From child prodigy to chief scientist

On Saturday morning, Xu Xiao got up early to prepare for the upcoming school-wide selection exam for the competition.

Because some students will take exams in multiple subjects, the exam times for each subject are staggered, and a certain amount of rest time will be left in between.

The math competition was the first subject, and Xu Xiao arrived at the exam hall half an hour before the exam.

After several days of study, Xu Xiao has basically memorized the core content of those materials.

Thanks to his photographic memory, Xu Xiao's knowledge of these entries remains very strong to this day.

Because she had been studying so hastily, Xu Xiao hadn't had much time to do practice problems, and she was still a little unsure of herself.

"Sigh... I don't know if this method will actually work."

Even if he doesn't get a spot in the competition, at least he has a clear conscience.

With fifteen minutes to go before the exam begins, the invigilator is preparing to distribute the exam papers.

Xu Xiao silently activated the focus entry in her mind, adjusting her state.

The math competition selection exam lasts two hours, and with Xu Xiao's energy, he can maintain a focused state for two consecutive hours.

After receiving the test paper, Xu Xiao first wrote her name and then glanced at the questions.

It's clear that at least the first few questions weren't particularly difficult, and some of them could be solved even by people who hadn't studied for competitions.

After the exam bell rang, the students immediately started writing their answers.

At this moment, Xu Xiao had already put himself into a state of focus and was concentrating on doing the questions.

The first question was a set problem, which wouldn't be considered difficult even in the college entrance examination. Xu Xiao had already solved it mentally when he received the test paper.

In regular league matches, there are usually one or two particularly easy questions, just to prevent some students from not being able to solve any of them.

The second problem was about finding the maximum value of a complex algebraic expression modulo 1. Xu Xiao quickly realized that it should be represented in trigonometric form, and after substitution, he obtained the final result.

After solving a few questions relatively smoothly, Xu Xiao soon encountered trouble.

Let x, y, z be non-negative real numbers, and x + y + z = 3. Prove that x²y + y²z + z²x ≤ 4.

This is a problem involving ternary symmetric inequalities, which is a fairly common type of question in high school math competitions.

Students who have systematically studied high school math competitions should be able to quickly find the standard proof method.

However, Xu Xiao had never done this type of question before, so he could only try to analyze it first.

"Why not first assume a relationship between the magnitudes of x, y, and z, and then use the AM-GM inequality to transform the equation..."

After some calculations, Xu Xiao arrived at an upper bound, but it was still far from meeting the requirements of the problem.

"No... how about trying the Lagrange multiplier method?"

In theory, this method can be used to solve for the extrema of multivariable functions under equality constraints.

However, after listing the system of equations, Xu Xiao realized that the system of equations was not easy to solve and might involve some knowledge beyond the scope of the curriculum.

After trying several methods, Xu Xiao couldn't help but feel a little anxious, and beads of sweat appeared on her forehead.

In a typical exam, if Xu Xiao gets stuck on a question, she would usually skip it decisively.

However, if you can't even solve the earlier questions, there's even less hope of solving the remaining difficult ones.

"It seems I can only try to find some useful knowledge in my mind."

Left with no other choice, Xu Xiao closed her eyes and quickly reviewed her memories in her mind.

"On page 153 of 'The Secret of Inequalities,' my senior marked several methods for solving ternary symmetric inequalities..."

"On page 256 of 'A Study of Mathematical Competitions,' if scaling fails, consider completing the square to form a sum of squares..."

"On page 87 of the 'Mathematical Olympiad Mini-Series,' when x + y + z = k, you can try setting..."

Having identified several relevant knowledge points, Xu Xiao opened his eyes and began to formulate a new approach.

"First, analyze the boundary case where one variable is 0. After differentiation, the maximum value should be able to be proven. Then, analyze the internal case where x, y, and z are all greater than 0..."

After some deduction, Xu Xiao finally completed the entire proof process.

"It was a bit slow, but at least we managed to get it done."

It turns out that memorizing the knowledge from those competition materials beforehand is indeed effective.

Having wasted a lot of time on this question, Xu Xiao quickly continued working on the next one.

For more than half of the following questions, Xu Xiao couldn't immediately find a solution, so he resorted to his old trick of searching his brain for the knowledge he had memorized over the past few days.

This feeling of having several books in your mind is truly amazing, even more convenient than simply flipping through a book.

However, this method is basically equivalent to Xu Xiao learning and applying what he learned on the spot in the exam room, which is indeed too slow in terms of problem-solving speed.

Before I knew it, there was less than half an hour left in the exam.

Those students who answered quickly had already finished all the questions and were having their answers checked.

At this moment, He Muzhou raised his hand to signal to the teacher that he wanted to hand in his paper early.

The questions in the school's selection exam were indeed not difficult for him, and were even less challenging than his usual training.

By this time, he had already checked twice, and there was no point in continuing to linger in the examination room.

"bell……"

As soon as the exam bell rang, the invigilator immediately ordered everyone to stop answering the questions.

Xu Xiao hadn't finished the last question, but since the exam time was over, she didn't continue writing and put her pen on the table.

"Phew... still a little short."

If Xu Xiao had just five more minutes, he should have been able to finish the problem.

However, Xu Xiao was quite satisfied with himself for managing to complete most of the questions despite the limited preparation time.

As for what kind of score he could get, he had no idea.

At noon, Yang Dun finished his physics exam and went to the cafeteria for lunch with Xu Xiao.

Judging from Yang Dun's relaxed expression, Xu Xiao could tell that he must have done well on the exam.

"Yandon, were the physics problems difficult?"

"It wasn't too difficult, even a bit easier than my usual practice tests. I handed in my paper half an hour early."

"Really? Then you should be able to get a spot."

"Hmm, there shouldn't be any problems. Xu Xiao, what about you?"

"I think I'm in big trouble. I haven't even finished the test. I got stuck on so many questions."

Xu Xiao didn't have high hopes for passing the school's selection exam; at worst, she would just focus on preparing for the college entrance exam.