I really just want to be a scholar

Chapter 37 Qin Ke is addicted to problem solving and cannot extricate himself
Ning Qingyun was the quickest to come back to her senses. She glanced at Qin Ke with a complicated expression, and then started her own topic silently, but she still glanced at Qin Ke from the corner of her eye from time to time.

After 5 minutes, Qin Ke, who knew nothing about the outside world, opened his eyes again, picked up a brush and began to write.

He has deduced that this question can be proved by mathematical induction and construction!
This can be regarded as an advanced multiplexing of mathematical induction and construction, and it is also the strongest and highest-level mathematical problem-solving method that he can use proficiently at his current mathematics level "High School Olympiad (Provincial Rematch)".

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

Assume that when m=k, the proposition holds, that is, there is a point set Sk, and for any A∈Sk, there are exactly k points in Sk whose distance to point A is 1.
Take each point in Sk as the center to make a circle with a radius of 1. There are finite number of intersection points between these circles. If they form a set Tk, then there are only a finite number of directions connecting any two points in Sk∪Tk.

Take any vector d in a direction that does not belong to this limited number of directions, and translate Sk along the vector d by one unit to obtain the point set Sk'.

It is not difficult to verify from the above method of vector d: on the one hand, Sk∩Sk'=Φ; on the other hand, the distance between two points A∈Sk and A'∈Sk' is 1 (if and only if A' is composed of A shifted).

When m=k+1, let Sk+1=Sk∪Sk', for any A∈Sk+1, without loss of generality, let A∈Sk, according to the inductive hypothesis, there are exactly k points in Sk and point A The distance is 1, and the distance between a point in Sk' and point A is 1, so it can be concluded that the distance between k+1 points in Sk=1 and point A is 1, and it can be proved that when m=k+1 , the proposition holds.

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

The original title 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 looking at his favorite 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 current provincial Olympiad mathematical thinking to the extreme !

The vice principal and a group of math teachers standing behind Qin Ke were completely shocked.

They witnessed Qin Ke's whole process from reviewing questions, thinking to the completion of the proof, but only about 7 minutes, no drafts were used, and the proof process was written in correct handwriting on the answer area of ​​the paper in one go!

This was the first time they saw a student who didn't use scratch paper for the Mathematical Olympiad questions, and some young teachers even had to ponder over and over again to understand the proof process and barely follow Qin Ke's train of thought!

Genius, this is real genius!
And he is a genius with explosive mental arithmetic ability!

Vice-principal Wen and a dozen teachers looked at Qin Ke with fiery eyes!

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

He flipped through the test paper and made sure that his name was written on the test paper. He stretched in satisfaction, folded the test paper casually, took out Ning Qingyun's English notebook in the drawer, and opened it to read.

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

If it were someone else, he might be yelled in fright, but Qin Ke has always been very thick-skinned. He put away the notebook and put it back in the drawer as if nothing had happened, and waved to Old Zheng to say hello: "Mr. Zheng, why are you here?"

Old Zheng put on airs and tried his best to maintain the muscles on his face so that he would not laugh out loud.He said calmly and gently: "Qin Ke, you spent 25 minutes on three questions this time, and the speed is a bit slow."

He said "slow down", but that face, that look of being so proud that he was about to laugh, all betrayed how proud and happy the old man was in his heart.

The math teachers next to me all yelled "pretending to be a criminal" in their hearts!
After completing three quasi-rematch difficult questions in 25 minutes, the spread of the news will shock the math teachers in the city and even the province, and "slower"... Slow down, girl!Wait for you to "beep——" (the sound is silenced)!

"This time I got carried away. In fact, if I don't use the 'mathematical induction construction method' for the last question, and use another conventional proof method, I don't have to think about it for so long. I just write it out. It should be two to three minutes earlier... But the other method of proof is too routine, not challenging, and the process takes a whole page, so I simply spent some more time thinking about the present proof method..."

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

All the teachers secretly took a deep breath.

When this student closed his eyes and meditated just now, did he actually think of two proof methods?
If he wasn't so diligent, he could even finish this paper in 23 minutes?

Even Ning Qingyun, who was doing the questions next to him but kept paying attention to this side, couldn't help but stop writing, feeling a sense of frustration that he couldn't continue.

Old Zheng tried his best to keep his face sullen, so as not to let out a proud pig cry in front of all the teachers and leaders: "Well, the current proof process is indeed much simpler. But, you boy, why don't you hand in the test paper after finishing it?
Qin Ke happily listened to the system notification sound of "Ding! Due to the violent mood swings of the teachers and Ning Qingyun, the host gained double 40 points of shock value! The experience value of the god of learning +40..." said with a smile : "Isn't it because I'm afraid of getting up and handing in the paper and making a noise, disturbing other students' answering questions?"

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

Qin Ke showed a shy smile: "My English foundation is relatively poor. As the saying goes, hard work can make up for weakness, so I have to spend more time after school..."

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

Qin Ke hurriedly got up and said, "Ah, so it's Principal Wen. Of course I know Principal Wen, but I'm afraid he won't recognize me."

Old Zheng snorted: "Don't you know each other now?"

He turned his head to Vice Principal Wen, with a changed look on his face, and said with a smile: "Principal Wen, I have no way to teach disciples, this kid is spoiled by me, and his behavior is somewhat inappropriate, Principal Wen... ..."

Vice President Wen's originally tense face now showed a relieved smile. He patted Qin Ke on the shoulder and said kindly: "Qin Ke, right? Yes, next weekend's Olympiad preliminaries take a good test and try to get in the top three Come back, do you have confidence?"

"Top three?" Qin Ke smiled embarrassedly: "I still want to get No.1, is it possible?"

(End of this chapter)