A genius? I just love studying.

Chapter 119 A sudden smile of relief

At 8 o'clock, the test papers were distributed.

The test questions were not much different from yesterday's, still consisting of three questions.

Once he got into problem-solving mode, Li Zehan instantly focused all his attention on the questions, as if he were a completely different person.

The question is quite easy to understand. It means that there are 2025 walnuts that have been shuffled and placed on a circle, and the number of each walnut is known.

Then, in the next 2025 operations, each operation will operate on the two walnuts to the left and right of the k-th walnut. It is necessary to prove that there must exist at some point that the walnut numbers on both sides of the k-th walnut are such that one is greater than k and the other is less than k.

Upon seeing this question, Li Zehan already had an idea in mind.

As we learned in junior high school, when faced with proofs of existence, the first thing to think of is proof by contradiction.

Assume that in these 2025 operations, the walnut numbers on both sides of k are either greater than k or smaller than k.

This relationship is rather difficult to describe, and at this point, the staining method naturally comes to mind.

This is also a common method for solving existence problems. After coloring, the quantity and properties of the points, lines, faces, angles, etc. that make up the problem can be analyzed to simplify the problem and make it more intuitive.

In this problem, we can color the kth walnut in the kth operation, for example, by coloring it yellow.

After this operation, all walnuts smaller than k will be dyed yellow, while walnuts larger than k will not be dyed. This makes it easy to distinguish between the two categories of walnuts, which are larger than k and smaller than k.

The final proof then becomes that in these 2025 operations, there must be at least one operation that swapped two walnuts of different colors.

Using proof by contradiction, we assume that each operation swaps walnuts of the same color.

"So, what kind of contradiction will this ultimately lead to?"

Li Zehan frowned and began to think.

Initially, none of the walnuts were dyed. After the process was completed, all the walnuts were dyed yellow.

There is a state transition in between.

If it were just dyeing individual walnuts, there would be no problem. But now that it involves dyeing and exchanging walnuts of the same color, the state transition is likely to fail.

Furthermore, given that the problem requires proof, it is clear that the operation of coloring plus same-color swapping will lead to coloring failure.

After a brief moment of thought, Li Zehan found the key to solving the problem.

But a crucial step is still missing.

How can you prove that staining will fail?

Li Zehan racked his brains.

Obviously, simply dyeing walnuts is not enough to prove the final conclusion.

"I see!"

After a series of mental calculations, Li Zehan had a sudden inspiration.

If dyeing walnuts isn't enough, then dye the connecting edges of adjacent walnuts as well, and you'll be all done!

If two adjacent walnuts are both yellow, then color the edge connecting the two walnuts yellow as well.

So initially, all edges are uncolored. After 2025 operations, all 2025 edges are yellow.

If the walnuts swapped each time are of the same color, then the color of the k-th walnut and the two edges adjacent to it will not change, and the swapping operation will not cause any state transition.

Only coloring the k-th walnut may cause a change in the edge color. If two adjacent walnuts are not colored, then this coloring operation will not cause a change in the edge. If both walnuts are colored, then there will be two more colored edges.

In other words, each operation either adds 0 colored edges or adds 2 colored edges. It is impossible to have an odd number of 2025 edges, which contradicts the problem statement. The proof is complete!

"I'm such a genius!"

Li Zehan chuckled inwardly. Even though he knew that the problem was only at a junior high school level and could be easily solved once he mastered the method, it didn't stop him from thinking he was amazing.

I glanced at the time; only twenty-odd minutes had passed since eight o'clock.

The overall approach to the problem was quite clear. He spent most of his time thinking about how to prove the final contradiction, but in just over twenty minutes, that was already extremely fast.

Upon thinking of this, he subconsciously looked up at Chen Hui's position.

Then he heard a whooshing sound as the contents were turned over!

"?"

"The boss has already started on the third question?"

"What the heck!"

Li Zehan didn't know how to describe his feelings at that moment.

To be honest, even though he had come to terms with the fact that he could never compare to that kind of monster, he still felt a blow when this cruel reality came to pass.

But true warriors dare to face the bleakness of life and the dripping blood!

"I, Li Zehan, am not so easily defeated!"

With renewed spirits, Li Zehan turned his attention to the second question.

The problem is concise and elegant, and the conclusion to be proved is also very clear: the difference between two terms in the sequence must be less than one factorial of n, and n must be greater than or equal to 2.

Upon seeing an inequality, even elementary school students... oh no, middle school students should know that they should use the construction method!

The construction method mainly involves introducing identities, dualities, functions, graphs, and sequences to make the problem more intuitive. If a regular term like 'n' appears in the inequality, then you should think of sequences.

For example, when proving the sum of the terms of a sequence, one should think of constructing a new sequence by transposing and subtracting terms, and then analyze the monotonicity of the new sequence.

For this problem, the form of an nth power means that we can split both sides of the inequality into the product of n identical or general expressions, and then compare their sizes.

Li Zehan's ideas came naturally. He had focused on high school math competitions for years, so his basic knowledge was incredibly solid. Almost the instant he saw the question, the solution would appear in his mind. He just needed time to transform those ideas into the final answer.

The square root sign is obviously conspicuous in inequalities, so we can consider dealing with it first. By observation, we can easily find that we can remove the square root sign by squaring or cubed each term on the left side of the expression.

这就很容易能够想到a^(2*3*……*n)-b^(2*3*……*n)这种形式，即可将全部根号去除，并且相减后能消去多余的项，得到(n+1)√(n+1)。

Therefore, we need to construct a new sequence, ai =
bi=
所以题目要求的不等式就是a2-b2，同时a(i+1)-b(i+1)=(ai)^i -(bi)^i=(ai-bi)(ai^(i-1)+ai^(i-2)bi+……+aibi^(i-2)+bi^(i-1))
The power expansion of (ai)^i -(bi)^i has a ready-made formula, and any high school student should remember this expansion. Also, because the expressions following the power expansion follow a pattern, it can be denoted as Cn.

所以有，    a3-b3=(a2-b2)c2
a4-b4=(a3-b3)c3
……

a(n+1)-b(n+1)=(an-bn)cn
将式子两边相乘，约去相同的项，就能得到a(n+1)-b(n+1)=(a2-b2)(c2*c3……cn)，所以(a2-b2)=[a(n+1)-b(n+1)]/(c2·c3……cn)。

而a(n+1)-b(n+1)=(an)^n -(bn)^n，所以a(n+1)-b(n+1)=(a2)^(n*n-1……3*2)-(b2)^(n*n-1……3*2)=(n+1)√(n+1)
Finally, let's deal with Cn.

Li Zehan knew without even thinking that this formula required scaling.

Because an > bn ≥ n√n = n^(1/n)
所以an^(n-1)+an^(n-2)bn+……+anbn^(n-2)+bn^(n-1)式子中每一项都大于等于n^((n-1)/n)，而Cn有n项，所以cn≥n*n^((n-1)/n)>n*n^((n-1)/(n+1))。

这时再回到刚才的式子，c2*c3……cn=n!*（一坨），当n>2时，n^((n-1)/(n+1))都是大于1的，所以可以只保留第n项，即c2*c3……cn=n!*n^((n-1)/(n+1))。

所以，a2-b2<1/n!*[(n+1)√(n+1)]/n^((n-1)/(n+1))。

显然，(n+1)√(n+1)]/n^((n-1)/(n+1)=((n+1)/n^(n-1))^(1/(n+1))，当n>2时，前面的式子小于2n/n^2<1，所以a2-b2<1/n!。

call!
Li Ze let out a long sigh of relief. Although he had solved the problem, he felt a bit dizzy and his head was spinning. Even for him, this problem was quite difficult. It required him to construct a unique sequence, master the power expansion formula, be familiar with the product cancellation pattern, and master scaling.

If you don't have a solid grasp of any part of the process, you'll get stuck and won't be able to continue solving the problem.

He glanced back at the clock at the back of the classroom; it was already 9:30. That meant he had spent a full hour solving the problem!
Looking up again at where Chen Hui had been, the place was now empty.

He was so engrossed in solving the problem that he didn't notice when Chen Hui had already handed in his paper.

"How on earth did he do it?"

Li Zehan inwardly groaned. He hadn't forgotten that Chen Hui had already started working on the third question while he was finishing the first one.

At the time, he thought the second question was very simple, but now it seems that even if he didn't get stuck at all and his thinking was smooth, it would still take at least half an hour to write out the whole process.

monster!

Li Zehan suddenly smiled with relief.

His competitive spirit completely subsided, and the sense of urgency finally disappeared. Everything returned to calm, and he began to look at the third question.

……

8:42
Chen Hui walked out of the exam room again. Today's questions were indeed more difficult than yesterday's, but as Teacher Yuan said, now that he is participating in the CMO, the IMO seems to have no meaning anymore.

Of course, from the perspective of simply doing the questions, it is still meaningful for Chen Hui, after all, the CMO school has promised a bonus of 50,000 yuan.

Outside the examination hall, teachers from various provinces still gathered. Chen Hui didn't understand why they didn't go back to rest but chose to wait outside Zhihua Building, just like parents waiting outside the examination hall during the college entrance examination.

This deep affection and care truly moved Chen Hui.

He didn't realize that the teachers were looking at him like he was a monster, especially when he walked out of the exam room, many of them took out their phones to check the time.

Normally, they would find a place to rest first and come over when the exam was almost over. In fact, many team leaders would not come over at all.

But what happened yesterday changed their minds; they wanted to see if that person could still create miracles today.

The result did not disappoint them.

Forty-two minutes!
Without a doubt, China is about to produce another extraordinary mathematical genius.

Of course, many mathematical geniuses have emerged in Chinese history, but in the end, it seems that none of them were able to go any further.

I just don't know how far this little guy will go in the future.

As Chen Hui walked out of the examination room, he didn't see Teacher An, but instead saw Yuan Xinyi holding a stack of A4 papers.

"These are the research materials I've prepared for you. Take a good look at them. If you don't understand anything, feel free to ask me. The lecture will be held the day after tomorrow."

The two had already added each other on WeChat yesterday, so after handing the paper to Chen Hui, Yuan Xinyi turned and left. He was busy verifying the delocalization phenomenon derived from higher orders.

This stack of papers was at least a thousand pages long and felt quite substantial in his hands. However, Chen Hui was eager to devour them. In his eyes, these were not just papers; they were clearly a test of proficiency, a golden road to the Big Four accounting firms and to the 500,000-dollar salary!

He used to have to search for information on his own, but now he's mastered it all on his own. It's true, the difference between having a teacher and not having one is huge!
He easily found an empty study room and began to examine the stack of papers in his hands.

The paper at the very top is titled "Cohomological Theory on Layers-Based Fractional Chern Insulators: Non-Abelian Topological Classification Theory," which translates to "Cohomological Theory on Layers-Based Fractional Chern Insulators: Non-Abelian Topological Classification Theory."

This is a recent paper published by a renowned professor at Stanford University who studies condensed matter physics, and also the professor Yuan Xinyi said he would be giving a lecture on.

Generally, the CMO has a seven-day schedule. Apart from the three days for the opening ceremony and exams, the remaining four days are used to hold lectures to broaden the horizons of these young mathematical geniuses selected from all over the country, let them learn about cutting-edge mathematics, and see what problems mathematicians are currently researching.

Below this paper are several other papers by the same professor published earlier on condensed matter physics.

In addition, at the bottom are some of Yuan Xinyi's own published papers, which are related to the Langlands Program. Chen Hui has also learned about his teacher in the past two days and found that the information he had previously learned seemed to be somewhat inaccurate.

Although this teacher works at Jiangcheng University, he has already published two papers in the top four journals at a young age, won the Ramanujan Prize in 16, received the Breakthrough Prize in Science in 21, and was invited to give a 45-minute report at the International Congress of Mathematicians in 22.

Such an achievement is already second to none in the country.

Moreover, the research focuses on the promising field of the Langlands program. It's worth noting that Ng Bao Chu, the 2010 Fields Medal winner, won the award for proving the fundamental lemma of the Langlands program.

Judging from my teacher's ecstatic reaction the day before yesterday, he seems to have made a breakthrough in this area. Perhaps, Professor Yuan will become the first mathematician in China to win the Fields Medal!

Fields Medal!

Chen Hui couldn't help but feel a sense of longing.

Ignoring the papers before him, he went straight to reading Langlands's papers.

The Langlands Program was proposed by Canadian-American mathematician Robert Langlands and aimed to connect two major branches of mathematics—number theory and representation theory. The program contains a series of conjectures and insights, which eventually developed into the Langlands Program.

As it happens, number theory and representation theory are areas of expertise for Chen Hui. He did not encounter much difficulty reading papers related to the Langlands program. On the contrary, his English proficiency caused him considerable trouble.

Improving my English level is now an urgent matter!

The level of proficiency in humanities subjects is closely related to memory, so the next free attribute point can be added to memory.

It's not far anymore.

Chen Hui was confident that, based on his previous experience, after winning the gold medal as CMO this time, he would gain another free attribute point. He could then improve his memory and start working on his English level, which would be much more efficient.

As for the CMO gold medal, it's not that Chen Hui is being immodest. He doesn't know what the cutoff score is this year, but he's confident in himself. Even if some points are deducted for certain steps, he believes that winning the gold medal is still a sure thing.

Now, let's get back to the paper!
Looking at the thick stack of papers in front of him, Chen Hui felt a mixture of joy and sorrow.

The lecture starts the day after tomorrow, so he doesn't have much time left!
(End of this chapter)