"What you said is true?"

Pang Shaoan was slightly stunned, he knew the level of his grandson, in the same age, Pang Xuelin can be called a leader, even if he wins the IMO gold medal, Pang Shaoan will not feel much surprised.

But now Pang Xuelin told himself that he had not only integrated the mathematical knowledge of high school, but even taught some of the college-level courses such as calculus, mathematical analysis, and abstract algebra on his own, which surprised him a little.

I haven't seen him study the content of the university on his own on weekdays, could it be that he secretly studied it in private?

Pang Xuelin smiled and said: "Grandpa, if it's true or not, I won't know if you test it!" ”

"Okay, then I'll make this bet with you, we'll go to the study now, I'll make the question on the spot, and you can answer it on the spot!"

Pang Xuelin had nothing to do, and Yao Jianzhong on the side also came over with great interest to watch the excitement.

When he came to the study, Pang Shaoan found the manuscript paper and quickly wrote the first question on it:

Let R be the set of all real numbers, and find all the functions f:R→R, so that for any real number x and y, there is

f(f(x)f(y))+f(x+y)=f(xy)。

This question seems to be sketchy, and it does not go beyond the scope of high school mathematics, but the question itself is very difficult for high school students, at least as an IMO test question, there is no problem at all.

Pang Xuelin took the exam question, just glanced at it, and then without thinking about it, he picked up the pen and wrote the answer in the blank part below the manuscript paper.

Solution: If the equation of the problem is written as P(x,y), then it can be known from P(0,0).

f(f(0)^2）=0 （1）

For any real number x≠1, there is a real number y, satisfying x+y=xy, then y=x/x-1, as can be seen from P(x,x/x-1),

f(f(x)f(x/x-1))=0，x≠1 （2）

F(0) can be discussed in two cases.

……

Pang Shaoan originally thought that Pang Xuelin would have to think for more than ten minutes, and after finding the idea, it would take ten more minutes to solve the problem.

Unexpectedly, the other party only glanced at the problem, and then solved the problem directly, and it didn't take five minutes before and after.

"Grandpa, look, am I right to solve it this way?"

Pang Xuelin confidently handed the answer to Pang Shaoan.

The Philippine Award boss does the IMO test questions, even easier than college students doing the primary school test questions, and at the first glance of the questions, Pang Xuelin has already seen through the intention of the questioner.

If you really want to talk about the level of mathematics, he is already more than a step higher than Pang Shaoan.

Pang Shaoan took it and looked at it, nodded and said, "Yes, I didn't expect your level to rise recently, and the speed of solving the problem is much faster." However, this is only the content of the high school competition, since you have already studied the courses at the university level, then my next topic is not limited to high school knowledge. ”

Pang Xuelin smiled and said: "Grandpa, even if you come out, if you can't answer, I will lose!" ”

"Stinky boy, when did you get so crazy?!"

Although Pang Shaoan scolded on his lips, he was very happy in his heart, if Pang Xuelin really had the level he said, wouldn't it prove the success of his education?

"You said earlier that you studied linear algebra, so the second question is related to matrices.

Soon, Pang Shaoan wrote the second question on the manuscript paper:

Proof: Any nth-order real square matrix A can be decomposed into A=A0+A1+A2, where A0=aIn, a is a real number, and A1 and A2 are both power-zero squares.

Like the first question, Pang Xuelin also didn't think about it for long, and wrote the answer directly on the manuscript paper:

Proof: Let's start by proving a lemma.

Lemma: Let A be a real square of order n and tr(A)=0 is satisfied, then there is an invertible real square matrix P, so that the diagonal elements of P^-1AP are all 0.

Summarizing n, when n=1 and A=(0), the conclusion is obviously true, let n≥2 be, and we consider two cases.

Scenario 1: All non-zero vectors in Rn are eigenvectors of A......

……

Similarly, it only took five minutes for Pang Xuelin to complete the problem.

This time, not only Pang Shaoan was surprised, but even Yao Jianzhong was amazed: "Old Pang, your Xiaolin is amazing, this question may not be solved by ordinary undergraduates!" ”

"Come again!"

Pang Shao'an said.

Next, questions 3, 4, 5......

From calculus, differential equations, to deriving conjugate gradients for solving systems of linear equations, to group theory, differential geometry, and functional analysis.

The difficulty also began to rise sharply, but Pang Xuelin's speed of solving the problem was moderate and there was no change, no matter how difficult the problem was, in his hands, it would take almost three times to divide five by two, and it would take ten minutes at most to get it.

Before I knew it, more than an hour had passed.

And Pang Shaoan and Yao Jianzhong also went from surprise and admiration at the beginning, to shock and numbness later.

"Xiaolin, when did you start learning all this?"

Pang Shaoan looked at Pang Xuelin and said a little incredulously.

Pang Xuelin was a little embarrassed and said, "Aren't there a lot of college-level math books in your study?" I secretly studied two years ago, grandpa, what do you think of my current level, can I not go to high school? ”

Pang Shaoan said: "In other words, it took you two years to not only learn the undergraduate mathematics system by yourself, but even basically integrate it?" ”

Pang Xuelin nodded.

Pang Shaoan and Yao Jianzhong looked at each other, what else could he say?

Most of the exam questions he came out just now were the content of undergraduate mathematics competitions, and some of them were difficult questions at the graduate level, all of which were easily solved by Pang Xuelin.

Yao Jianzhong gave a thumbs up on the side and said, "Lao Pang, your kid is really a demon, according to his level, it really doesn't make much sense to go to high school again." In this way, Xiaolin, I will help you go to the School of Physics of Jiang University to talk about it, you can go directly to the Department of Physics to audit, and you can go through the admission procedures for the spring semester early next year! ”

Pang Shaoan glared at him and said, "Old Yao, this is my grandson, don't think about digging the foot of the wall!" Xiaolin, let's do it, I'll go to your school tomorrow to talk to your teachers, and then I'll go to Liu Tingbo, and you can go directly to the Department of Mathematics of Jiang University next! ”

Pang Xuelin couldn't help but laugh and said: "Grandpa, but I want to study materials!" ”

Pang Shaoan glared and said: "How can this be done, with your mathematical talent, it is a waste to learn materials!" ”

Pang Xuelin said: "Grandpa, Grandpa Yao, I understand your kindness." However, I still want to study materials engineering, in this era, it is not meaningful to do pure number research, but materials have a strong application prospect in the field of engineering, which is more suitable for me. ”

"But ......"

"Grandpa, don't persuade me, if I study other majors, I might as well wait for high school graduation to take the college entrance examination and choose my own major!"

Pang Shaoan frowned, saw Pang Xuelin's determined look, and sighed: "That's okay, I won't force you anymore, the material is the material!" ”

Pang Xuelin said: "Grandpa, do you want to talk to my parents about going to Jiang University to listen to this time!" ”

This is the first time that Pang Xuelin has mentioned his parents after entering the plane.

He wanted to ask before, but he couldn't find a suitable entry point.

When I was in middle school in the real world, my parents were busy with business and didn't have much time to take care of themselves, but in the Wandering Earth plane, the whole world line has changed greatly, and building an earth engine requires a lot of resources.

Pang Shaoan said: "Your parents and Xiao Xia's mother are both working in Shanghai No. 2 engine, how can I have time to care about you, I will just talk to them when the time comes!" ”

"Shanghai No. 2 engine?"

Pang Xuelin was slightly stunned and couldn't help frowning.

There was a scene in the movie where Shanghai was swallowed up by a tsunami before setting sail on the earth, and the sea water even reached the spire of the Oriental Pearl Tower, causing heavy casualties.

It stands to reason that if the normal human plan had been followed, everyone on the surface of the earth would have retreated into the dungeon before the Earth set sail, and it was unlikely that such casualties would have occurred.

In this case, such a disaster can only occur if there is an uncontrollable disaster in the coastal areas, triggering a large-scale tsunami and the influx of seawater into the downtown area of Shanghai.

Jiangcheng is not far from Shanghai, and I am afraid that a disaster of this level will also be affected.

Pang Xuelin was shocked!