Chapter 25-Can't Keep a Low Profile Anymore
Something might happen during this exam.
Chen Fan was sure.
However, from the looks of it, if there was no problem with this question, then the problem was with this group of people.
About twenty minutes later, two students had finished making it.
There were only two people left, a boy and Chairman Zhou.
The ranks were pulled apart.
"Chen Fan, where did you find this question?" asked Qian Donghai.”
"I saw it casually and forgot about it. I thought it was very interesting, so I saved it. What's wrong, Teacher?”
"This question is a little interesting. It's really interesting. Have you solved it?”
"I…made it."
"That's fine. Send the answer.”
Chen Fan nodded and uploaded his answer.
"Let a=b=c=0 get 3f(0)2= 6f (0)2, which means f(0)=0. Now we let b=a,c=0 get f(a)2 f(a)2=2f(a)f(a), which is (f(a)f(a))2, then f(a)=f(a), which is f(n)
Even function.
Assuming that for some integral a such that f(a)=0, then for any integral b we have a b (ab)=0, so
f(a)2 f(b)2 f(a b)2=2f(b)f(a b),
This is equivalent to (f(b)f(a b))2=0, that is, f(a b)=f(b). Therefore, for an integral a such that f(a)=0,f is a function with a period.
Chen Fan's answer was relatively simple. To put it bluntly, he had simplified a lot of places. This question was done when Chen Fan was bored.
For Chen Fan, there was no point in doing the questions. He knew the answer to the basic questions at a glance, and he could quickly calculate what he needed in his mind.
Self-study classes were relatively long, and Chen Fan had nothing to do, so he chose to set questions.
It was to set a difficult problem for himself.
Chen Fan had set this question for himself.
There were many similar questions.
When Chen Fan finished writing, Qian Donghai took a look, rubbed his head and rubbed his temples.
He felt that this question was very subtle.
It was Mathematical Olympiad thinking, but it didn't seem like Mathematical Olympiad thinking.
However, Chen Fan's solution to this question was not traditional, but it was not beyond the outline.
"Chen Fan, have you read Advanced Mathematics?”
"No, why are you looking at that? I can't even finish reading high school math books. If I read those books when I have nothing to do, wouldn't I be adding to my burden? I don't want it. 0”
He understood. He had found the reason why these children were not intelligent. These children were using tricks.
"Teacher, I don't quite understand the formula written by Chen Fan.”
What do you mean you don't understand? You're stupid. Did a donkey kick your head?
The boy called Cheng Cheng was very active.
The first guy was called Zhao Mingtian, and the second was her sister, Zhao Mingling, a twin.
Zhao Mingtian said," Chen Fan wrote it very clearly. You don't understand. You are trying to solve the Toth problem. That solution is in the category of advanced mathematics. This is against the rules.”
"Ah, is it against the rules to use advanced mathematics to solve it?”
" Yes," said Qian Donghai." That's why the Olympics are so difficult. You can't exceed it too much."”
Zhao Mingtian continued,"Substitute b=a and c=2a into the equation f(2a)(f(2a)4f(a))=0. Taking a=1, we get f(2)=0 or f(2)=4f(1).
If f(2)=0, then f has a period of 2, and for any odd n, f(n)=f(1). It is easy to verify for any c.
Do you understand? Forget it. Judging from your eyes, I don't think you understand. This is just the most basic odd-even transformation of a function.
f(x)={0,c,2∣n,2n
The conditions of the question were met.
Now suppose f(2)=4f(1) and f(1) 0. If f(n)= n2f (1) for any number n, then the problem is solved. If there is an integral n such that f(n)= n2f (1), since f is an even function, we might as well treat n as a natural number, then obviously n3, we set n as the smallest positive integral such that f(n)= n2f (1)."
Zhao Mingtian was very serious. This number one person was indeed something. Chen Fan was a little relieved that there was finally someone with an IQ that was on the line.
Zhao Mingling nodded at the side, indicating that although this question was difficult, it was still possible to try hard.
Chen Fan could not count on the other three. He was just chatting with Cheng Cheng. Chen Fan could roughly guess how these three people had advanced. To put it bluntly, they had learned advanced mathematics in advance, used advanced mathematics to solve the problem, obtained the answer, and then used the answer to reverse promote it.
Was this the right method? The answer was yes.
In the basic Mathematical Olympiad competitions, there would always be trickery. However, in the process of conversion, there was one crucial thing.
If any of the steps you wrote were slightly out of line, then you would face the setting of zero points for this question.
There were always risks and opportunities.
"The three of them don't seem to be very interested in the international competition. This year's team is probably going to be gone.”
Chairman Zhou sent Chen Fan a private message.
Chen Fan was very curious. This guy, you are not my underling, why did you send me a message?
"That's cool. It's fine if we don't play together.”
"I don't want to give up. Chen Fan, think of a way.”
Chen Fan looked at the message from Chairman Zhou and said,""Why do you trust me so much? My results are not as good as yours. Don't be like this. I'm nervous and scared. I have a young lady. I'm not interested in boys.”
"I'm not interested in you. I just think that you can save this situation.”
Cheers.
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