Translator: Dreamscribe
"Marcello, isn’t this a bit too much?"
In the problem-setting committee room, Professor Roberto spoke with a serious expression.
A professor of mathematics at the University of Milan in Italy, he was a veteran who had served as an IMO problem-setter for the past 10 years.
"More than half of the students couldn’t even touch the problem. This isn’t a test, it’s abuse."
Marcello looked out the window.
Of course, it couldn’t be true, but it felt as if the students' screams were echoing all the way to the room.
"I admit it was a bit excessive. But it was necessary."
Though Roberto was a mathematician, he was also an educator. He believed that instilling a passion for mathematics in students was extremely important.
To such a Roberto, Marcello’s approach seemed overly extreme.
"Wasn’t our job to plant dreams in the students and help them discover the beauty of proof? Not to make them despair."
Marcello turned his body and looked straight at Roberto.
"Roberto, you're mistaken."
"Mistaken about what?"
Marcello took a step closer. His voice was low, but each syllable carried weight, and everyone in the committee room could hear him clearly.
"We are, in a sense, like gem appraisers.
A stone remains a stone no matter how much you polish it. A stone pretending to be a gem is actually harmful to the mathematical community, because it deceives the world."
Roberto looked up in shock.
It was not something one would expect to hear from a scholar who had spent his life as a professor.
"What kind of outrageous remark is that? Do you think such words are acceptable from an educator?"
"Then tell me. What are the IMO gold medalists from the 2000s doing now?"
"Th-that’s..."
The question cut through Roberto like a dagger.
It was a sore spot, a touchy subject for the IMO.
Systematic education was actually filtering out true geniuses. Can you distinguish between students who excel at pattern recognition and those with true mathematical intuition?
These are doubts that have been raised in the mathematics community regarding the IMO for years. And the IMO has yet to provide a convincing rebuttal to these claims.
"The Olympiads of the past were not like this. We used to discover shining talents. Grigory Margulis, Elijah Cronen, Grigori Perelman, Vladimir Drinfeld..."
Marcello’s eyes grew distant, as if reminiscing about a far-off past.
"They would cling to a single problem for years without ever tiring. Material gain or fame meant nothing to them. They were pure souls thirsting only for the truth itself."
A cold gleam passed through Marcello’s eyes.
"But what about now?
Most IMO gold medalists head off to finance or the IT industry after graduation.
I won’t just sit and watch this happen. Isn’t that why Cambridge brought me here in the first place?"
"B-but this was too much. We’re going to be criticized for repeating the mistake of 1988."
"Mistake?"
"Yes. Zero percent correct answer rate. Only Elijah Cronen managed to get a single point. I heard the problem-setters at the time were heavily criticized afterward."
"And why was that considered a mistake?
Didn’t Elijah Cronen, who was only 13 years old at the time, grow into an outstanding mathematician?
He proved Reed’s Conjecture, a problem unsolved for 50 years in the mathematical world, and won both the Fields Medal and the Abel Prize. And it was the problem-setters of that time who discovered such an Elijah."
"Are you saying it’s okay to sacrifice the majority for the sake of one or two geniuses?"
Marcello looked at Roberto with an expression of disbelief.
"Sacrifice?
Is that what you call a situation where students can't solve a difficult problem?
I’m doing what’s most necessary for the Olympiad and the field of mathematics. Even now, at this very moment, a genius might be born somewhere in the world. But if we insist on setting only easy problems, we’ll never know it."
"Then what are we supposed to do with all those children?"
Marcello’s intentions were understandable.
But no matter how generously he thought about it, it seemed clear that most of the students wouldn’t even get to look at Problem 6.
“They will find their appropriate place. Not everyone needs to become Perelman.
Even if they mess up the test, they can still become excellent teachers or applied scholars. If they work hard, they might go to Wall Street, Silicon Valley, or wherever they want.
But I don’t want to give those students the title of IMO gold medalist.”
Roberto remained silent.
A heavy silence filled the committee room.
However, the eyes that had been filled with resentment toward Marcello were no longer there.
***
Tap tap.
Seo-ha’s fingers tapped the desk unconsciously.
His slightly quickened breathing tickled the tip of his nose. His lips felt slightly dry.
It was a side of Seo-ha that hadn’t been seen during this IMO period.
Normally, as soon as he read a problem, an approach would form in his mind, but this time was different.
As he read the first line, a slight smile tugged at the corner of Seo-ha’s lips.
It was an interesting combination.
Prime number theory, harmonic series, and quadratic residues.
It wasn’t easy to interweave various fields of number theory this intricately.
From the way maliciousness and delicacy coexisted, the problem-setter must have been a meticulous and extremely ill-natured person.
In fact, Marcello hadn’t created this with the idea of setting a problem for students. He wanted to present a research-level topic that would make even mathematicians tear their hair out. Just like Problem 6 in 1988.
‘n ≥ 3… distinct prime numbers… sum of reciprocals equals (n-1)/n…’
Formulas poured into Seo-ha’s brain like a waterfall.
But Seo-ha could intuitively sense that every path he followed would eventually lead to a dead end.
The distribution of prime numbers, the divergence of infinite series, structural constraints of perfect square numbers, and the complex relationships entangled among all these.
Thirty minutes had passed.
Seo-ha’s scratch paper was already covered with erased calculations and fresh attempts.
Sweat began to trickle down.
At that moment, Seo-ha’s pencil stopped.
‘No solution?’
‘No, that can’t be. I must have missed something.’
Seo-ha decided to reconsider everything from the beginning.
‘Sum of reciprocals, harmonic series, divergence, but here it converges to a specific value, and the product must be a perfect square…’
‘Wait.’
Something flashed through his mind. A faint but unmistakable intuition.
‘What if a solution really doesn’t exist? Or if only a single one exists?’
Seo-ha’s heart began to race.
Normally, there’s no solution- but what if it’s a very special case?
Seo-ha’s brain started racing furiously.
He backtracked from the blocked path and began searching for all possible side routes.
‘Found it!’
If n is 4, and if the four primes are of a very special form, it might be possible. But even then, it was merely the starting point.
‘What a nasty personality.’
The more he tried to find a solution, the deeper it led into a mire. The problem was designed so that you could never reach the destination by that method.
The correct answer was not a proof of existence, but a proof of non-existence.
But in that process, exceptions could appear.
In other words, the problem-setter, almost certainly someone with a deranged personality, was demanding that the participants prove the exception to a non-existent solution.
***
‘Pathetic.’
Marcello had come down to the main hall and was watching the students working on the problems.
Low moans could be heard here and there.
Most students didn’t even know where to begin.
‘They’re like novice cooks who’ve memorized recipes word for word.’
The dishes they prepared might come out edible. But does the world call such people chefs?
The recent increase in perfect scores at the IMO wasn’t because modern mathematics had become easier.
In fact, quite the opposite.
‘The boundaries between domains of mathematics are disappearing.’
Number theory, topology, geometry, analysis, it had become an era where meaningful research was no longer possible by digging deeply into just one field.
‘So if one is truly devoted to mathematics, it’s good to experience this early on.’
Marcello observed the students with the sharp gaze of a hawk searching for prey.
And in a voice only he could hear, he murmured softly,
"Mathematics is, of course, the art of creative thinking."
What is the biggest difference between mathematics and physics?
Unlike physics, which is bound to interpreting natural phenomena, mathematics allows all propositions that are logically consistent.
Therefore, imagination is the most essential virtue a mathematician must possess.
Problem 6 was meant to test exactly that.
The goal, just like in 1988, was a score of 1 out of 7.
‘Minimal progress on the problem.’
That was the maximum Marcello expected from the students.
Marcello walked slowly through the exam hall, observing the students. Most were as he had expected.
Problems 4 and 5 were only just barely worthy of being called fusion mathematics.
But even that had already left the students groggy and overwhelmed.
They were stuck, not knowing where to begin, or bravely repeating brute-force substitutions.
‘Tsk tsk. How is that mathematics?’
There were a few students who were at a somewhat decent level.
Participants who had placed 1st and 3rd last year.
Marcello’s eyes sparkled.
They were the most talented in past competitions, and he was curious to see how they would overcome the problem he had set.
Marcello checked his watch.
There wasn’t much time left. At a glance, both seemed stuck at the entrance of the problem.
He couldn’t hide his disappointment.
At that moment, one student caught his eye.
The Korean participant he remembered as the youngest contestant this year.
He was furiously filling in his answer sheet.
Marcello’s steps stopped near Seo-ha’s desk.
He stood a little distance away, careful not to disturb the student, and watched as the answer was being written.
Marcello’s expression, as he examined the answer, shifted from seriousness to surprise, and finally to shock.
‘He’s going to solve this all the way?’
Problem 6 was not set with the intention of having it fully solved. Even just laying out the essential logic for the proof would be an excessively massive task.
He had intended to award points just for presenting the minimal approach.
But this student had already written over five pages. And in an astoundingly creative and beautiful manner.
Marcello felt like his heart was about to burst.
The core ideas he had kept in mind while setting the problem were being executed, one by one, at the boy’s fingertips. No, beyond that, even traces of intense reasoning that the problem-setter himself hadn’t considered.
‘A truly great mathematician is about to be born.’
Surely, God had not abandoned mathematics just yet.
For 20 years, there had been no notable breakthroughs in the mathematical world. In the past, theory had driven the development of technology but now, the situation had reversed. Technology had surged ahead.
The field of mathematics was currently in a major stagnation.
‘Where on earth did this treasure come from?’
If Euler or Gauss were to be resurrected, would they look like this? Marcello unknowingly swallowed hard.
Seo-ha wrote the final line.
「Q.E.D.」
The Latin abbreviation meaning "proof complete" was written at the end of the answer sheet.
Marcello momentarily lost his sense of reality. His mind went blank and then came back.
Thirty years as a professor at Cambridge, he was now witnessing a talent greater than anyone he had ever seen in his life.
Seo-ha scanned over his answer one last time.
‘Finally succeeded!’
And in that moment, he realized. There were only ten minutes left.
‘Ah....’
He had gotten so immersed in Problem 6 that he had approached it in excessive detail.
He hadn’t even touched Problems 4 and 5 yet.
Seo-ha’s face turned pale.
‘I’m screwed.’