May 17, 2024.
After May Day, Star City entered the rainy season, and it hadn't stopped raining for two consecutive weeks.
However, Lan Jie, the head of the Mathematics Research Group of the Star City Railway First Middle School High School Department and the Olympic Competition coach, didn't care much about this depressing weather. He was holding an umbrella, following two students in white school uniforms from a distance.
This is what happened.
About ten minutes ago, after Lan Jie finished teaching the Olympic Competition tutoring class to the math prodigies at the school, he didn't choose to go home for dinner because he had an evening study session with the Rocket Class in the second year of high school later that evening.
Plus, the gloomy and rainy weather was too stifling, and he didn't want to eat at the cafeteria, so he simply went to a wonton noodle shop outside the school and had a bowl of wontons.
After eating the wontons, just as he stepped out, he saw the two first-year students who had recently made the fastest progress in the Olympic competition class, Lu Jia and Ma Yufei, nibbling on buns while sneaking into a small alley outside the school gate.
Just two buns to satisfy themselves while still growing?!
Originally, Lan Jie wanted to call the two kids to stop and ask why they weren't staying in school patiently waiting for the evening study session but instead ran outside the school. But suddenly, he had a thought: since he was idle anyway, he might as well follow them to stretch his legs after eating.
And so, this scene unfolded.
Soon, Lan Jie saw the two guys walk straight into an Internet café in the alley and go inside.
Ha…
Now he could catch them red-handed.
Although the two kids had made great progress recently, sorry, there's only an hour and a half designated for dinner in the evening, and they still came here to surf the internet—they needed to be criticized.
Moreover, this Internet café was over the line.
The banner banning minors from accessing the internet was still hanging on the door, yet they dared to let these high school freshmen in freely. Clearly, the owner requires further education, and Lan Jie felt it necessary to report this with real names to the Cultural Bureau.
So Teacher Lan didn't hesitate to take out his phone, activate the recording mode, and followed them into the Internet café.
...
Although the computers in the small alley's Internet café seemed newly replaced, it was still as smoky and chaotic as Lan Jie remembered.
Even though Huaxia had long issued a ban on smoking in public indoor spaces, and there were notices posted everywhere in the Internet café, this small café wasn't very thorough in enforcement.
But thinking about it, if they dared to allow minors, how would they care about smoking?
But what surprised Lan Jie was that after the two kids entered the Internet café, they didn't go to the bar to log on but walked straight to the booth in the back corner.
Lan Jie hid his phone and walked openly past the two, just in time to hear Ma Yufei, resting his hand on the edge of the booth, say: "Brother Qiao, we're counting on you again today."
"The usual deal."
"Sure, recharging fifty as always, right?"
"Mm!"
At this point, Lan Jie had walked past a seat where someone was playing a game. The previous conversation piqued his curiosity, so he casually sat down at an empty seat and then watched Ma Yufei turn to the bar.
Soon, Lan Jie heard the bar announcement that a member card was successfully recharged with fifty yuan.
Well then, no wonder the two only had buns for dinner—they're spending money to recharge someone else's card!
Lan Jie became even angrier and couldn't help but directly turn around to look into the booth.
No, it didn't seem to be some older guy; instead, a boy who looked even younger than the two of them was in the booth, seriously playing a game on the computer inside.
He looked youthful from the side, probably just a middle schooler. At this moment, his young face, wearing headphones, was staring intensely at the screen with a lollipop in his mouth. Lu Jia, sitting next to a blackened-out computer, was also seriously watching the screen, murmuring, "To the left, there's someone to the left... Hey, how did you still get shot like this?"
By this time, Ma Yufei had come back, blocking Lan Jie's view. Lan Jie only saw the boy raise his hand to make a motion to remove his headphones.
"Brother Qiao, it's recharged."
"Let's hear it."
"The problem is like this: Set integers a and b, such that ab+1 divides a²+b². Prove that (a²+b²)/(ab+1) is the square of an integer."
Upon hearing this, Lan Jie almost spat up blood. So the recharge was to ask him to solve a problem?
This was precisely the problem he assigned for after-class contemplation during today's Olympic Competition lesson.
It can also be said to be a very classic Olympic Competition problem.
Just listening to the prompt might make it seem simple, but try doing it, and you'll know it's incredibly hard.
This type of question had appeared in the International Mathematics Olympic Competition before, and only two contestants scored full marks with it back then. Even many judges who attempted the problem didn't get full marks.
Although he had slightly transformed it, the difficulty remained high.
In the end, these two guys actually brought this problem to consult a delinquent teenager hanging out in an Internet café?
As Lan Jie expected, the boy sat there, trapped in thought, with no sound for a while. After all, this problem isn't easy to solve.
About two or three minutes passed like this, and the boy suddenly spoke: "Let me see the problem again."
"Sure." Ma Yufei immediately handed over a sheet of A4 paper.
And so Lan Jie watched as the boy stared at the problem, pondering. After a few more minutes, Lan Jie finally couldn't hold back anymore. Just as he was about to stand up and go over, the boy suddenly spoke again: "Do you know about Vieta's Jumping?"
"Vieta's Jumping? The teacher taught Vieta's formula today. Does it have anything to do with Vieta's Jumping?"
"So what did I say? You need to apply your math knowledge flexibly. Vieta's Jumping uses Vieta's formula combined with the method of infinite descent to solve problems.
First, using proof by contradiction, assume that (a²+b²)/(ab+1) is not the square of an integer. Then let c and d be a set of numbers satisfying the condition, with c+d being the smallest among all satisfying numbers.
Assume further c is greater than or equal to d; then c squared plus d squared divided by cd plus 1 equals k. By factorizing, you get that c squared minus k times cd plus d squared minus k equals zero..."
"Wait a second, Brother Qiao, could you slow down? I can't keep up with taking notes..."
Already dumbfounded to the point where he couldn't bother hiding, Lan Jie stood up and walked toward the booth.
Luckily, at this moment, Ma Yufei and Lu Jia were busy taking notes and hadn't noticed the movement beside them.
At this time, the rather young and unknown guy had retracted his eyes from the problem and started playing the game on the screen again, which seemed to be a shooting game...
"Okay, Brother Qiao, continue."
"Assume another solution for the corresponding system of equations is e, using Vieta's formula directly, e plus c equals kd..."
Lan Jie listened in awe to the guy sitting in front of the computer, explaining the complete solution to the problem while playing a game. It sounded without errors, except that this guy seemed to use letters randomly when setting unknowns.
But the train of thought was definitely correct!
Vieta's Jumping plus proof by contradiction!
Back then, the two full-score participants used this method to solve the problem.
However, it wasn't over yet...
Because that guy was still rambling on...
"Actually, there's also a geometric method. This problem, at its core, is about the point distribution on a geometric figure. Construct points and slopes on the plane, find slope and midpoint, and use distances and ratios to prove it. But drawing the graph is too cumbersome."
"No problem, Brother Qiao; you can talk about it anyway, and I'll jot it down and study the geometric method later."
"I told you it requires drawing! And explaining the geometric method is too cumbersome. Since your teacher taught Vieta's formula today, they surely want you to solve it using that. However, if you top up another fifty for my membership..."
Before Qiao Yu could finish his sentence, he heard the two high school students beside him exclaim in terror almost simultaneously: "Teacher Lan..." Then he felt a strong force at the back of his neck, his clothes tightening painfully around his throat, so he had to stand up...
Then a voice exploded by his ear: "Who are you?"