A fat man who was reborn from the dead? He transforms into a perfect male god in a week.

Question 3 (45 points)

Let f: ℕ→ℕ be a strictly increasing function, satisfying f(1)=1 and for any positive integer n, f(f(n))=f(n)+n. Find the expression for f(n).

Jiang Chen glanced at it.

"Function equation problems, a classic problem type."

He wrote without much thought:

「解：先求前几项。由f(1)=1，代入n=1得f(f(1))=f(1)+1，即f(1)=1+1=2，矛盾？因为f(1)=1，所以f(f(1))=f(1)=1，右边f(1)+1=2，1=2矛盾？」

Jiang Chen was stunned.

"Is there an error in the question?"

He read the question again: "f:ℕ→ℕ is a strictly increasing function, satisfying f(1)=1 and for any positive integer n, f(f(n))=f(n)+n."

代入n=1：f(f(1))=f(1)+1 → f(1)=1+1=2，但f(1)=1，矛盾。

"this……"

Jiang Chen frowned.

Three seconds later, he understood.

「哦，f(1)=1，但f(f(1))=f(1)吗？不，f(1)=1，所以f(f(1))=f(1)=1，右边应该是1+1=2，确实矛盾。」

"Unless... the ℕ in the problem is a positive integer, but might include 0? Or f(1)=1 is an initial condition, but the function equation holds for n≥2?"

He continued to think about it.

"Let's assume the question is correct. Then the contradiction means my reasoning is flawed. f(f(n)) = f(n) + n. When n = 1, f(f(1)) = f(1) + 1 = 2. Therefore, f(f(1)) = 2."

「而f(1)=1，所以f(1)=1，那幺f(f(1))=f(1)=1，但需要等于2，所以必须f(1)≠1？可题目明确说了f(1)=1。」

Jiang Chen felt his mind was a bit of a mess.

"Is there... a problem with this question?"

He decided to skip ahead and look at question four.

Question 4 (45 points)

Let p be an odd prime number, and a₁, a₂, ..., a_p be integers. Prove that there exist integers k (1 ≤ k ≤ p) such that ∑_{i=1}^p (a_{i+k} - a_i)² is divisible by p². Here, the subscript modulo p is understood (i.e., a_{p+1} = a_1, etc.).

Jiang Chen's eyes lit up after reading it.

"A combinatorics problem in number theory, that's interesting."

"∑(a_{i+k}-a_i)² = ∑a_{i+k}² - 2∑a_{i+k}a_i + ∑a_i² = 2∑a_i² - 2∑a_{i+k}a_i, because ∑a_{i+k}²=∑a_i²."

"So we need to prove that there exists a k such that 2∑a_i² - 2∑a_{i+k}a_i ≡ 0 mod p², which is equivalent to ∑a_i² ≡ ∑a_{i+k}a_i mod p²/2? No, it's mod p²."

"That is, prove that there exists a k such that ∑a_{i+k}a_i ≡ ∑a_i² mod p²."

"Let S_k = ∑a_i a_{i+k}, we want to prove that there exists k such that S_k ≡ S_0 mod p², where S_0=∑a_i²."

"This is equivalent to proving that there exists k such that S_k - S_0 ≡ 0 mod p²."

"Consider the sum of S_k for all k? Or use a polynomial method..."

Jiang Chen's mind raced.

Ten seconds later, he had an idea.

"Use Fourier transform (discrete Fourier transform)."

"Let A(x) = ∑a_i x^i (polynomial), then S_k is the coefficient of the x^k term in A(x)A(x^{-1})? No, it's circular convolution."

"In fact, S_k = ∑a_i a_{i+k} is the autocorrelation function of the sequence {a_i}."

"We need to prove that there exists a k such that S_k ≡ S_0 mod p². If all S_k ≡ S_0 mod p² are false, then all S_k - S_0 ≡ 0 mod p² are false..."

"Proof by contradiction: Suppose that for all k, S_k ≢ S_0 mod p², then S_k - S_0 ≡ r_k mod p², where r_k is a number between 1 and p²-1."

"Consider ∑_{k=1}^p (S_k - S_0) = ∑_k ∑_i a_i(a_{i+k}-a_i) = ..."

Jiang Chen quickly calculated on the draft paper.

Two minutes later, he found the key equation.

"Got it! ∑_{k=1}^p S_k = p∑a_i², so ∑_{k=1}^p (S_k - S_0) = p∑a_i² - p∑a_i² = 0."

"But if none of S_k - S_0 are divisible by p², their sum modulo p² cannot be 0, which is a contradiction."

Therefore, there exists a k such that p² divides S_k - S_0.

"For rigor: Let b_i = a_i mod p. Consider the sequence {b_i} modulo p. Using a similar proof, we can find that there exists k such that ∑b_i b_{i+k} ≡ ∑b_i² mod p. Then we can elevate this to modulo p²..."

Four minutes, problem four solved.

Jiang Chen glanced at the time: 9:50.

After finishing the fourth question, there is still the third question left.

He turned back to look at the third question.

"The function equation is f(f(n)) = f(n) + n, f is strictly increasing, and f(1) = 1."

"Substituting n=1 gives f(1)=2, which is a contradiction. So either the question is wrong, or my understanding is wrong."

Jiang Chen thought for a moment, and then a sudden inspiration struck him.

"Wait, f is ℕ→ℕ, where ℕ usually refers to a positive integer, but sometimes it also includes 0. If it includes 0, then f(0) may exist."

"Let f(0) = c, then f(f(0)) = f(c) = f(0) + 0 = c, so f(c) = c."

"Since f is strictly increasing, f(0) = c, f(1) = 1. If c < 1, then f(0) < f(1), but 0 < 1, so it is possible. c must be an integer, so c = 0."

「那幺f(0)=0，f(1)=1，代入n=1：f(f(1))=f(1)+1=2，所以f(1)=2？但f(1)=1，矛盾。」

"It's still a contradiction."

Jiang Chen frowned.

"Unless... f(1) is not 1? But the question clearly states that f(1) = 1."

He decided to try a different approach.

「假设f(1)=1，那幺f(f(1))=f(1)=1，但方程要求f(f(1))=f(1)+1=2，矛盾。」

"So there must be a typographical error in the question? Or is it that the set of natural numbers includes 0, and f(0) = 1?"

「设f(0)=1，那幺f(1)是多少？由严格递增，f(1)＞f(0)=1。」

「代入n=0：f(f(0))=f(1)=f(0)+0=1，所以f(1)=1，但f(1)＞1，矛盾。」

"That won't work either."

Jiang Chen felt that this question was a dead end.

He glanced at the time: 9:52.

"Forget it, let's solve it using the standard method first. Assume that f(1)=1 holds true and ignore the contradiction of n=1."

"Let g(n) = f(n) - n, then the equation becomes f(f(n)) = f(n) + n → f(n + g(n)) = n + g(n) + g(n) = n + 2g(n)."

"But f(n+g(n)) is also equal to f(f(n))=f(n)+n=n+2g(n), which is self-consistent."

"Is g(n) strictly increasing, non-negative, and increasing? Not necessarily."

「尝试求前几项：设f(1)=1，则f(f(1))=f(1)=1，但方程要求等于2，所以矛盾。跳过n=1。」

「从n=2开始：f(f(2))=f(2)+2，设f(2)=a，则f(a)=a+2。」

"Since it is strictly increasing, f(1) = 1 < f(2) = a, so a ≥ 2."

"If f(3) = b, then f(b) = b + 3, and b > a ≥ 2."

"Continue the derivation..."

Jiang Chen wrote down a series of equations on the draft paper.

Two minutes later, he guessed the answer.

"f(n) = φn + ψ, where φ is the golden ratio (1+√5)/2? No, it must be an integer function."

"In fact, the classic solution is f(n) = ⌊φn⌋ or a similar form, but it needs to be verified."

He quickly verified it.

「设φ=(1+√5)/2≈1.618，则φ²=φ+1。」

"If f(n) = ⌊φn⌋, then f(f(n)) = ⌊φ⌊φn⌋⌋ ≈ φ²n = φn + n = f(n) + n, which is approximately true."

"But it requires number theory properties to be strictly true, which is the classic Beatty sequence."

Jiang Chen decided to write the answer directly:

Solution: f(n) = ⌊φn⌋, where φ = (1 + √5)/2, and holds true when n ≥ 1. Verification: By Beatty's theorem, ⌊φ⌊φn⌋⌋ = ⌊φ²n⌋ = ⌊(φ + 1)n⌋ = ⌊φn + n⌋ = ⌊φn⌋ + n = f(n) + n, which holds true when φn is not an integer. Since φ is an irrational number, φn is never an integer, therefore the equation holds true.

"But we need to verify that it is strictly increasing: obviously."

"f(1)=⌊φ⌋=1, which satisfies the condition."

After finishing writing, Jiang Chen breathed a sigh of relief.

"This question is fucking tricky."

I glanced at the time: 9:55.

Four questions, done in fifteen minutes.

Is it too early to hand in the paper now?

And what if the examiners don't find it impressive enough?

No, I still have to show off my skills.

Jiang Chen did not hand in his paper immediately, but instead turned his attention to the last and most challenging question.

"There are many ways to solve this problem; I only used the most common one just now!"

"There's a second option..."

"The third type..."

Five minutes later.

10:00 sharp.

He raised his hand.

"Teacher, I'm handing in my paper."

……

The examination room was completely silent.

Everyone looked up at Jiang Chen, their eyes filled with complex emotions.

There was surprise, there was doubt, but more than anything... disdain.

"Handing in your paper again?"

"What can he do in twenty minutes?"

"I guess I won't be able to do any of them, so I might as well give up."

"What a waste of a spot..."

Qin Mo sat in the first row, glanced up at Jiang Chen, and looked indifferent.

He just finished the first question and is now working on the second.

Twenty minutes later, Jiang Chen handed in his paper?

He couldn't think of any other possibility but to give up.

"It seems I overestimated him before."

Qin Mo sneered inwardly, "So what if I got a perfect score on the monthly exam? Competitions are a completely different world."

He lowered his head and continued working on the problems.

……

Teacher Liu Yue took Jiang Chen's test paper, her hands trembling.

She glanced at it quickly.

The first question was answered correctly.

The second question was answered correctly, and the solution was better than the standard answer.

For the third question, although I had some doubts at first, I eventually found the correct solution.

The fourth question was answered correctly. It used Fourier transform and modular arithmetic, and the approach was brilliant.

Furthermore, there is a second solution, a third solution, and so on...

And... four questions, twenty minutes.

"Classmate... classmate," Liu Yue's voice trembled, "you... you're really handing in your paper?"

"Okay." Jiang Chen nodded. "Can we go now?"

"Yes...yes."

Jiang Chen turned and left.

Liu Yue watched his retreating figure, then looked at the exam paper, feeling as if she were dreaming.

……

Outside the examination hall.

When Zhou Dapeng saw Jiang Chen come out, his eyes almost popped out of their sockets.

"Jiang Chen? You... why are you out again?"

"It's done," Jiang Chen said.

"Twenty minutes? Finished?" Zhou Dapeng's voice changed. "Jiang Chen, this is the second round! Four major questions, 180 points!"

"I know," Jiang Chen said innocently. "The questions weren't difficult, I just finished them a little too quickly."

Zhou Dapeng opened his mouth, but couldn't say anything for a long time.

Nearby, Mr. Wang from Jiangcheng No. 1 Middle School and Mr. Li from Jiangcheng No. 2 Middle School exchanged a glance and smiled.

"Teacher Zhou, your students... they're quite unique," Teacher Wang said sarcastically.

"Twenty minutes to finish the paper, that speed... you probably didn't even finish reading the questions, right?" Teacher Li mocked.

Zhou Dapeng looked grim, but did not refute.

He was also feeling uneasy.

Jiang Chen... could it be that he really didn't finish?

"Jiang Chen," Zhou Dapeng pulled Jiang Chen aside and lowered his voice, "Tell the teacher the truth, did you really finish? Or... did you give up because you didn't know how?"

"It's finished," Jiang Chen said. "Teacher Zhou, don't you believe me?"

"It's not that I don't believe it..." Zhou Dapeng smiled wryly, "it's just that this speed... is too exaggerated."

"Don't worry," Jiang Chen patted Zhou Dapeng on the shoulder, "you'll know when the results come out."

After saying that, he turned and went back to the hotel.

Zhou Dapeng stood there, his feelings complicated.

He both hoped that what Jiang Chen said was true and found it too unbelievable.

……

Time passed second by second.

Inside the examination hall, the candidates were still working hard.

After Qin Mo finished the second question, he glanced at the time: 10:50.

70 minutes have passed.

He wiped the sweat from his forehead.

The difficulty of this second test... far exceeded his expectations.

He still has no ideas for questions three and four, especially.

"Never mind, let's make sure the first two questions are all correct." Qin Mo took a deep breath and began to check.

Meanwhile, Jiang Chen was already in his hotel room, leisurely scrolling through his phone.

The system notification sounded intermittently.

[Ding! You have received +20 Amazing Points!]

[Ding! You have received +30 Amazing Points!]

The proctor alone contributed a lot of amazing points to his performance.

"This is just the beginning," Jiang Chen said with a smirk. "When the results come out... that'll be truly spectacular."

……