”
"Suppose the other set of solutions corresponding to the system of two-dimensional equations is e, directly use Veda's theorem, e plus c equals kd..."
Lan Jie listened blankly while sitting in front of the computer. While playing the game, the guy explained the complete solution to this problem. It sounds like there is nothing wrong with it, but this guy seems to like to use letters randomly when setting up unknowns.
But the idea is definitely fine!
Vedic jump plus proof by contradiction!
The two children with perfect scores used this solution.
However, the matter is not over yet...
Because the guy is still chattering...
"In fact, there is a geometric method. To put it bluntly, this problem is the points on the geometric figures. Distribution problem. Constructing points and slopes on the plane, finding the slope and midpoint, can also be proved by using distance and ratio, but it is too troublesome to draw. ”
"It's okay, Brother Qiao, you can also talk about it. I'll write it down and think about the geometric method later."
"I said I wanted to draw a picture! And the geometric method is too troublesome to talk about. Since your teacher taught it today Vedic Theorem, I definitely need you to use Vedic Theorem to solve the problem, but if you give me another fifty members..."
Before Qiao Yu finished speaking, he heard two guys next to him who were in the first year of high school shouting in horror almost at the same time: "Teacher Lan..." Then a strong force came from the back of his neck, and the clothes strangled his neck. It hurt a little, so I could only stand up...
Then a voice rang in my ears: "Who are you?"
Newcomers, new books, asking for everything.
(End of this chapter)