This top-tier academic genius

The National College Entrance Examination (NCEE) officially began at 8:10 AM on October 2nd.

The first session, the first test, will last from 8:00 to 9:20, for a total of 80 minutes.

After receiving the exam paper, Lin Yan glanced at it briefly.

The first test paper has only 11 questions, 8 of which are multiple choice questions, with a total score of 120 points.

The exam scope does not exceed the high school syllabus, and the content is consistent with the college entrance examination, but the problem-solving methods and overall difficulty are higher.

Before the exam, everyone was saying that this year's exam papers would be much more difficult than in previous years.

Lin Yan glanced at it and didn't really think so, mainly because many of the questions involved a lot of calculation.

These types of questions primarily test students' basic high school knowledge. If their foundation is not solid, they may get stuck and run out of time, failing to finish the questions.

Lin Yan doesn't really like these kinds of questions because there aren't many shortcuts.

Fortunately, his knowledge system was comprehensive enough that apart from calculating a few more steps, there was no difficulty. He finished the test paper with 15 minutes left before the exam.

After the first attempt, take a 20-minute break before starting the second attempt.

Compared to the first round, the second round is the real competition, an exam that differentiates scores.

If the first test assesses speed, calculation, and basic knowledge, then the second test assesses intelligence, thinking ability, and construction ability.

It can be said that the results of the second round of testing determine whether you can win first place in the province and enter the provincial team.

The second test had only six questions, all of which were problem-solving questions, with a total score of 280 points.

The first four questions are fixed in format, almost always consisting of 1 geometry question, 1 algebra question, 1 number theory question, and 1 combinatorics question each year. The last two questions are bonus questions with varying formats, but the difficulty increases dramatically.

The first question was a geometry proof, which was a mere appetizer for Lin Yan. He thought about it for a moment and came up with the answer.

He solved the problem easily in no more than 5 minutes.

The second question tested algebraic inequalities. This question is quite difficult for ordinary students because it involves Cauchy's inequality, fractional expansion and contraction, and the piano inequality, which need to be combined to solve the problem.

However, this was a piece of cake for Lin Yan, as this type of question was his forte and he had solved many of them.

The third question tested number theory, which Lin Yan answered quite easily.

The fourth question was a little more interesting; it was a combination question.

Given S = {1, 2, ..., 2026}, choose any 1014 numbers. Prove that there must be two numbers, one of which is a multiple of the other.

After reading the question, Lin Yan thought for a moment and realized that this was the pigeonhole principle.

He picked up his pen and began to answer:

For any positive integer x, it can be uniquely written as: x = 2k⋅m, where m is an odd number.

Divide set S into drawers based on odd numbers m: odd numbers can be 1, 3, 5, ..., 2025...

It only took about half an hour to solve all four problems.

It's said that this year's questions were very difficult, but they didn't seem that hard.

Next, let's tackle the bonus questions.

It is said that the difference between math whizzes and academic geniuses lies in the bonus questions.

Appendix 1: [Given positive real numbers a, b, and c, prove that: a/√(a² + 8bc) + b/√(b² + 8ac) + c/√(c² + 8ab) ≥ 1]

As the saying goes, the fewer words in a question, the more difficult it is.

Clearly, this question is not simple at all.

Lin Yan looked at the question, gently twirled the pen in his hand, and suddenly, a smile appeared on his lips.

Got it!

High-order fractional inequalities like the one before us cannot be solved by simply applying the ordinary Cauchy formula; they are simply unsolvable.

To solve this problem, we need to use the power-means inequality, homogenization, and scaling, all three working together.

Lin Yan wrote down:

Solution: Since the inequality is homogeneous, let's assume abc = 1.

First, prove the local scaling lemma: a /√(a² + 8bc) ≥ a^(4/3)/(a^(4/3) + b^(4/3) + c^(4/3)).

The simplification by crossing the squares of both sides proves the statement to be true.

Adding the three equations together: ∑ a /√(a² + 8bc) ≥ ∑ a^(4/3) / ∑ a^(4/3) = 1

The original expression is thus proved.

The last bonus question really stumped Lin Yan.

Find all positive integers n such that there exist positive integers a and b such that n | aⁿ + bⁿ, gcd(a, b) = 1.

This question is worth 50 points, but in Lin Yan's opinion, it should be worth 100 points.

After thinking and working out on the draft paper, Lin Yan suddenly had a flash of inspiration and thought of a solution. This problem could not be solved by brute force; it had to be classified according to odd/even and prime factor structures.

At 11:30, with one hour left before the exam ended, Lin Yan got up to hand in his paper.

Xin Yi, who was struggling with the last bonus question, glanced up but didn't pay any attention.

Because Lin Yan was not the first person to hand in his paper early.

At this time, there were only about half of the candidates left in the examination room; many candidates had already handed in their papers and left.

The proctor glanced at Lin Yan, assuming he was just another student filling in the numbers who couldn't answer the questions and chose to hand in his paper early. But when he looked at the exam paper, he was surprised to find that it wasn't blank as he had imagined, but completely filled with writing.

Meanwhile, a tea shop not far from the school gate was packed with people.

Many of them were teachers leading teams from various schools.

Four teachers from Xincheng No. 1 Middle School and No. 2 Middle School also sat at a table. They were all from the same place and knew each other. They took the opportunity of waiting for the candidates to have some tea and chat.

Gong Luosheng glanced at the original questions from the second round of testing sent to his phone and remarked, "This year's questions are indeed challenging!"

He also teaches mathematics, but in recent years he has been more involved in management and has neglected his studies in mathematics. He really couldn't think of a suitable solution to this bonus question for the time being.

"It is a bit difficult." Qin Zhengyan nodded, he was also looking at the questions.

At this point in time, although the exam is still in progress, the teachers have already received the original questions.

"However, Director Gong, you don't need to worry. That kid named Wei Chuan in your experimental class is more than capable. He'll definitely win the provincial first prize."

Many teachers in Xincheng were aware of Wei Chuan's math scores; he had ranked first in several city-wide mock exams and was considered one of the top math students in his cohort.

Last year, everyone thought he would definitely make it into the provincial first team, or even the provincial team, but he was unlucky and suffered from appendicitis.

Hearing Qin Zhengyan praise the students below, Gong Luosheng couldn't hide his smile.

"Your No. 1 High School isn't bad either. Li Jia and Wang Mingchao are both promising talents, and Zhou Yuelin is also quite good. I heard they even added an extra spot this year; that must be your trump card, haha~"

"I wouldn't call it a killer move, more like... a bit of talent. Oh, the person's here."

Just then, Lin Yan walked in from outside.

Gong Luosheng glanced at him and saw that he was quite handsome, but he had actually handed in his paper and left early. All the capable candidates were still in the examination hall, while this one had come out early, so his ability probably wasn't that great.

Qin Zhengyan waved to Lin Yan, "Over here."

"Greetings, teachers."

"Come, have a seat." Qin Zhengyan poured Lin Yan a cup of tea and asked, "How did the exam go?"

Lin Yan picked up the teacup, took a sip, and her eyes lit up. "Good tea indeed! The teachers really know how to enjoy it." She then replied jokingly, "It's alright. Anyway, I've finished everything. I should get full marks."

Upon hearing this answer, the teachers from the other schools nearby looked over in surprise, and Gong Luosheng almost spat out the tea he had just drunk.

Good heavens, someone actually dared to claim they got a perfect score.

Then, everyone shook their heads and chuckled.

This student is really good at joking!

No one believed Lin Yan was telling the truth. Not only was he unknown and hadn't produced a perfect score student in the past two years, but this year's questions were also clearly more difficult.

Qin Zhengyan was the only one who smiled and glanced at Lin Yan. Compared to the others, he knew Lin Yan's abilities best. If he said that Lin Yan could get a perfect score, then it meant that he must have done quite well.

He really wanted to ask about the situation in detail, but there were too many people here, so he didn't ask.

Of course, he wouldn't explain anything to anyone else. When he poured Lin Yan another cup of tea, he said, "Rest here for a while after the exam, and wait for them to come out together."