This top-tier academic genius

Lin Yan had just finished writing the question and immediately came up with the answer, a result that Sun Yutong hadn't expected.

Her first thought was that Lin Yan had done this question before and remembered the answer, since the question was very short and easy to remember if you had done it before.

So she turned to another page and found a proof problem for which she couldn't remember the answer.

[Proof: The function f(x) = x + 1/x is monotonically increasing on the interval (1, +∞).]

This is an original question from the chapter on the monotonicity of functions. It's not particularly difficult, but the proof in the textbook uses the standard definition method, which is a bit long. If you don't really understand how to do it, you won't remember it.

Sun Yutong pushed the paper and pen aside a bit, "Come on, let's prove it."

She didn't believe that this somewhat pretentious junior could remember the proof process in the book.

Lin Yan didn't start writing immediately, but chuckled and said, "Senior, I'm really not lying to you. Why don't you give me something a little harder?"

"Prove it first."

"Alright."

Helpless, Lin Yan picked up his pen and began to write.

Since f'(x) = 1 - 1/x², x > 1 ⇒ x² > 1 ⇒ 1/x² < 1 ⇒ f'(x) > 0, therefore it is monotonically increasing.

It was completed in just a few simple steps, and it was completely different from the answer in the book.

Looking at the simple proof on the paper, Sun Yutong said, "You used the derivative method!"

"Yes, the definition method is too complicated; this is simpler. Of course, proving it using the properties of the check function, or the method of complex monotonicity analysis, would also be quite fast."

Lin Yan spoke casually, but Sun Yutong was a little unsettled by his words.

From this sentence, she learned an important piece of information: the junior in front of her seemed to be able to prove this problem in four different ways.

Four kinds! Even I, a math major, could only think of three at first, but a mediocre student who scored 17 points knew four kinds.

Is this really a poor student?

Sun Yutong adjusted her thick glasses and looked at this somewhat attractive junior with a suspicious gaze.

Was what he just said true? That he has already mastered high school math?

But his mother's report card showed that he only got 17 points in math.

"Actually, you're very good at math?"

"I wouldn't say it's great, but it's alright."

"Then why did you only get 17 points on your math test last time?"

"17 points? Oh, you mean your grades at the end of last semester? Actually, I only recently figured it out, and my math grades suddenly improved. Even my parents don't know."

Sun Yutong: "..."

Why does he seem so pretentious?

Sun Yutong's temper flared up, and we had to curb his arrogance, otherwise he might not be able to continue tutoring.

She remembered that the other person seemed to be looking at "Towards the IMO: A Collection of Mathematical Olympiad Problems" when they came in, so she simply picked out another problem from it and wrote it down on a piece of scrap paper.

Let a, b, c > 0, and a + b + c = 3. Prove that √(a/(b+2)) + √(b/(c+2)) + √(c/(a+2)) ≤ √3.

This question is very difficult. Sun Yutong thinks that if it were her senior year of high school, she would probably have been stuck for more than ten minutes.

"Come!"

I'd like to see if you're up to the task.

Lin Yan glanced at the problem, his mind racing: This kind of cyclic inequality with a square root can't be calculated by brute force. First, he had to use Cauchy to remove the square root, then use the mean to reduce the denominator, and combine that with the given conditions to reduce it to √3 in one step. Hmm, not too difficult.

He only thought for about ten seconds before putting pen to paper.

By Cauchy's inequality (Σ√(a/(b+2)))²≤(Σ a)(Σ 1/(b+2))......, therefore ∑√(a/(b+2))≤√3, and the equality holds if and only if a = b = c = 1.

Watching Lin Yan write down the solution almost without hesitation, without much thought, Sun Yutong asked incredulously, "Have you done this problem before?"

"No, I've only just started reading this book."

Sun Yutong took a deep breath and glanced at some books on the corner of the table, including "Linear Algebra and Its Applications", "Combinatorial Mathematics", and "Higher Algebra".

At that moment, she finally believed it.

The junior student in front of her was far better at math than she had imagined. His math skills were three or four stories high, and he was even better than she had been in high school.

How can there be such a person? He could only score 17 points at the end of last semester, but now he can solve math olympiad problems. Is he really a genius?

She had to tear off the "stupid" label she had attached to Lin Yan and replace it with "genius".

Just then, Wang Lanyin walked in carrying a plate of fruit.

"You're learning, here, have some fruit."

"Thank you, Aunt Wang."

"You can eat fruit and study at the same time, I won't disturb you."

After Wang Lanyin left, Sun Yutong readjusted her tone and said, "With your math skills, you're perfectly capable of participating in COM. It's a pity the preliminary registration period has already passed."

CMO is the National Mathematical Olympiad, and the preliminary round is the National High School Mathematics League.

"Did you participate in CMO before, senior?"

"Yeah, I only went to the preliminary rounds, and I came in eighth in the province, but I didn't make it to the CMO."

Sun Yutong excelled in mathematics during high school, but unfortunately, she ranked eighth in the provincial preliminary round and failed to qualify for the CMO.

"Oh, that's impressive. I've already registered for the preliminary rounds; my math teacher helped me with a special application."

"Hmm? Who is your math teacher?"

Sun Yutong knew that special applications were not easy to obtain; they required intervention from school leaders.

The school actually made a special application for this junior, which shows that the teachers also discovered this gem.

"Teacher Qin Zhengyan is also the one leading the team in this competition."

"It's him. He's really good at math. I heard he graduated from the Mathematics Department of Nanjing University. I was Professor Cai back then."

The two found common ground, which immediately brought them closer, and they started chatting.

Wang Lanyin, standing outside the door, glanced at it and nodded in satisfaction.

After a while, Sun Yutong was silent for a few seconds before saying, "You just said you wanted to understand the knowledge framework of university mathematics, so I'll explain it to you in detail."

"Actually, there are only two lines in university mathematics."

"One approach is analysis. Starting with limits and calculus, we gradually move towards more refined and abstract concepts, eventually reaching real analysis, complex analysis, and functional analysis to study what functions and infinity really are."

"The other path is algebra. Starting with systems of equations and matrices, we move up to linear spaces, groups, rings, and fields, studying structure and symmetry."

"Add probability and statistics to handle randomness and data."

"There are also branches like differential equations, geometry, and topology..."

Sun Yutong, as expected of a mathematics graduate, organized her university-level mathematics knowledge in an exceptionally logical and organized manner, explaining it very clearly and thoroughly, and analyzing each knowledge point in detail.

With Lin Yan's current logical abilities, he can understand it completely.

Suddenly, the framework of university-level mathematics knowledge appeared in his mind.

"So, if you want to learn university-level math courses in advance, I suggest you start with calculus. It's the universal language for all subsequent math and major courses. Whether you study linear algebra, probability, statistics, mechanics, circuits, economics, computer science... almost everything eventually boils down to rates of change, cumulative quantities, limits, and approximations, and these are what calculus governs."

"I can recommend a few books to you: *The Princeton Calculus Textbook*, *Thomas Calculus*, *Advanced Mathematics*, and *Stuart Calculus*. You can start with the most basic *The Princeton Calculus Textbook*..."

Sun Yutong offered many suggestions.

Lin Yan nodded. Calculus, I have to learn it!