After taking the advice, I became a top student.

The second question was a geometry problem about the properties of a cyclic quadrilateral.

Chen Mo did not use analytic geometry to establish the coordinate system; with his current abilities, he could see the essence of the problem almost at a glance.

With the combined power of mathematical intuition and insight, he provided a geometric proof in just five minutes.

Then comes the third question.

Chen Mo turned to the last page of the exam paper and frowned slightly when he saw the questions.

【Question 3】Let n be an integer greater than 1. Define

(1) Proof: When n is odd, F(n) = 2^−φ(n).

(2) Let p be an odd prime number. Prove that:

(3) Using the above results, prove that for any positive integer n>1, F(n)=√n/2^φ(n)⋅μ(n)

Where μ(n) is the Möbius function, and we discuss the relationship between the values ​​of μ(n) being 0 and ±1 and F(n).

Sure enough, the questions in the first round were easy, but the questions in the second round were bound to be more difficult, especially the third question, which was almost on a completely different level from the questions we had done before.

This number theory problem tests extremely advanced knowledge; even a typical undergraduate mathematics student might not be able to solve it.

Not far ahead of Chen Mo, Lin Zhiyuan rubbed his hands excitedly. As a contestant who had won the CMO gold medal last year, he felt that the questions he had done earlier were not enough.

Because he can do these problems, and so can others.

But upon seeing this question, he finally smiled happily. He knew that the main event was finally here, and the provincial competition would use this question to select the spots for the winter camp.

After reading the problem several times, Lin Zhiyuan finally began to derive the solution on the draft paper.

The first question is relatively simple. We only need to use the complex root method. Consider the root ω_k=e^{πik/n} of x^{2n}-1=0, then sin(kπ/2n)=∣ωk−1∣/2. Then, by decomposing x^{2n}-1=(x^n-1)(x^n+1), we can extract the factor corresponding to gcd(k,n)=1 and obtain F(n)=2^−φ(n).

After spending more than twenty minutes completing the first question, Lin Zhiyuan felt refreshed and straightened his back. He stood there smugly for a while before turning to the second sub-question.

This time, he thought for a while and then started writing. Using x^p-1=(x-1)(x^{p-1}+……+ 1), and letting x=1, he got p=∏_{k=1}^{p−1}∣1−e^{2πik/p}∣. He only needed to change this expression to sine to get the result of the second question.

After finishing the problem, Lin Zhiyuan had a few beads of sweat on his forehead, and even he felt a little strained.

"It's obvious that this year's question was indeed set by a professor from Peking University!"

Lin Zhiyuan thought to himself, "This difficulty is completely different from what I did last year. If I hadn't been tempered by the CMO last year and had studied for another year, I probably would have stumbled on this problem too."

He subconsciously glanced back, but unfortunately, due to the angle, he couldn't see Chen Mo and Qiu Mingyuan behind him. He also couldn't go too far, otherwise the proctor would think he was cheating, and then there would be no reason to argue.

But he knew without even looking that the two guys must also be stumped by this question.

Without wasting any time, he looked at the third sub-question.

After some deduction, Lin Zhiyuan quickly had an inspiration.

"We only need to use the inclusion-exclusion principle to write F(n) in the form F(n) = ∏_d∣nG(d)^μ(d), where G(d) is some known product, and then substitute it into the calculation..."

Muttering to himself, Lin Zhiyuan's eyes grew brighter and brighter, and his hands moved swiftly as he wrote line after line of derivation process on the draft paper.

Splash splash splash...

Lin Zhiyuan had written several sheets of draft paper but still couldn't arrive at the result. The proof process required a lot of algebraic operations and classification discussions.

Lin Zhiyuan's forehead was already beaded with sweat, but he was completely immersed in it, doing it with great enjoyment.

At that moment, he had no thought of winning, no other thoughts whatsoever, only the problem in front of him!

In the same examination room, the other students looked in Lin Zhiyuan's direction, secretly amazed, "No wonder he is a top student, his thoughts flow so freely."

They looked at the question but their minds went completely blank; they had absolutely no idea what to do.

I feel so hopeless.

On the other side, Chen Mo was also writing furiously, deriving the proof process for the third sub-question.

If this were a school exam, he might have already given up on the derivation, because he felt that the difficulty of the question exceeded the knowledge he had learned, which is why he had such a complicated derivation process.

Mathematics shouldn't be this complicated.

Based on Chen Mo's understanding of mathematics over the past few days, he felt that mathematics should be a simple and elegant thing.

If a conclusion is too complicated, it must be because the person who arrived at that conclusion did not find the correct method.

His intuition kept telling him to stop, that this shouldn't be happening.

But this was a provincial competition, so he decided to try deriving it.

After deriving for a while, Chen Mo suddenly stopped writing. During the derivation, he suddenly realized that if F(n) is written as F(n)=1/2φ(n)∏_{1≤k≤ngcd(k，n)=1}∣1−e^{πik/n}∣, then the eigensum can be introduced.

Let χ be a Dirichlet eigenvalue modulo n, and define the Gaussian sum G(χ) = ∑_{k=1}^nχ(k)e^{πik/n}.

Then, through a series of derivations, we can obtain an equality: for any positive integer m, χ_m is a certain characteristic modulo m, then ∑^mχ_m(k)e^{πik/m}=√m⋅ (a certain factor with a value of ±1 or 0).

Finally, by verifying the case of smaller n, we can find that this summation is actually equal to μ(n)⋅√n.

This way, the proof of the last sub-question can be completed without having to perform a lot of tedious algebraic calculations and classification discussions.

I checked the time; there was still an hour and a half left.

Remembering Teacher Zhou's instructions, Chen Mo buried his head again and carefully checked the three questions once more.

Ten minutes later, Chen Mo raised his head again.

No problem.

So he packed up his stationery, got up and walked out of the classroom.

Under other circumstances, he might not have been so hasty; he hadn't handed in his paper early that morning either. But now it was different; the effect of his heightened mental clarity was still there.

This was a one-time bonus, and with more than an hour left, it would be a waste to just sit in the exam room doing nothing. He decided to go back and study first.

"Hey, classmate..."

The proctor was about to ask some questions, but when he saw the exam paper filled with writing, he chose to keep quiet.

The other candidates in the same examination room stared wide-eyed.

"I was still working on the second big question when he handed in his paper?"

"Was the question too easy this time, or was my approach wrong?"

Many students felt flustered and confused, making it difficult for them to concentrate on the questions.

"Trying to look cool in the provincial competition?"

"This is so childish!"

Some students also thought that Chen Mo was just pretending. They were working on the questions, so how could they not know how difficult they were? They said that even if it took two and a half hours, let alone an hour, they might not be able to finish.

Lin Zhiyuan and Qiu Mingyuan both looked up, their eyes filled with confusion.

After these two days of interaction, they realized that Chen Mo was not a high-profile person, and they also recognized Chen Mo's strength, believing that it was impossible for Chen Mo not to be able to solve the third major problem.

So, that's all he did?

How could it be so fast?

With the amount of calculation required for this question, it would take at least an hour to complete. How did this guy hand in his paper early?

Did he give up?

Lin Zhiyuan kept guessing, but he still shook his head. He didn't think Chen Mo would give up.

Then there's only one possibility: Chen Mo really did finish.

Lin Zhiyuan felt a sense of dread, as if facing a formidable enemy.

He refocused his attention and returned to the draft paper.

The next second, Lin Zhiyuan almost lost his composure.

"Damn, where was I in my calculations?"

"Chen Mo has misled me!"