Chapter 211

The time came to the fifteenth day of the first month.

Today is the Lantern Festival, and it is also the day of the annual National College Student Mathematics Competition.

First-year students are scheduled to start school on the 18th day of the first month.

Therefore, in the dormitory, there is still only Ma Zhengxuan.

The competition started at 9 a.m. in a teaching building at Yan University.

Got up early in the morning, Ma Zhengxuan washed up, and after breakfast, he came to the library for final preparations.

During this week, Ma Zhengxuan listened to the competition coaching class and went to Gu Lu's office to ask questions from time to time, and he had made the most adequate preparations.

Ma Zhengxuan is not like Bi Qi, Ma Zhengxuan pays attention to safety.

Since you have chosen to participate in the college student mathematics competition, it is natural that you can win the award steadily.

In the past few days, Ma Zhengxuan has been sleeping very late, and he has read the real questions of previous years' competitions and the ten sets of mock questions from Gu Lu over and over again.

Now, it's time to test the results of his preparations.

At half past eight, Ma Zhengxuan left the library and walked towards the teaching building where the examination hall was located with steady steps.

At nine o'clock, the National College Student Mathematics Competition officially began.

There are a total of 26 questions in the examination paper, including two additional questions.

A total of 200 points.

According to the situation in previous years, a score of 190 points or more is required to win the national first prize.

After all, it's the highest math competition in the country.

Even students from Yan University and Tsinghua University will participate in this competition, which is enough to prove how difficult it is to win awards in this competition.

Ma Zhengxuan's goal is naturally to run for the first prize.

In this National College Student Mathematics Competition, more than 30 students from the Department of Mathematics participated in the competition, most of whom were sophomores and juniors.

First-year students, plus Ma Zhengxuan, there are only three of them.

Ma Zhengxuan felt a lot of pressure.

However, during this time, under Gu Lu's crazy indoctrination, Ma Zhengxuan realized that he might not be weak with those high-adhesion seniors.

Ma Zhengxuan has a calm personality, but it does not mean that he does not fight or grab.

"I can't be sorry for Teacher Gu's expectations!" Ma Zhengxuan clenched his fists, took a deep breath, opened the test paper, and swept his gaze over the questions one by one.

Decent!

This was Ma Zhengxuan's judgment for a moment. The question types and test points of the test questions are not much different from those of previous years, but there are slight changes in the specific questions, and the whole can only be regarded as decent.

Moreover, there are a few questions, which are similar to the questions in Gu Lu's ten sets of mock papers, and Ma Zhengxuan can directly and easily analogy to solve them.

In an instant, Ma Zhengxuan's confidence increased a lot.

Then pick up the pen and start solving the problem.

Question 1: [Let the real square matrix H1=(0,1|1,0),Hn+1=(Hn,I|I,Hn),n≥1, where I is the same order square matrix of the same unit as Hn, then rank(H4)=______]

The test point of this question is related to the knowledge points of the diagonal square.

Whew!

Ma Zhengxuan wrote the steps to solve the problem on scratch paper: [Hn is a symmetrical square matrix of order m=2^n, then there will be an orthogonal square matrix P such that ...... To get the answer, rank(H4)=10. 】

Ma Zhengxuan's speed of doing the questions is not very fast, but it is still less than five minutes to complete the first question.

In half an hour, Ma Zhengxuan completed the first ten choices, leaving only the last sixteen questions.

And there are still three hours left until the end of the exam.

This time, enough.

Ma Zhengxuan picked up the pen and began to do the first question of the sixteen questions.

[Let the Ma series of α∈(1,2),(1-x)^α be ∑akx^k, the real constant matrix A of n x n is the power zero matrix, I is the identity matrix, and let the matrix value function G(x) be defined as...... , the sufficient and necessary condition for the existence of 1≤i, j≤n, and integral ∫g(ij)(x)dx is A^3=0.]

This is a question of proof.

There are many things to examine, including integrals, matrices, and inequalities.

But that doesn't stump Ma Zhengxuan.

These three aspects of knowledge are very basic content, and Ma Zhengxuan has no reason not to know.

For a question of this difficulty, Ma Zhengxuan didn't even need to perform calculations on the scratch paper, but for the sake of safety, Ma Zhengxuan still calculated on the scratch paper and then freed up on the answer sheet.

[A is a power zero matrix, so there is A^n=0, denote f(x)=(1-x)^α, when j>k, denote ...... , G(x) is directly expressed by the Jordan standard type, so that the sufficient and necessary condition for the existence of the integral ∫g(ij)(x)dx is A^3=0.]

When there was still an hour and a half left, Ma Zhengxuan only had the last two additional questions left.

Additional Question 1: [Set X1, X2...... Xn, which are independent and identically distributed random variables, have a common distribution function F(X) and a density function f(x), and now for random variables, X1...... Xn, rearrange ,...... in order of size]

Additional Question 2: [Proof: If f∈S, then in Δ:|z|≦1, there is |z|/(1+|z|) ^2≦|f(z)|≤|z|/(1-(x))^2.】

The first additional question is not difficult, but the second additional question has made Ma Zhengxuan stuck for a long time.

After thinking about it for a long time and recalling it for a long time, Ma Zhengxuan kept recalling a little knowledge point that Gu Lu told him when he was training and preparing for IMO in the winter camp this time last year.

"It's ...... Koebe Deviation Theorem!" Ma Zhengxuan's eyes lit up, and he recalled what Gu Lu had told about the 'Koebe Deviation Theorem'.

The so-called Koebe deviation theorem, also known as the stem of the second additional question, is a defined theorem used to describe the single-leaf function on a unit disk.

"How did the teacher prove this theorem at that time?" Ma Zhengxuan closed his eyes and recalled carefully.

"de Branges theorem!" After a long time, Ma Zhengxuan slowly spit out the term.

He remembers that he used de Branges' theorem to derive the Koebe deviation theorem.

de Branges' theorem is a theorem in the complex function course of the university, and its main content is that if the power series of a function is expanded to f(z)=z+a2z^2+a3z^3+...... anz^n, then |an|≦n and the equal sign is true if and only if the function z/(1-z)^2 or its rotation.

At that time, in Ma Zhengxuan's memory, Mr. Gu used de Branges' theorem to deduce the range of f(z) when |z|<1. Since f(0)=0,...... , |f(z)|=|∫f(ζ)dζ|≤|z|/(1-z)^2, and finally, the Koebe deviation theorem is obtained.

At that time, when I was in the winter camp, Mr. Gu clearly said that this is the content of the super-curriculum, and it is very unlikely that IMO will use it, so let everyone listen to it.

Although it would not be used in IMO, Ma Zhengxuan at that time still wrote it down in his notes and occasionally looked through it a few times.

But I didn't expect that it was not used in the IMO, but this part of the knowledge was used in the National College Student Mathematics Competition.

If it weren't for Ma Zhengxuan's frequent review of the content in his notes, Ma Zhenxuan would definitely not be able to remember this part of the content after a year.

Now that you know the process of proof, the rest is easy to do.

In ten minutes, Ma Zhengxuan completed the answer to the second additional question.

At this point, the whole set of test papers Ma Zhengxuan has been completed, and there is still more than half an hour before the paper is handed in.

In the exam rules, early submission is allowed.

But Ma Zhengxuan did not have the habit of doing this, and after carefully checking it many times, he waited until the bell rang at the end of the exam before Ma Zhengxuan handed in the paper.

The rest is to wait for the results to come out.

The grading speed of the college student mathematics competition is very fast, as short as ten days, and as long as half a month, the ranking and awards will be announced.

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