Chapter Thirty-Seven: Critical Strike

After Mr. Zhang led Wu Tong into the school, he first took her to the place where she would rest tonight.

"This time, we're heading to Hainan for the competition. The provincial team has ten participants in total, all of whom ranked in the top ten at the provincial contest: two from No. 1 High, three from Foreign Language School, plus you and two others from the city districts—all staying temporarily in the Young Teachers' Apartment here. The two from No. 7 High are living in their own dorms."

"The apartments are two-person rooms. Your roommate is Feng Yueyue, from Shangdu Foreign Language College, who was also the runner-up in this math league. She has already checked in. Among those selected for the provincial team, only you and Feng Yueyue are girls."

As they walked, Mr. Zhang gave Wu Tong a brief introduction to the provincial team. This year, although the team remains overwhelmingly male, it was two young women who claimed first and second place, making it clear that no one could underestimate them.

Feng Yueyue had excelled during the week-long provincial team training, outperforming all the boys in her class. She came from a family with a strong academic tradition and had been trained in Olympiad mathematics from a young age. After the provincial team's training, she had made obvious progress compared to the math league, and was regarded as one of the province's top contenders for the national team—a favorite of the training teachers and the provincial mathematics society, their own protégé.

Wu Tong had dominated the provincial contest with a perfect score, demonstrating her absolute excellence. However, she hadn't attended the early training, which left some teachers and society members somewhat critical, suspecting that Wu Tong, coming from a small town, was perhaps a bit arrogant after achieving a good result.

But when they sent her the subsequent training problems—whether simulated past national CMO questions or actual International Mathematical Olympiad problems—none could stump her in the exchanges.

With her level, attending centralized training was hardly meaningful. Truly gifted students aren't shaped by force-feeding or cramming.

In his heart, Mr. Zhang favored Wu Tong. This year's math league was no trivial feat—not just anyone could achieve a perfect score, with a forty-point gap between her and the second place.

Moreover, according to their post-contest review, the additional questions included material beyond the syllabus. This girl had handed in her paper two hours early, proving not only that the test was difficult, but that it simply wasn't difficult for her. Her mathematics skills likely already touched on university-level material, and her grasp of high school fundamentals was solid and unshakeable.

"I live here too, in Room 102. If you need anything, you can always find me." He wasn't one of the regular teachers, but a member of the provincial mathematics society, currently teaching at Zhongyuan University. He had been specially seconded to train the provincial team members.

As he spoke, Mr. Zhang took Wu Tong to Room 106 on the first floor of the Young Teachers' Apartment and swiped his card to open the door. The apartment was quite nice: a private bathroom at the entrance, two single beds inside, complete bedding, desks at the head of each bed, the decor simple and bright, clean and tidy.

At this hour, her roommate Feng Yueyue was likely in class, so the dorm was empty. It was clear Feng Yueyue had chosen the bed by the window, and Wu Tong placed her suitcase at the bed near the door.

She hadn't brought much, only some books and a few changes of clothes. Worried she might need money, Teacher Jin had specially put a thousand yuan in cash and a bank card in her wallet before she left, repeatedly reminding her to buy whatever she needed, and if money was short, to ask the accompanying teacher to help her withdraw more.

"Wu Tong, would you like to rest and wander around a bit, or head to the training classroom? Everyone will gather tonight for a simple dinner to celebrate the end of training."

"Please take me to the training classroom," Wu Tong replied without hesitation, picking up her backpack and preparing to leave with Mr. Zhang, seeing no need to waste time in the dorm.

The training classroom was in the first teaching building outside the dorm area, not far from the apartments—very convenient. At this time, the students attending the training were frowning in concentration over the problems set by the teachers on the blackboard, with only a few actually writing.

Besides the provincial team members at the front, there were other students at the back of the classroom, whose expressions showed even more confusion at the problems. According to Mr. Zhang's explanation, Wu Tong knew these were students who had received sponsorship to participate in the training.

Some had ranked well in this year's math league, others were preparing for the next one. With the high quality of this training camp, naturally some parents eager for their children's success had managed to send them in.

When Mr. Zhang brought Wu Tong in, the atmosphere in the training room suddenly grew tense.

None of these students were fools—in fact, those who excel in mathematics tend to be highly intelligent. At this moment, the only student who could be brought in by the teacher was the newly emerged provincial champion with a perfect score—the one they'd all heard would arrive today.

The instructor for the class was another team leader, Zhang Ping, the chief coach of the provincial team this year. He was tall and imposing, with a rough, heroic demeanor, forming a striking contrast to the refined, scholarly Mr. Zhang. To the students, they were like a pair of black-and-white guardians. Zhang Ping was a veteran, a provincial master teacher who had produced multiple national prize winners, even members of the national team.

Seeing Wu Tong, he raised his eyebrows. He'd seen plenty of provincial champions, but perfect scores were rare. As a newcomer, she ought to show her skills—perhaps she could win the respect of the room.

"Wu Tong, right? Put your bag somewhere and come up to solve the problem on the blackboard."

Today was the last day of training, and the problem he’d chosen was a formidable final challenge. If Wu Tong could solve it on the spot, he'd have no complaints about her absence from the previous training.

If students have real ability, he didn’t mind being ignored. Old Zhang’s tolerance was broad.

There weren’t many students in the training—apart from the provincial team members and ordinary trainees, everyone worked freely. There was an empty seat in the first row by the aisle.

The blackboard problem was succinctly written; Wu Tong read it as she set her bag aside, finishing it as she walked to the front.

Let n be a positive integer, S = {x, y, z, ...}. Find the minimum number of planes containing S but not (0, 0, 0) in their union.

At first glance, Wu Tong recognized the problem’s difficulty—it was quite high, comparable to the classic sixth question from the 1988 IMO she had once solved.

Instantly entering a deep-learning state, Wu Tong’s mind rapidly extracted the key points, her thoughts whirling in an invisible storm. Suddenly, inspiration struck—she could introduce the Lagrange mean value theorem. Picking up a piece of chalk, she began writing on the empty space of the blackboard.

Solution: Let the degree of the polynomial p(x) be N. Define the difference operator Δ such that...

Let I be the identity operator. By the Lagrange mean value theorem, we have:

Δp(x) = p(x + 1) – p(x)...

...

Taking x = y = z = 0, we get f(0, 0, 0) = ...

This contradicts f(0, 0, 0) ≠ 0, thus m ≥ 3n, with equality as shown previously...

After fluently writing nearly a full blackboard’s worth of proof, Wu Tong explained softly, "This is a fairly straightforward solution. There’s also a more complex approach, but the blackboard isn’t large enough, so I won’t elaborate here."

A gentle laugh...

The other provincial team members felt as if they’d been struck by a multiplying blow. The goddess on stage—could she spare a thought for the hearts of those below? Was this the gap between them and the perfect-score champion?

They had struggled fruitlessly with the problem, still unable to find a way in, while she, after a casual reading, wrote out a full blackboard of proof that they found tough even to comprehend, and then told them she had another solution, but the blackboard was too small to fit it?

How were they supposed to cope with this!