Due to the limited space on the whiteboard, the final steps presented were as concise as possible, which inadvertently meant that the thought process was overlooked.
But anyone who studies Olympiad mathematics knows that the solution steps are merely the external presentation; what truly matters is the ability to think.
Tan Jiaxu also wanted to hear how she solved the problem—how she approached it, how she came up with the idea of nesting one lemma within another to prove the conclusion.
Jiang Fuyue: “Does everyone want to hear it?”
“Yes—”
“Alright.”
Tan Jiaxu returned to his seat and looked up at the podium, his eyes showing, for the first time, a genuine recognition of Jiang Fuyue.
Zou Hao clicked his tongue in amazement.
Although Jiaxu usually hung out with them and they had a good relationship, geniuses always carried a certain pride. He never deliberately displayed it, but Zou Hao could sense it.
Yet now, he was looking at Jiang Fuyue with such an expression…
This could only mean that the only thing that can make a genius bow is another genius of an even higher caliber.
At this moment, Jiang Fuyue had already picked up a marker and stood before the whiteboard—
“When faced with this problem, many people’s first instinct is to try to find the relationship between vertices and edges.”
A large portion of the students below nodded vigorously.
That’s exactly how they had approached it.
“We know that in a connected graph, if it has V vertices and only V-1 edges, a cycle is impossible—otherwise, the graph would not be connected.”
“But once we have V edges, a cycle must appear. And for every additional edge beyond that, a new cycle is created. Suppose we have V+1 edges and only one cycle in the graph. If we remove one edge from this cycle, then the graph will no longer have a cycle. However, the graph still has V vertices, meaning that another cycle must exist. Therefore, our assumption does not hold, and there must be at least two cycles in the graph…”
Tan Jiaxu’s gaze sharpened.
Zou Hao originally wasn’t planning to listen, but the girl’s voice was both clear and cool, like a mountain spring flowing through a deep ravine, crisp and soothing.
He found himself unconsciously drawn in.
Jiang Fuyue: “…At this point, we need to introduce a new definition called the surplus, which is defined as the number of edges (E) minus (V-1), resulting in E – V + 1. This quantifies the excess of edges relative to the number of vertices, which in turn determines the approximate number of cycles in a graph.”
Tan Jiaxu’s eyes lit up.
Yes, this was it!
He had been thinking about how to circumvent this proof point, while Jiang Fuyue had directly introduced a new definition.
He avoided risk, while she confronted it head-on.
This was the fundamental difference in their thought processes, which ultimately led to their respective outcomes.
Thus, he lost—and there was a clear reason for it.
On the podium, Jiang Fuyue stood tall, her figure elegant and poised. But at this moment, almost everyone unconsciously ignored her appearance, seeing instead her radiant intellect, her impeccable logic, her rigorous approach, and her deep respect and reverence for knowledge.
“…Going further, as the number of edges in a graph increases, the number of cycles will also increase, and these cycles will begin to intersect, with their lengths progressively decreasing. Returning to the problem statement, which mentions ‘a finite number of counterexamples,’ it’s almost a direct hint that this approach won’t work. So, instead, we can take a different angle: proving a lemma that leads to a contradiction, allowing us to employ proof by contradiction…”
Some people’s souls truly shine.
So brilliantly that one cannot look away.
Standing beside her, Ling Xuan slightly tilted his head—just enough so that the students below couldn’t see the dazed admiration in his eyes.
His heartbeat pounded, one beat after another, an overwhelming sensation threatening to burst from his chest. It filled him with both fear and excitement.
He knew that some things could not be suppressed.
For example—liking someone.
But he didn’t even have the courage to say it out loud.
Jiang Fuyue was too dazzling.
He once thought he was bright enough, only to realize now that Jiang Fuyue was the light itself.
Her brilliance made him shrink back. Her radiance outshone everyone, including himself.
He wanted her, yet he didn’t dare. That feeling of longing for something unreachable turned into a kind of torment.
Ling Xuan lowered his gaze sharply, forcing himself to endure. He warned himself to be patient.
At this moment, through the slightly ajar back door of the classroom, another person was also watching the girl on the podium, mesmerized by her brilliance.
Jiang Fuyue: “…But the problem states that there exist only a finite number of counterexamples, which suggests that this assumption is false. Therefore, instead of directly proving the statement, we introduce a lemma and use it to derive a contradiction, thereby validating our proof…”
Her crisp, cool voice carried an undeniable rationality, underpinned by powerful logic.
The man listening outside watched, a faint smile slowly surfacing in his eyes.
“Professor Xie?” Zhao Tiejun called softly.
Xie Dingyuan snapped back to reality but didn’t withdraw his gaze.
Following his line of sight, Zhao Tiejun quickly understood and chuckled. “This is the Olympiad math class. They’re currently in intensive training for the national competition.”
Xie Dingyuan gave a soft “Mm.”
Seeing his interest, Director Zhao eagerly continued, “The girl speaking on the podium is the top student in the senior class. She’s also the gold medalist and special prize winner in both the IPhO and IOI competitions this year, with perfect scores in both subjects. A top candidate for B University and Q University…”
As he spoke, a sense of pride filled his chest, making him stand up a little straighter.
“Is that so?” Xie Dingyuan responded casually, giving him space to elaborate.
Sure enough, Zhao Tiejun enthusiastically went on—
“This kid is truly exceptional. Not a single teacher has anything but praise for her intelligence.”
Xie Dingyuan: Mm, her brain does seem quite sharp—whether it’s for academics or for roasting people.
“She gets along well with her classmates too. She often takes the time to explain problems and go over key points for them. Everyone calls her ‘Sister Yue.'”
Xie Dingyuan: Sister*? She’s just a little girl.
* they use the older sister word, not little sister
“Her athletic abilities are also outstanding. She’s trained in high jump and shot put, and after school, she often plays basketball with her classmates.”
Xie Dingyuan: Classmates? Are they male or female?
“Not long ago, she even had a research paper published in the main issue of PRL. The topic was… uh, something related to bacterial movement. I can’t quite remember.”
He didn’t need to say it. Xie Dingyuan already knew.
He might have been one of the first to see the draft of that paper.
“Overall, Jiang Fuyue is an all-around talent—academically, physically, and artistically outstanding. She’s the pride of our school!”
Xie Dingyuan nodded slightly, amusement flashing in his eyes. But at that moment, inside the classroom, Jiang Fuyue was already saying—
“…So, the lemma is as follows: For any positive real number a and any natural number N, we must prove that for a graph with N vertices and at least (1 + a)N edges, there exists a cycle with a length smaller than aN, with only a finite number of exceptions…”
Just hearing her voice lifted his mood.
“The only downside,” Director Zhao sighed, “is that she’s a bit stubborn and headstrong.”
“Oh?” Xie Dingyuan raised an eyebrow.
“A while back, she received guaranteed admission offers from both B University and Q University, allowing her to choose any major she wanted. But she flat-out refused, leaving no room for negotiation. Her parents even went along with it…”
Director Zhao looked regretful.
That was B University and Q University!
It was said that Xu Jing, Meng Zhijian, and Yu Wenzhou were all losing their minds over her refusal.
And Jiang Fuyue?
She remained steadfast—no one could persuade her.
Xie Dingyuan was hearing this for the first time. “Why did she refuse?”
Director Zhao: “Apparently, she wants to take the national college entrance exam instead.”
Hearing that, a hint of pride resurfaced on his face.
How many students in other schools had the guts to turn down B University and Q University?
None.
But in their school?
They had one.
Actually, they had two—Ling Xuan had rejected them too. What a duo.
This year’s batch was truly something else!
Xie Dingyuan withdrew his gaze, suppressing the small smile at his lips. “Let’s go, Director Zhao.”
“Ah, sure, sure. Is Zhong Ziang’s leg recovering well?”
“Yes.”
“That’s good, that’s good. Actually, you didn’t need to come in person—a phone call would’ve been enough. Or you could’ve just had Student Zhong bring a written leave note signed by the parents when returning to school…”
Anything would’ve been better than showing up in person, right?
Honestly, Zhao Tiejun didn’t even know why this big shot suddenly decided to show up.
When the school assigned him to go receive him, he was completely bewildered.
Xie Dingyuan stepped forward, while behind him, the girl’s voice gradually faded—
“If a graph has N vertices and at least (1 + a)N edges, and all the cycles in the graph are of different lengths, then when N is sufficiently large, we need to construct a contradiction to prove the assumption is false. So we can proceed like this: first, find the shortest cycle in the graph…”
When Xu Jing came back from a teaching seminar, he suddenly noticed that the atmosphere in the class had become noticeably more harmonious.
He asked around to find out why, but the students were all tight-lipped—every single one of them.
“?” His curiosity was killing him.
But he didn’t dwell on it too much—after all, harmony is a good thing.
That peaceful vibe lasted all the way until the end of the two-week semi-closed training camp.
And soon, the National Math Olympiad arrived…
No particular reason—just wanted to see my wife.
Zhong Ziang: Using me as a cover? Have some shame, will you?
