This top-tier academic genius

Qin Zhengyan frowned as he walked to the office door, where he saw Lin Yan sitting at his desk, completely absorbed in his work.

He slowly walked over and found Lin Yan doing a test paper, so he didn't call out to him, but watched from the side.

Upon seeing this, Qin Zhengyan's eyes widened in surprise.

"The Mock Exam for the National Higher Education Institutions? Why is he doing this exam?"

In fact, Qin Zhengyan had prepared a test paper, a paper selected from past college entrance examination questions, which he intended to use to test Lin Yan's math level.

To my utter surprise, he was actually working on the "High School Mock Exam," and had already started working on the more challenging questions.

Given that a, b, and c are positive real numbers, and a + b + c = 1, prove that (a²/b) + (b²/c) + (c²/a) ≥ 1.

This problem is somewhat difficult. In Qin Zhengyan's opinion, it is possible to solve it by brute force of the mean inequality, but the steps are very long and there is nothing particularly remarkable about it.

If it were him, he would choose the more efficient Cauchy inequality to solve it, which is the most orthodox approach.

Of course, it would be faster and more ingenious to solve it using Cauchy's weighted sum inequality.

However, this method is something that most students wouldn't think of.

However, the solution that Lin Yan wrote down completely exceeded Qin Zhengyan's expectations.

[Solution: According to the weighted sum of squares, a²/b + b²/c + c²/a ≥ (a + b + c)²/(a+b+c) = 1]

It uses the weighted square and inequality.

Qin Zhengyan nodded in surprise, not expecting this kid to be so clever, coming up with such an ingenious way to solve the problem so quickly.

Then, what surprised Qin Zhengyan even more was yet to come.

Lin Yan did not solve the next problem, but wrote down another solution below.

[Solution 2: Let x = a, b, c. The tangent inequality a²/b ≥ 2a - b. Adding the three equations together, we get 2(a+b+c) - (a+b+c) = 1, thus proving the statement.]

"Is this... tangent scaling?"

Qin Zhengyan never expected that Lin Yan would use the tangent scaling method.

However, Lin Yan, engrossed in his problem-solving, did not notice the person standing next to him. Seemingly not satisfied, he wrote down another solution method next to him.

[Solution 3: Construct f(x) = x²/t (t>0), the second derivative f''(x) = 2/t>0, the function is convex, by Jensen's inequality, (f(a)+f(b)+f(c))/3≥f((a+b+c)/3)=f(1/3)=1/3, therefore the sum ≥1.]

Lin Yan had no intention of showing off; he simply thought of these problem-solving methods and wrote them down to confirm his understanding of the knowledge points.

Standing next to him, Qin Zhengyan stared wide-eyed, his eyes filled with disbelief, as if he were looking at a monster.

In all his years of teaching, this was the first time he had ever seen someone solve a problem using three methods simultaneously: weighted summation, tangent scaling, and the piano inequality.

In his view, anyone who could solve this problem using Cauchy's inequality was considered quick-witted; someone who could use the weighted sum of squares was considered very clever; someone who could use the tangent bounding method was truly a genius; and someone who could think of using the second derivative to determine convexity and the piano inequality was simply... a genius.

This is bizarre! He actually knows a solution to a problem that even he himself couldn't have come up with.

After solving one problem, Lin Yan moved on to the next, solving problems at an astonishing speed, as if nothing could stump him.

Before he knew it, he had finished the test paper.

Looking up, Lin Yan suddenly realized something was wrong. Why were there teachers standing around him, all looking at him with strange eyes?

"Teacher Qin, you..."

"Great, great, excellent!"

Qin Zhengyan picked up the test paper he had just finished and sat down at the next table. "Wait a minute."

Then, physics teacher Huang Nanfang placed a test paper in front of his desk. "Here, Lin Yan, do this one too."

"Okay, sure."

Upon seeing the questions, Lin Yan realized that he had started answering them without even thinking, and hadn't even glanced at the title of the test paper, which read: "Selection of Physics Olympiad Questions"!

He looked at the question and pondered silently.

"Angle momentum is conserved, so the answer to this question should be... C."

For the first multiple-choice question, I answered instantly, almost without thinking.

Amidst the astonished gazes of the surrounding teachers, Lin Yan solved the problem at an unbelievable speed.

Several non-physics teachers couldn't even understand the questions, but Lin Yan finished them with just a few strokes.

"So fast, did he do it right?"

"I don't know, but it looks like there shouldn't be any problem."

Twenty minutes later, Lin Yan suddenly stopped writing.

"Teacher Huang, I've finished writing. I don't know how to do the rest."

Huang Nanfang pointed to a problem and asked, "Your solution to several of these problems is correct in its approach, but why do you stop halfway through and just plug in the formula to get the result?"

"I'm sorry, Teacher Huang, I forgot the formula."

"You mean you don't remember the formulas in the book?"

"Hmm, I haven't had a chance to see it yet."

Huang Nanfang: "..."

He never expected this answer.

It's simply unbelievable that someone who can't even remember the most basic formulas in the textbook can see through the essence of a physics Olympiad problem at a glance, avoid the traps, and arrive at the correct solution.

How evil!

I've never encountered a student like this before.

The other teachers also had very strange expressions. So you were just relying on pure talent to solve the problem?

The chemistry teacher came over and took the test paper too. "Why don't you take the chemistry test paper as well?"

"I don't know chemistry."

Biology teacher: "And what about biology?"

"I don't know anything about biology either."

The teachers breathed a sigh of relief. Thankfully, they had grasped the point, but not by much—only two points.

However, this is a bit outrageous.

"380! 380!"

Qin Zhengyan held the test paper as if it were a priceless treasure, overjoyed.

The teachers' expressions varied; some were surprised and amazed, while others didn't quite understand what it meant.

So a teacher explained: "The mock exam covers questions from both the first and second rounds, and the total score is 400 points. Last year, the top scorer in the provincial math competition also scored 380 points."

Once this explanation was given, the teachers who were unfamiliar with the competition understood, and their expressions towards Lin Yan immediately changed.

This handsome student actually possesses the ability to rival the top student in the entire province.

While this statement is somewhat inaccurate, as exam papers vary in difficulty and other factors such as performance on the day also play a role.

But that's already pretty impressive.

Even more outrageous is the rumor that this student previously had the worst math scores, scoring only 29 out of 150 on a test.

Qin Zhengyan looked at Lin Yan with fervent eyes, even more so than when he looked at his own son.

I was just worried about not having a star athlete in this year's competition, but here's one right in front of me.

"Lin Yan, come, come, have some fruit. Your teacher brought it from home."

"Teacher Qin, please get to the point."

"Alright, let's go outside and talk about something."

Qin Zhengyan led Lin Yan to a quiet corner of the balcony.

The two stood side by side. Qin Zhengyan was thin and less than 1.6 meters tall. Standing next to Lin Yan, he looked rather short, like a little old man.

"Are you interested in participating in this year's preliminary round of the Mathematical Olympiad?" Qin Zhengyan looked up at the boy in front of him, as if he had seen a rough gem.

Lin Yan looked troubled: "Teacher Qin, the registration period has long passed, can I still participate?"

"As long as you're willing to go, I'll figure out how to register." Qin Zhengyan, thinking Lin Yan wasn't very keen, continued to persuade him, "Listen to me, you're very talented in mathematics..."

"Okay, I'm willing."

"Great. Here's what we'll do: go back and wait for my message. I'll go and get your registration sorted out." With that, Qin Zhengyan hurried off.

Watching the old man jog away, Lin Yan rarely showed a smile.

Math Olympiad, here I come!