Xueba, I beg you to go and escort him quickly!

But today, Xu Heng didn't stop!

After turning the page, there is the answer.

The fifth question (that is, the second question today):

求所有的函数f:Z→Z使得对任意满足a+b+c=0的整数a,b,c恒有f(a)2+f(b)2+f(c)2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)。

In the eyes of the female examiner, even Xu Heng's action of turning the page is so smooth.

Turn the page!

Pick up the pen!

Start answering questions.

All in one go!

During the whole process, Xu Heng was extremely calm, as if everything was well planned!

of course……

He is confident!

In Xu Heng's view, today's topic is almost the same as yesterday's, and even simpler than yesterday's.

So he had finished reading these two questions 10 minutes before the official exam!

It was precisely because he understood these two questions, and in order to prevent them from making such a fuss, Xu Heng chose to slow down.

Write slowly!

Answer the questions slowly!

In the eyes of the female examiner, Xu Heng showed the elegance of the Orientals!

Every word written is enjoyed!

It is a sacred visual feast, which is fascinating.

The female examiner couldn't help but moved closer to the other two examiners and nudged them with her elbow.

It was only then that they noticed that in the entire examination room, only Xu Heng stood straight between the sky and the earth like a sharp sword.

Looking at the other five, they were either bowing their waists, scratching their ears or scratching their heads, or their brows were furrowed, and their spirits were listless.

nonsense!

Yesterday's shock, their mentality collapsed, coupled with the fact that they didn't sleep well last night, and today's topic is still very confusing, how could they behave so calmly and with ease?

Only Xu Heng!

Xu Heng was methodical.

令a=b=c=0可得3f(0)2=6f(0)2,这说明f(0)=0。现在我们令b=a,c=0可得到f(a)2+f(a)2=2f(a)f(a)即(f(a)f(a))2,于是f(a)=f(a),即f(n)为偶函数。

假设对某个整数a使得f(a)=0,则对任意整数b我们有a+b+(ab)=0,因此f(a)2+f(b)2+f(a+b)2=2f(b)f(a+b),这等价于(f(b)f(a+b))2=0,即f(a+b)=f(b)。

Therefore, when f(a)=0 for an integer a, f is a function whose period is a.

……

现在假设f(2)=4f(1)并且f(1)≠0。

If f(n)=n2*f(1) holds for any integer n, then...

……

To sum up, the solution of the functional equation is: f(x)=cx2, where c∈Z; f(x)={0,c,2∣n,2n where c∈Z.

as well as……

The three examiners were immersed in Xu Heng's answering process.

Xu Heng sat on the seat with his back straight, and every stroke was very serious.

No matter which angle they look at, they feel very enjoyable.

Perfect handwriting!

Perfect temperament!

Perfect appearance!

cough cough...

The female examiner was immersed in Xu Heng's natural handsomeness for a moment, unable to extricate herself.

At this moment, they all ignored it.

Xu Heng only spent 10 minutes on these two questions last night!

indeed!

This time, Xu Heng made one stroke at a time, slow enough, and enough to kill time!

But why can't these two questions be so simple!

Finally, it was done two days ago.

Xu Heng breathed a sigh of relief, and leisurely turned his pen.

And the other five, all frowning deeply, are still answering the first question!

Only then did the three examiners realize...

Even though Xu Heng wrote so slowly, it only took 10 minutes!

Even buy Karma! ! !

At first, they were suspicious of Xu Heng's attitude!

But now it seems that they have been slapped in the face!

Xu Heng's operation is even more powerful than last time, even worse!

What can you give an answer by looking at a question? !

We almost believed it!

This is obviously two questions to read together!

After reading it together, they gave accurate answers one by one!

"." Hiss...` ~! "

Can't help but make a sound.

As for Xu Heng, he turned a deaf ear and looked at the last question.

Let f be a function defined in a set of integers that takes positive integers. It is known that for any two integers m≠n, f(mn)|f(m)-f(n).Proof: For any integer m, n, f(m)≤f(n)→f(m)|f(n).

Look at the title, 10 seconds!

Think, 30 seconds!

Xu Heng started writing again.

The three examiners looked sincerely, with anticipation on their faces.

They were looking forward to Xu Heng's writing, and looking forward to Xu Heng showing them how he answered the questions again!

They have completely switched from being an examiner to being a fan!

They looked at Xu Heng with a feeling of adoration and admiration.

Xu Heng replied:

Proof: f(m-0)|f(m)-f(0) is known, so f(m)|f(0) holds true for any integer m.

Since f(0—n)|f(0)-f(n), so f(-n)|f(n), n→-n can get f(n)|f(-n), so for any Integer n has f(-n)=F(n).

If the conclusion is not established, then there are m, n...

……

In summary, the conclusion holds.

For example f(odd)=a, f(even)=ab, f(0)=abc.

As it was written, Xu Heng gave the answer.

And it's still the correct answer.

"Forehead……"

(good)

Xu Heng was stunned for a moment.

He was really stunned for a moment, he didn't expect the last question to be so simple!

too easy!

No matter what others think, anyway, Xu Heng thinks that this question is not as interesting as the previous two!

Hey……

This is the so-called dwarf who is tall in the middle!

But compared with yesterday's whole, the difference is too big.

Xu Heng felt unsatisfied.

After all, all these problems have been solved, and these problems have not been solved in half an hour!

Xu Heng originally wanted to write more slowly, more slowly!

But no matter how slow you are, you can't hold back your simple topic!

Forget it...

Write more!

Xu Heng continued.

If f(n)|f(kn), since f(n)|f((k+1)n)-f(kn), there is also f(n)|f((k+1)n), Also since f(n)|f(kn)...

……

As Xu Heng wrote, the three examiners watched.

They don't know Chinese characters, but they do know the letters and mathematical symbols!

They wondered, why did Xu Heng's writing go back a bit?

These letters, these numbers, always feel somewhat similar to the previous ones.

???

Before coming to invigilate Xu Heng, the first three examiners especially emphasized that when reading Xu Heng's answers, one must pay attention to (2), and at the same time, there are five Chinese characters: "the second solution".

But they all stared at Xu Heng's test paper, and (2) and "the second solution" did not appear.

It's so weird...

Didn't Xu Heng himself pay attention?

I was caught by my own problem-solving ideas?

The three examiners looked at each other.

But just when they were confused, Xu Heng stopped, and then drew brackets in front of "If f(n)|f(kn)", and then...

PS: Thank you [ii] for supporting the factory!

Kneel for all support!There will be updates later! .

Chapter 215

……

If f(yn)≤f(xm), then only f(d)=f(yn), since f(d)≤f(n)≤f(yn), so f(n)=f(d )≤f(m), so there can only be f(n)=f(m), and the conclusion holds.

If f(yn)>f(xm), then f(d)=f(xm), so f(m)=f(d), because f(d)|f(n), so f(m)| f(n).