stjorthotic Chapter 16, Math Olympiad Problem 6
Chapter 16, Math Olympiad Problem 6
Top-floor studio in the heart of Beijing's business district.
A blinding spotlight shone on the center.
Standing in front of the camera, Shen Yuxuan wore his signature calm smile as he gazed at the teleprompter screen.
Today he wore a casual yet far from cheap haute couture suit, and his hair was styled meticulously with hair gel.
His entire body, from every pore, exuded the absolute authority of a "national gold medal mathematics teacher".
Recently, he launched a new course called "Framework" for top students.
Unsurprisingly, it exploded in popularity as soon as it was released, with the screen filled with enthusiastic praise such as "As expected of Shen Yuxuan".
In the past, the education sector has criticized his classes quite a bit.
Why?
Because of an excessive focus on test-taking efficiency.
Too materialistic!
That's too domineering!
That kind of "even if you're an idiot and don't understand the essence of mathematics, you can still find the correct answer by applying my formula" rogue technique is the core factor that made this young teacher quickly become popular and a star lecturer.
Of course, this is often criticized by old-school educators who say, "This is destroying the foundation of mathematics and cannot truly improve students' mathematical abilities."
But within high school circles, Shen Yuxuan has only one name:
"A tyrant who has an extreme distrust of students' intelligence."
His original words are:
"I don't believe in your brains at all."
So don't try to understand.
Your intelligence doesn't support you doing this kind of thing.
Just follow my steps like an emotionless machine!
mad!
That's insane!
Arrogant, jarring, yet extremely effective.
His fan base and student base are extremely stable, so stable that even when he is being verbally abused, countless people still wave their money and beg him to give them a spot in his class.
Those criticisms?
It's a piece of cake.
At this moment, the live stream chat was scrolling wildly.
Attending Shen Yuxuan's lectures can easily cause high blood pressure; always keep some nitroglycerin tablets on hand!
[It's true I'm a bad student, but did you have to put it so harshly? Don't I have any pride?]
[But your grades improved, right?]
【……really.】
So you still have to hold your nose and listen!
Try to put yourself in Shen Yuxuan's shoes. What kind of deity is he?
Graduated from Princeton University's Mathematics Department with a full scholarship! Studied under a Fields Medal winner!
In his eyes, how could a bunch of math dunces like us, who can't even figure out derivatives, be considered the same species?
No, no, no, the person upstairs is too optimistic.
In Shen Yuxuan's eyes, even if you could rank in the top 10% of your grade, he would still consider you a nobody.
[If I get into the top 1% of students, can I eat at the same table as him?]
The top 1% can listen to his "Framework" series of courses.
Remember, from that point on, you were barely considered a person in his eyes.
During the Q&A session after class, Shen Yuxuan, looking at the live chat comments, tried to liven up the atmosphere:
"There's a rumor going around that I think people who don't get into the top 10% aren't even human?"
Because there are simply too many students, the physical venue can't accommodate them all, so he hardly ever teaches in-person classes anymore.
But he enjoys the live-streaming format, where he can see the students' reactions and responses in real time.
Isn't this the truth?!
[Science Fact: I actually heard him say something like, "Why can't you do this? Just follow the steps and you'll get it right. How can you get this wrong? Are you even human?" Such insane things!]
Woof woof woof!
[Awooo! Snoring!]
[Meow—]
【Meow meow meow!】
The screen was instantly filled with comments, turning into a scene of a large-scale zoo celebrating the return of ancestors.
"Calm down! It's just a misunderstanding."
Shen Yuxuan rubbed his temples helplessly.
"It was only a few years ago that I realized that, in essence, there isn't that much difference between you and me."
【? ? ? 】
【Seriously, teacher? What's wrong with you? Have you been possessed?】
[We're definitely going to get thoroughly humiliated again, haha! Everyone stay calm, it's not like we've fallen for this before!]
Shen Yuxuan is at it again, subtly showing off. Get ready to take screenshots.
Shen Yuxuan sighed and shook his head:
"Sigh. I've committed too many sins, and now no one believes me even if I speak the truth from the bottom of my heart. It's my fault, I admit it."
That's right! Admitting mistakes starts with correcting your attitude!
Hahahaha!
Stop talking nonsense, why don't you tell us your first love story!
"I want to state upfront that this is entirely my highly subjective personal opinion."
Shen Yuxuan's smile faded, and his eyes became deep and thoughtful.
"There are too many self-proclaimed or self-proclaimed 'geniuses' in this world; they're practically everywhere."
If we only consider the mathematics community, there are probably around a thousand people.
These are all individuals who have published significant papers in academia.
He paused, his tone suddenly turning cold.
"But if you ask whether those people are geniuses, I will tell you without hesitation, no."
[Holy crap? Then who really deserves the title of genius?]
"A genius. A talent bestowed by God!"
In Latin, Genius means a person endowed with special abilities by God!
Throughout history, the East and West have never crossed paths, yet they have created words with completely identical meanings!
Do you think there are thousands upon thousands of people who are blessed by the gods?
[So, what kind of genius do you see in your eyes, teacher?]
I often emphasize that genius must speak for itself through its achievements!
Being able to solve calculus at age five makes you a genius? Don't be ridiculous!
Many high school students can solve calculus problems.
There are always some media outlets in China that mistakenly portray precocious children as geniuses and hype them up.
What's the point of taking a step or two earlier than others?
An IQ of 180? Mensa certification? So what!
Did they achieve anything world-changing with that level of intelligence? No!
Does having a good memory make someone a genius?
Is it impressive to memorize information that can be easily found online? That's just being a human USB flash drive!
[Heh... I actually fell for this bizarre logic.]
Comrades, don't be fooled! He's brainwashing us!
To be honest, there's nothing wrong with that. These days, anyone, no matter how good or bad, dares to call themselves a genius.
So who really is the genius?!
Shen Yuxuan took a deep breath and recited a string of names:
"Geniuses are rare, appearing only once every hundred years, or perhaps every fifty years if you're lucky."
Historically, there were long periods when there were none, and there were also times when several geniuses emerged in the same era, creating a veritable battle of titans.
From ancient Pythagoras, Euclid, Archimedes, and Aristotle, to modern Newton, Euler, and Gauss, and then to the present day…
Only those at the level of Einstein and Ramanujan deserve to be called geniuses!
[Holy crap! This standard is insane!]
Ultimately, unless you reach the level of a great person, are we all just ordinary human beings?
[Holy crap, why do I suddenly feel so much better?]
"Below this god-like level, there is another echelon that can serve as 'assistants' to geniuses."
Although they were not geniuses, they were still extremely outstanding.
This refers to the thousand people I mentioned earlier.
This certainly includes numerous Fields Medal and Nobel Prize laureates.
Shen Yuxuan chuckled self-deprecatingly.
"I know I can't even get into this group, so I gave up becoming a mathematician and came here to teach you."
From this perspective, you and I are indeed no different; we are all mortals.
This is what I wanted to say today, and I hope everyone understands.
Does that mean that humanity hasn't produced a genius in decades since Einstein?
"Yes, because modern mathematics and physics have not yet been able to break through the boundaries of Einstein's theory."
Why was Einstein so great? Because his theory of relativity encompassed the law of universal gravitation.
Newton discovered the laws that applied to Earth, while Einstein extended them to the entire universe. So what's next?
[Quantum mechanics. 】
When in doubt, consult quantum mechanics!
Wow, is this the world of top students?
I was inexplicably drawn in by this old man's inspirational quotes!
"That's something you'll have to figure out for yourselves after you get into university."
At this point, Shen Yuxuan suddenly changed the subject.
"However... I know a Chinese man from Xia who has a very high chance of becoming a genius."
But I won't reveal who he is. You know why.
If there are no results, it's all nonsense.
[People from Xia Kingdom?? Holy crap, if that's true, that's amazing!]
Could he be his student?
[It could also be his teacher.]
Even Shen Yuxuan, who's so arrogant and looks down on us, praises us so highly! I'm really looking forward to it!
"Alright, alright! I'll reveal the secrets to you guys when that person achieves something."
Ugh, it's all your fault for stirring things up! We used up all our Q&A time today! get out of class dismissed!
"Smack."
The live stream has ended.
Shen Yuxuan let down his guard, leaned wearily against the back of the leather chair, and stared at the spotlights on the ceiling.
"A genius..."
Even he himself couldn't help but feel a surge of fervent anticipation.
From the moment Shen Yuxuan met Su Hao, his keen intuition, as sharp as an old dog's, had a vague premonition of this.
There's one thing he didn't tell the students during the live stream.
That is, too many geniuses, because they are bound by mundane rules, fail to fully display their talents and perish early for various reasons.
Shen Yuxuan made up his mind and clenched his fist tightly.
Even if it costs me my life, I absolutely cannot let this kind of shitty thing happen to Su Hao!
......
a few days later.
At Su Hao's home in Daji County.
"What are you doing?"
Having made the rare decision to turn down high-end business deals and personally travel to this remote and impoverished area, Shen Yuxuan's forehead veins throbbed as he looked at the scene on Su Hao's desk, and he finally couldn't help but speak.
The first round of the math olympiad preliminary rounds is about to begin!
However! The practice exam set that he had gone to great lengths to obtain through connections and even personally annotated and sent over was now being carelessly piled up on the corner of the table like waste paper!
At first glance, everything looks brand new! Not even a crease!
Clearly, they haven't even turned the page on their ancestors' history!
"Ah! You mean that past exam paper?"
Su Hao, who was sitting at his desk, looked up. His handsome face showed a hint of confusion at being interrupted. He then scratched his head somewhat awkwardly.
"Yes! Why didn't you do it?" Shen Yuxuan suppressed his anger.
"Actually, what I was originally going to do was..." Su Hao said casually, pointing to the center of the table.
"But I just happened to find something interesting, so I did the math on the spot."
smoothly?
Fun?
Shen Yuxuan's brows furrowed even deeper as his gaze fell on Su Hao's desk.
There was a thick stack of cheap draft paper there, densely covered with derivation formulas.
He stepped forward, wanting to see what this kid was messing around with instead of doing serious math problems.
Immediately afterwards, Shen Yuxuan's gaze fell upon the question at the top of the draft paper.
His expression suddenly froze, as if a needle had pierced the depths of his pupils.
My heart skipped a beat, and I knew something was wrong!
【IMO 1988 Problem 6】
【Let a and b be positive integers such that ab+1 divides a^2+b^2. Show that frac{a^2+b^2}{ab+1} is the square of an integer.】
boom!
It was as if a thunderbolt had exploded in Shen Yuxuan's mind!
The infamous sixth problem from the 1988 International Mathematical Olympiad!
He had clearly deleted that nightmarish question before sending the materials...
It seems it got mixed in accidentally during the binding process!
Ruined!
This kid's Dao heart is probably about to collapse!
It's important to know that math competitions will never present problems that are "unsolvable".
This is an ironclad rule that has remained unchanged since Romania hosted the first competition in 1959!
The only exception was 1988, which was a catastrophic one in the history of mathematics!
What kind of question is this?
This is a monster-level exam question that even the top experts from the Australian exam committee couldn't solve within the allotted time!
Even later, four top mathematicians specializing in number theory joined forces and spent a full six hours, sweating profusely, but still couldn't find the answer!
Therefore, this question was marked with an extremely dangerous double asterisk and even had a serious warning attached: "Too difficult, not suitable as an exam question, it will destroy the contestants' confidence!"
But the committee at the time, for some unknown reason, claimed to "believe in human potential" and forcibly included this question in the exam.
The results of it?
The result was a brutal massacre of the world's top geniuses!
The exam halls back then were practically littered with corpses; countless gifted teenagers wept bitterly at this question, questioning the meaning of life!
But now... this elementary school student actually says it's "pretty fun"?!
"Su, Su Hao... how far have you gotten?"
When Shen Yuxuan opened his mouth, he realized that his voice was trembling, and his pride in composure and high-class demeanor shattered instantly.
He couldn't resist moving closer, his face almost touching the draft paper, staring intently at Su Hao's problem-solving process.
On paper, there is no blind trial and error, nor is there a lengthy and messy exhaustive list.
There is only one suffocating, surgically precise logical progression.
[...especially when b is large enough, it can be proven that a' < a.]
[Further proof is needed to demonstrate that a' > 0.]
Therefore, (a', b) becomes a completely new solution smaller than the original solution (a, b). This contradicts the minimization of (a_0, b_0)...
The profound mathematical symbols seem to come alive under the boy's pen; they split and recombine in his mind, eventually transforming into an impeccable chain of logic on the paper.
"Whoosh!"
Shen Yuxuan felt all the blood rush to his head in that instant!
The surrounding air seemed to be sucked away in an instant, and even the noisy car horns outside the window completely disappeared from Shen Yuxuan's ears at this moment.
His eyes were so wide they almost bulged out of their sockets, and his pupils trembled violently.
Holy crap!!!
"Really?!"
"What kind of international joke is this?! What... what did this kid do?!"
Shen Yuxuan's mind was screaming wildly, feeling as if his values were being repeatedly crushed by an invisible road roller!
At this moment, Su Hao tilted his head, looking at Shen Yuxuan, whose face was full of horror as if he had seen a ghost, with a clear look of doubt in his eyes:
"What's wrong, Teacher Shen? Is there a problem? Is there a logical flaw?"
His tone was so innocent, so nonchalant, and even carried a hint of disdain, as if to say, "This kind of idiotic question wasted ten minutes of my lunch time."
Shen Yuxuan stiffly twisted his neck, looking at Su Hao as if he were a monster.
The core of this proof lies in a problem-solving approach known as quadratic form theory.
Su Hao had clearly figured this out with his terrifying intuition.
But that's nothing.
But what made Shen Yuxuan's scalp tingle even more was that he had taken an incredible step forward on this basis.
"No..." Shen Yuxuan swallowed hard, his voice hoarse and unlike his own, "It's perfect."
It seems that they felt that the conventional proof of quadratic forms was not enough, and that using a sledgehammer to crack a nut was not sophisticated enough.
Su Hao directly used an extremely rare method in the mathematical world, known as the "Vieta jump," to deduce the answer.
Without any hesitation or delay.
It uses the simplest, most forceful, and most beautiful form, just like a work of art that is perfectly natural!
The ink on the last line was not yet completely dry:
Therefore, k must be a perfect square.
The proof is complete.
Perfect. Flawless.
Shen Yuxuan swallowed hard, feeling his throat tighten.
Why did you choose this method to prove it?
Faced with Shen Yuxuan's question, Su Hao scratched his head and answered in a tone that seemed matter-of-fact, even as if he thought the question was stupid:
"Assuming there exists a minimum element, then constructing an element smaller than it to deduce a contradiction—that's the cleanest and most efficient method. Don't you think it's beautiful?"
Clean and neat.
Very beautiful.
Shen Yuxuan seemed to hear the sound of his pitiful pride shattering into pieces.
This method has been hailed by countless top scholars in the history of mathematics as "the most exquisite proof technique".
Su Hao was completely drawn to the beauty of this mathematical structure by his monstrous intuition, and directly embarked on the most perfect solution path.
This is fundamentally different from Shen Yuxuan's own utilitarian mindset of "as long as I can find the answer and get the points, I don't care about the process!"
This is a true dimensional reduction attack!
"thump."
Shen Yuxuan's legs went weak, and he slumped down on an old wooden chair, as if all his strength had been drained away, and he couldn't come to his senses for a long time.
Looking at the stack of discarded practice exam papers, Shen Yuxuan closed his eyes bitterly.
"Su Hao doesn't need that set of practice questions at all."
Using human test questions to test a god who can see through the laws of mathematical structure?
This is blasphemy against genius!
At this moment, he could not suppress the intense desire in his heart that was almost bursting his chest!
I really want to introduce this genius to the world as soon as possible!
"Let those arrogant Western academics taste what it's like to be utterly crushed and left questioning the meaning of life!"
Shen Yuxuan took a deep breath and suppressed his wildly beating heart.
"Just wait a little longer, just a little while longer."
He was plotting viciously in his mind.
If next year's International Mathematical Olympiad questions are extremely difficult, so difficult that they make geniuses all over the world weep... that would be wonderful.
I'm going to make all the mathematicians in the world taste what it's like to have their worldview shattered!