Jo Taejin always sat in the front row.
No one told him to. Like a habit, he entered the classroom thirty minutes before class and set down his bag. When choosing his seat, he picked one where the blackboard was directly in front of him rather than by the window where sunlight streamed in at an angle.
From a young age, Taejin had instinctively known how to rise to the highest position within any group.
His father was a chief judge; his mother was the daughter of a local dignitary.
That fact alone allowed Taejin to always stand in a position of superiority among his peers. But he never bullied others or raised his fists.
"Don't cause trouble and block your own future. I believe my son isn't someone so pathetic."
His father had always been cold toward Taejin.
When he was young, there was a time he had polished his father's shoes before his father left for work. There was no other reason. He had simply been imitating a heartwarming anecdote he had seen somewhere.
He just wanted to hear a small word of praise.
He had imagined the scene of his father saying, "Our son is so kind," while patting his head.
The image of his younger self, holding the gleaming shoes and looking up at his father with pride.
But what returned was a cold gaze.
"Never do something like this again."
His father's voice was as cold as a blade.
"The moment you do something menial even once, it becomes a habit. If you aren't going to do it for the rest of your life, don't even touch it from the start. What pleases parents is the child's achievement."
After that, Taejin stopped saying more than necessary in front of his parents. Because he realized that no matter what he said, it held little meaning.
Instead, Taejin spent most of his time in front of his desk.
A blessed environment and DNA—combined with personal effort, they produced a terrifying synergy.
As time passed, Taejin's grades rose endlessly. First in class, first in school, and even first in the district.
When exams ended, Taejin's name was always at the very top. He was also the first student name teachers memorized. Friends flocked around him. In an instant, he became the center of the group.
However, Taejin's life at home hadn't changed at all. His father remained expressionless, and even when presented with a report card, all he did was nod.
Then one day, his mother invited a few acquaintances over. Perhaps they were important guests, as his father joined them.
Taejin, who had left his room to go to the bathroom, happened to hear his name pop up in the adults' conversation.
"I heard Taejin is such a good student? And that he got into the gifted high school... You must be so happy, Taejin's mother."
His mother, who rarely showed emotional fluctuation, let out an exaggerated laugh and shook her head.
"It's nothing special. He still has a long way to go."
His father also raised his glass and added.
"He must take after me; the child works very hard. Achievement only becomes real when it continues. Right now, it's merely a process."
Though they were words of modesty, Taejin didn't miss the expressions that flickered across their faces.
Pride, satisfaction, delight, affection—everything he had ever wanted from his parents was there.
'So my parents have been watching me properly all along.'
It felt as though his efforts all this time were being rewarded.
Taejin formed a faint smile, returned to his room, and closed the door. Then he opened his book again.
The first day after entering the gifted high school,
Taejin felt an instinctive sense of familiarity the moment he stepped into the classroom.
'Why?'
What had changed compared to middle school?
Taejin looked around.
About twenty children were forming small groups throughout the classroom.
Neatly groomed hair, clothes and bags bearing small luxury logos. Even their trivial gestures were imbued with the composure of those born with privilege.
He turned his head the other way.
Students he was seeing for the first time were engaged in discussion.
"Isn't that approach a bit simplistic?
"Ah, that was refuted in a paper I saw before."
Soft voices, yet brimming with confidence.
'Ah! Was this it?'
All the children here exuded a similar atmosphere.
If forced to express it in words, would it be a consciousness of being the elect?
A sense of kinship from having grown up in the same world—they were projecting reflections of themselves.
Taejin found it both distasteful and strangely reassuring.
'Except for a few...'
Every group had its irregulars.
Taejin turned his head to look at the boy who was the talk of the class.
He had been quite the stir since the entrance ceremony.
It was because a girl who appeared to be his younger sister had cried for the school to disappear, not wanting to be separated from her brother.
But Taejin had noticed the parents' attire before the sister.
They had dressed meticulously, yet something was off.
The man's suit didn't fit his body well, as if bought long ago.
The woman's two-piece was of a tacky design that seemed to follow an old trend.
Nevertheless, there was no trace of shame on their faces.
Amidst parents clad in luxury brands and tailored suits, they smiled brightly as they looked at their son.
That sight somehow made Taejin uncomfortable.
* * *
The next day, too, Taejin found himself keeping his eyes on Yu Seoha.
He didn't know why.
When he entered the classroom, his eyes naturally sought him out first.
"Hello."
Without fail today as well, despite lacking any particular context or prior friendship, Seoha lowered his head first whenever they passed each other.
In response, Taejin lightly pointed to his own earphones.
People without a sense of distance made him uncomfortable.
Most children in this school always maintained a calculated distance.
Whether to greet someone was determined only after considering the other's background, standing, and future relationship.
But Yu Seoha didn't seem interested in such things.
Today, too, he entered the classroom before anyone else, straightened crooked desks, and adjusted the lowered window blinds, touching them several times to align them precisely.
'An unexpectedly neat personality?'
Swish, swish.
Even though he wasn't on duty, he erased the blackboard clean.
Tap.
Tap.
Only after aligning the chalk in rows by color, one by one, did his morning routine end.
Honestly, Taejin was intrigued.
He still didn't fully understand how impressive a first-place individual finish at the IMO was. Now that he was a high school student, he would be able to check soon by competing himself.
He sat for a moment, and then class began.
And soon, math period arrived.
Rattle.
Teacher Park entered the classroom holding test papers.
"We're having a quiz."
Taejin tensed up.
Even if it was just a quiz, it was the first test since admission. It would be an important measure of whether he could rise to a superior position here as well.
Papers were distributed.
The moment the flipped test papers were placed on the desks, the classroom fell silent at once.
"Begin."
Pencils began moving simultaneously.
Scratch, scratch.
The sound of pencils friction against paper spread in a uniform rhythm.
Students were hunched over, concentrating on the problems.
Watching them, Park Seongdae formed a satisfied smile.
'The kids are good this year, too. They won't be easy to rattle, but I like the spirit of at least trying.'
Question 1.
[Assume there is a 2n × 2n grid. We move from coordinate (0, 0) to (2n, 2n). Each move can only be to the right or upward. Prove that the number of possible paths is always even.]
Taejin swallowed when he saw the problem.
'Prove it is even.'
It wasn't simply a problem of counting paths. He immediately thought of binomial coefficients.
He needed to form an equation using binomial coefficients and prove that it was even.
Taejin's pencil moved busily. He first tried plugging in small values to find a pattern.
A pattern clearly existed.
But calculation alone couldn't constitute a proof.
'What should I do? Ah!'
He recalled the structure of mathematical induction.
Scratch, scratch.
Equations continued onto the paper.
He wrote Pascal's identity and expanded the binomial coefficient. He decomposed it into product terms and examined the evenness of the denominator and numerator.
Line by line, solid logic built up like stairs.
Taejin stopped for a moment and looked around.
'As expected of a gifted high school.'
A difficult problem had appeared right from the start, yet the students were finding answers in their own ways.
Everyone was filling their papers densely with tense faces.
'Hm?'
Then one student who wasn't doing anything caught his eye.
It was Yu Seoha.
His flustered, lost expression evoked sympathy.
'So this is a problem that even an IMO first place can't solve? Then just what level is this school...'
A small doubt arose in his mind, but Taejin erased it and focused on the problem.
* * *
Seoha was flustered.
Very much so.
Everyone was solving problems diligently. Many students were asking for extra paper because they ran out of space.
But Seoha couldn't figure out what to write on his answer sheet.
Still, he had to solve the problem.
Seoha squeezed his eyes shut, then picked up his pencil and wrote down an answer.
["P ↔ P', therefore the total number of paths is even."]
'Is this enough? Professor Park said to write it out in as much detail as possible, even if it seems obvious to me...'
After the controversy at the Olympiad, Seoha had gained a realization.
It was that he shouldn't expect everyone to understand the logic he established. But that was only when logic was actually necessary.
To Seoha, the problem before him was an endpoint where the equation could no longer be expanded.
When paired by symmetry, of course all paths came in pairs.
The proof ended there.
The students approached this problem through understanding and calculation. But Seoha viewed the problem through structural symmetry and reinterpreted it through a completely different concept.
Seoha stared at what he had written. It seemed far too lacking in effort.
'Deoki, I really can't do this. You figure something out.'
The anxious Seoha eventually sought help even from his uncomfortable cohabitant. But Deoki stubbornly refused to solve the problem, as if telling him not to be called for something like this.
Seoha despaired.
Tick, tock.
Time passed.
Having come this far, Seoha decided to write detailed proofs for the other problems no matter what.
But it was the same in the end. For every problem, he simply couldn't write more than a line.
The problems Teacher Park had created were of a difficulty beyond the high school curriculum, but they fell far short of IMO level.
Therefore, to Seoha, who had found even Olympiad problems too easy, they were no different from basic arithmetic.
"Stop!"
"Already? I just need a little more time..."
"Teacher! Humanely speaking, this is too short."
As the test ended, sounds of regret erupted here and there.
One by one, students came forward and submitted densely filled test papers without a single blank space.
Soon it was Seoha's turn. He carefully gathered his four sheets of paper.
And slowly walked to the front.
Seeing Seoha's pale face, some students stifled laughs.
"I did finish them all, but..."
The hand holding out the test papers trembled.
Teacher Park's pupils shook violently at the sight of Seoha's test paper, which was virtually no different from a blank sheet.