“Marcello, isn’t this going a bit too far?”
In the problem committee room, Professor Roberto spoke with a grave expression.
A professor of mathematics at the University of Milan in Italy, he was a veteran who had served on the IMO problem committee for the past ten years.
“More than half the students can’t even lay a hand on the problem. This isn’t an exam. It’s abuse.”
Marcello looked out the window.
It couldn’t possibly be true, of course, but he felt as though he could hear the students’ screams reaching even here.
“I admit it was a little excessive. But it was necessary.”
Roberto was a mathematician, but at the same time, he was also an educator. He believed that instilling a passion for mathematics in students was also extremely important.
To Roberto, Marcello’s methods seemed far too radical.
“Isn’t our work supposed to be planting dreams in students and helping them realize the beauty of proof? Not driving them into despair.”
Marcello turned and looked Roberto straight in the eye.
“Roberto, you are mistaken.”
“What do you mean?”
Marcello took a step forward. His voice was low, but each syllable carried such weight that everyone in the committee room could hear him clearly.
“We are, in a manner of speaking, like gem appraisers.
No matter how much you polish a pebble, it remains a pebble. A pebble pretending to be a gem is, if anything, harmful to the mathematical world because it deceives the world.”
Roberto jerked his head up in shock.
It was not something one would expect to hear from the mouth of a scholar who had spent his entire 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 since the 2000s doing now?”
“That… that’s…”
The question, sharp as a dagger, tore through Roberto’s chest.
This was a wound like a reverse scale to the IMO.
Systematic education is instead filtering out the true geniuses. Can one distinguish between students who excel at pattern recognition and students with exceptional mathematical intuition?
These were questions the mathematical world had been raising about the IMO for years. And the IMO had yet to present a convincing counterargument.
“The Olympiads of the past were not like this. We discovered shining talents. Grigory Margulis, Elijah Cronen, Grigori Perelman, Vladimir Drinfeld…”
Marcello’s eyes grew distant, as though recalling a faraway past.
“They could spend years clinging to a single problem and never grow weary. Neither material things nor honor mattered to them. They were pure souls thirsting for truth itself.”
A cold light flashed through Marcello’s eyes.
“But what about now?
Most IMO gold medalists leave for finance or the IT industry after graduation.
I have no intention of standing by and watching this. Wasn’t that why Cambridge brought me here?”
“B-but this has gone too far. We’ll be criticized for repeating the mistake of 1988 all over again.”
“Mistake?”
“Yes. A zero percent solve rate. Elijah Cronen was the only one who received even one point. I heard that the problem committee members who submitted that problem were heavily criticized afterward.”
“Why do you call that a mistake?
Didn’t Elijah Cronen, who was only thirteen at the time, grow into a great mathematician?
He proved the Reed conjecture, a fifty-year open problem in mathematics, and won both the Fields Medal and the Abel Prize. And it was the problem committee members of that time who discovered that Elijah.”
“Are you saying it’s acceptable to sacrifice the many for the sake of one or two geniuses?”
Marcello looked at Roberto as if he found the question absurd.
“Sacrifice?
Is that what you call a situation in which students fail to solve a problem because it is difficult?
I am doing what is most necessary for the Olympiad and for the mathematical world. Even now, at this very moment, a genius may be born somewhere in the world. If we insist only on complacent problems, we will never know.”
“Then what are all those children supposed to do?”
He could understand what Marcello meant.
But no matter how generously he thought of it, it seemed most of the students would not even get to see Problem 6.
“They will find the place that suits them. Not everyone needs to become Perelman.
Even if they fail the exam, they can become excellent teachers or applied scholars. If they work hard, they can go to Wall Street, Silicon Valley, or wherever else they want.
However, I have no desire to give such students the signboard of IMO gold medalist.”
Roberto fell silent.
A heavy silence settled over the committee room.
But no one was looking at Marcello with resentment anymore.
* * *
Tap, tap.
Seoha’s fingers unconsciously drummed against the desk.
His breathing, a little faster than usual, tickled the tip of his nose. His lips felt slightly dry.
It was a side of Seoha that had not been seen at all during this IMO.
Normally, the moment he read a problem, an approach would take shape in his head, but this time was different.
The moment he read the first line, the corners of Seoha’s mouth rose slightly.
It was an interesting combination.
Prime number theory, harmonic series, and quadratic residues.
It was not easy to weave together multiple areas of number theory so delicately.
Judging by how both viciousness and delicacy could be felt at once, the problem setter was likely meticulous and had an extremely nasty personality.
In truth, Marcello had not created this problem with the thought of setting something for students. He had wanted to propose a research topic that would make even mathematicians tear their hair out. Just like Problem 6 in 1988.
‘n ≥ 3… distinct prime numbers… the sum of their reciprocals is (n-1)/n…’
Formulas poured down like a waterfall inside Seoha’s mind.
But Seoha could intuitively tell that no matter which path he took, he would eventually reach a dead end.
The distribution of prime numbers, the divergence of infinite series, the structural constraints of perfect squares, and the complex relationships in which all of those were entangled.
Thirty minutes passed.
Seoha’s scratch paper was already covered with erased calculations and new attempts.
Sweat trickled down.
Then, suddenly, Seoha’s pencil stopped.
‘No solution?’
‘No, that can’t be. I must have missed something somewhere.’
Seoha decided to rethink everything from the beginning.
‘The sum of reciprocals, harmonic series, divergence, but here a convergent value, the product becoming a perfect square…’
‘Wait.’
Something flashed past. Faint, but unmistakable intuition.
‘What if there really is no solution? Or what if there is only one?’
Seoha’s heart began to race.
Usually, no solution exists. But what if there is a very special case?
Seoha’s brain began thinking furiously.
He returned from the blocked road and began searching every possible side path.
‘Found it!’
When n was 4, if the four prime numbers took a very special form, it seemed possible. Even then, however, this was only the starting point.
‘What a truly nasty personality.’
The more one tried to find a solution, the deeper it led into the mire. The problem had been designed so that such a method could never reach the destination.
The answer was not a proof of existence, but a proof of “nonexistence.”
However, an exception could appear in the process.
In other words, the setter of this problem—who was almost certainly a person with a ruined personality—was demanding that the contestants prove an “exception to the absence of solutions.”
* * *
‘Pathetic.’
Marcello had come down to the hall and was looking around at the students solving the problems.
Low groans could be heard here and there.
Most of the students looked as if they could not even grasp where to begin.
‘They’re like novice cooks who memorized recipes by rote.’
They could probably serve the dishes they had prepared to a passable degree. But did the world call such people chefs?
The reason there had been many perfect scorers in recent IMOs was not that modern mathematics had become easier.
Rather, it was exactly the opposite.
‘The boundaries between fields of mathematics are disappearing.’
Number theory, topology, geometry, analysis—an age had come when one could no longer conduct meaningful research by digging deeply into only a single field.
‘So if they have aspirations in mathematics, it would be good for them to feel this from early on.’
Marcello looked at the students with the eyes of a hawk searching for prey.
And in a low voice that only he could hear, he murmured,
“Mathematics is, of course, the art of creative thought.”
What was the greatest difference between mathematics and physics?
Unlike physics, which was bound to the interpretation of natural phenomena, mathematics permitted every proposition that contained no logical contradiction.
Therefore, imagination was the most essential virtue a mathematician had to possess.
Problem 6 existed to confirm precisely that.
The goal was one point out of seven, just as in 1988.
‘The minimum amount of progress on the problem.’
That was the maximum Marcello expected from the students.
Marcello slowly walked around the exam hall, observing the students. Most were as expected.
Problems 4 and 5 were barely at the level that could be called interdisciplinary mathematics.
But even that had already left the students groggy.
They were stopped in place, still not knowing where to begin, or bravely repeating simple substitutions.
‘Tsk, tsk. How is that mathematics?’
There were, at least, a few students who seemed decent.
The contestants who had placed first and third last year.
Marcello’s eyes gleamed.
He was curious to see how those who had been the most outstanding talents under the existing exams would overcome the problem he had set.
Marcello looked at the clock.
There was not much time left now. At a glance, both of them seemed stuck at the entrance.
He could not hide his disappointment.
Then one student caught his eye.
A Korean he remembered as the youngest contestant this year.
He was filling out his answer sheet with fierce momentum.
Marcello’s steps stopped near Seoha’s desk.
Worried that he might disturb the student, he stood a little distance away and watched him write.
As Marcello examined the answer, his expression went from serious to gradually astonished. And at last, Marcello was shocked.
‘He intends to solve this to the end?’
Problem 6 had not been set with the expectation that anyone would complete the proof. Even developing just the essential logic for the proof would require an excessively vast amount of writing.
He had intended to award points even for presenting only the minimum approach.
Yet this student had already written more than five pages of answers. And in a startlingly creative and beautiful manner at that.
Marcello felt as if his heart would burst.
The core ideas he had kept in mind while setting the problem were being realized one by one at the boy’s fingertips. No, beyond that, there were even traces of fierce argumentation that the problem setter himself had never considered.
‘A great mathematician is about to appear.’
Indeed, God had not yet abandoned mathematics.
For twenty years, there had been no notable progress in the mathematical world. In the past, theory had driven technological development, but now the situation had reversed. Technology had overtaken it.
The mathematical world was currently in a state of severe stagnation.
‘Where on earth did such a treasure come from?’
If Euler or Gauss were to return to life, would they look like this? Marcello unconsciously swallowed.
Seoha wrote down the final line.
“Q.E.D.”
The Latin abbreviation signifying the completion of the proof was written at the end of the answer sheet.
For a moment, Marcello lost his sense of reality. His mind went blank, then returned.
Thirty years as a professor at the University of Cambridge. He was now witnessing a talent greater than any human being he had seen in his entire life.
Seoha looked over his answer once more.
‘Barely made it!’
And in that instant, he realized it. There were only ten minutes left.
‘Ah…’
He had become so immersed in Problem 6 that he had approached it in far too much detail.
He had not yet touched Problems 4 and 5.
Seoha’s face turned pale.
‘I’m screwed.’