Ning Qingyun was the quickest to come to her senses. She glanced at Qin Ke with a complicated expression, and then silently started working on her own question, but she still glanced at Qin Ke from the corner of her eye.

Five minutes later, Qin Ke, who had no knowledge of the outside world, reopened his eyes, picked up a brush and started writing.

He has deduced that this problem can be proved by mathematical induction and construction methods!

This can be regarded as an advanced reuse of mathematical induction and construction methods, and it is also the strongest and highest level mathematical problem-solving method that he can skillfully use at his current mathematics level of "High School Mathematical Olympiad (provincial semi-finals)".

"Proof: When m=1, take the two endpoints of the line segment with length 1 to form a point set s. The original question can be proved.

Assume that when m=k, the proposition is true, that is, there is a point set sk. For any a∈sk, there are exactly k points in sk whose distance from point a is 1.

Take each point in sk as the center and draw a circle with a radius of 1. There are a finite number of intersection points between these circles. Assuming that they form a set tk, then there are only a finite number of directions connecting any two points in sk∪tk.

If the vector d in any direction does not belong to this limited number of directions, translate sk by one unit along the vector d to obtain the point set sk’.

It is not difficult to verify from the above method of selecting the vector d: on the one hand, sk∩sk'=Φ; on the other hand, the distance between the two points a∈sk and a'∈sk' is 1 (if and only if a' is determined by a

obtained by translation).

When m=k1, let sk1=sk∪sk'. For any a∈sk1, without loss of generality, let a∈sk. According to the inductive hypothesis, there are exactly k points in sk whose distance from point a is 1, and

The distance between a point in sk' and point a is 1, so it can be concluded that the distance between k1 points in sk=1 and point a is 1. It can be proved that when m=k1, the proposition is true.

It can be known from mathematical induction that for any positive integer m, there is a point set s in the plane that satisfies the meaning of the question.

The original question is proved.”

Putting down his pen, Qin Ke felt extremely happy and couldn't help but watch his proof process over and over again, just like watching his proud work.

In fact, he is indeed very proud. Here he first constructed the simplest m=1 situation, and then realized the inductive transition by translating the point set and then taking the union, applying the provincial Olympiad mathematical thinking he has mastered to the extreme.

!

The vice principal and the math teachers standing behind Qin Ke were completely shocked.

They witnessed with their own eyes Qin Ke's entire process from reviewing the question, thinking to completing the proof. It only took about seven minutes, without any drafts, and he directly wrote the proof process in correct handwriting on the answer area of ​​the paper!

This was the first time they had seen students solving Mathematical Olympiad problems without using scratch paper. Moreover, some young teachers present had to think carefully and repeatedly before they could understand these proof processes, and they could barely follow Qin Ke's ideas!

Genius, this is real genius!

And he is a genius with incredible mental arithmetic abilities!

Vice Principal Wen and a dozen teachers looked at Qin Ke with intense eyes!

Qin Ke didn't notice the dozens of blazing gazes behind him. He was still in the afterglow of excitement. Even the system prompts that kept ringing in his ears became like "white noise" because they were too frequent and were directly filtered by him.

Lost.

He turned over the paper and made sure that his name was written on the paper. He stretched with satisfaction, folded the paper, took out Ning Qingjun's English notebook from the drawer and opened it to read.

Suddenly there was a slight cough from behind, and Qin Ke turned around, only to see a group of big guys staring intently at him from behind, as well as the English notebook in his hand.

Anyone else would have been scared and screamed, but Qin Ke has always been very shameless. He put the notebook away and put it back in the drawer as if nothing had happened. He also waved to Lao Zheng and said hello: "Teacher Zheng, why are you here?"

Lao Zheng held his airs and tried hard to maintain the muscles on his face to prevent himself from laughing out loud. He said calmly and gently: "Qin Ke, you actually spent twenty-five minutes on just three questions this time. How fast is it?"

It’s a little slow.”

He said "slow down a little bit", but his face and his proud look on the verge of laughing betrayed how proud and happy the old man was.

The math teachers next to me all cursed in their hearts for being "pretentious"!

After completing three semi-finals-level questions in twenty-five minutes, it would shock the math teachers in the city and even the province if word spread about it. It was "a little too slow"... I'm too slow, sister! I'm too slow, "Beep——"

(silenced)!

"I got carried away this time. In fact, if I didn't use the 'mathematical induction construction method' for the last question and used another conventional proof method, I wouldn't have to think about it for so long and just write it out directly, which should be two to three minutes earlier...

But the other proof method was too routine and not challenging, and the process would fill a whole page, so I simply spent some more time to come up with the current proof method..."

Qin Ke said, scratching his head a little "embarrassingly".

All the teachers secretly took a breath.

When this student closed his eyes and meditated just now, he actually thought of two proof methods?

If he hadn't been so striving for perfection, could he have completed this paper in twenty-three minutes?

Even Ning Qingyun, who was doing the questions next to him but kept paying attention to him, couldn't help but stop writing, feeling frustrated that he couldn't continue.

Lao Zheng tried his best to keep his face straight to prevent himself from squealing proudly in front of the teachers and leaders: "Well, the current proof process is indeed much simpler. But, you kid, why don't you hand in the test paper after you finish it?

Qin Ke happily listened to the system prompt "Ding! Due to the violent mood swings of the teachers and Ning Qingyun, the host has received double 40 points of shock value! The God of Learning experience value is 40..." and said with a smile:

"Aren't you afraid that getting up to hand in papers will make a noise and disturb other students from answering questions?"

"Humph, so you just read the English notes?"

Qin Ke showed a shy smile: "My English foundation is relatively poor. As the saying goes, diligence can make up for weakness, so I have to use my spare time to..."

"Okay, okay, don't be shy, this is Principal Wen, do you know him?"

Qin Ke stood up hurriedly and said, "Ah, it turns out to be Principal Wen. Of course I know Principal Wen, but I'm just afraid that he doesn't recognize me."

Lao Zheng snorted: "Don't we know each other now?"

He turned to Vice Principal Wen, with a different expression on his face, and said with a smile: "Principal Wen, I am not a good teacher. This boy has been spoiled by me. There is something wrong with his behavior. Principal Wen, you...

…”

Vice Principal Wen's tense face now showed a relieved smile. He patted Qin Ke on the shoulder and said kindly: "Qin Ke, right? Not bad. Take the test well in the Mathematical Olympiad preliminary round next weekend and try to get the top three.

Do you have the confidence to come back?"
Chapter completed!