I really just want to be a scholar

Chapter 162 56 minutes until the start of the exam!

At this time, Qin Ke had already started reviewing the questions, and Yue Wendao also turned his attention to the seventh question which he was most proud of.

"7. As shown in the figure below, in the complete quadrilateral ABCDEF, O1, O2, O3, O4 are the circumcenters of △ACF, △BCD, △DEF, △ABE respectively, and H1, H2, H3, H4 are △O4O2O3, △O4O1O3 , △O2O4O1, the orthocenter of △O1O2O3, to prove:”

“（1）△O4O2O3∽△ACF，△O1O2O3∽△ABE，△O2O4O1∽△DEF，△O∽△BCD4O1O3。”

"(2) H1, H2, H3, and H4 are on BE, AE, AC, and CF, respectively, and the quadrilateral H1H2H3H4≌quadrilateral O2O1O4O3." (PS, the author of the figure put it in this chapter)
Yue Wendao is full of confidence in his favorite work.

This question involves a very rare knowledge point, Mickle point, Newton line, and the complete quadrilateral theorem in Mickle's theorem. Ninety-nine percent of high school students in the world have never even heard of these terms, let alone Don't say you have mastered it.

Even if you know the entry point of this proof, how to make the auxiliary line in the future is the key!

Of the nine auxiliary lines, if any one is missing, it is impossible to prove it.

And just seeing this complicated geometric figure, most students will feel dizzy. If they want to find out how to make auxiliary lines correctly, they need to try and ponder over and over again, so that they can find clues and gradually complete each auxiliary line, and finally complete the proof process.

Even if you are a mathematical genius, it is impossible to do this without spending a day.

Thinking of this, a smug smile appeared on the corner of Yue Wendao's mouth.

But as soon as his smile appeared, he froze, because he clearly saw that Qin Ke was directly drawing a line on the picture in the answer area!
Make vertical lines, draw rays, connect BO4, BO2, O4H'...

Qin Ke completed all the auxiliary lines almost a few seconds after reading the question, and it was extremely correct!

This cannot be happening! The GOD must be playing a joke!

Under Yue Wendao's stunned and unbelievable eyes, the proof process flowed out continuously from Qin Ke's correct font:

“证明方法一：（1）设M为其密克尔点，则BM为圆O2与圆O4的公共统，可得出O2O4⊥BM，同理可得O2O3⊥DM……”

“∠O2O1O4=∠CAF+∠ACF=∠DFE，由此可证△O2O4O1∽△DEF，同理可证，△O∽△BCD4O1O3。第（1）小问到此完成证明。”

“（2）自O2作O3O4的垂线交BE于点H'，连接BO4……”

Yue Wendao couldn't believe his eyes, this was his proud work that he was so confident that he even felt that he had a chance to be selected into the "Classic Puzzles of Mathematical Olympiad History Museum"!
It was actually broken by Qin Ke in this way!
It still needs to be cracked very cleanly!

It didn't cause Qin Ke any trouble at all, and Qin Ke didn't even need time to meditate!
Thinking of what I said "modestly" in the question group: "My question is really not too difficult, it is just an ordinary IMO question. It takes about a day for the top candidates in the national competition. Time to study, should barely have a chance to make it..."

Yue Wendao's face was hot, as if someone had slapped him loudly.

Especially seeing Professor Wei and Professor Han's expressions from shock to numbness to sparkling eyes made Yue Wendao, who has always been the most face-saving and pushy, feel uncomfortable, as if these experts and professors were mocking him !
Just when Yue Wendao couldn't stay any longer, Qin Ke had already completed the first proof, the most difficult goal of the whole volume, and started to write proof two again!

Yue Wendao completely froze in place!

Professor Wei and Professor Han, who were on the sidelines, were overjoyed. They had completely forgotten that they were here to see Qin Ke's jokes in the first place.

Qin Ke's wonderful proof method fully activated their hearts as a mathematician, allowing them to regain their love for mathematics when they were young, and the heartfelt joy and excitement when they saw the exquisite solutions to overcome problems!

Genius, real genius!

This is the real math genius student!
Those gifted students I have seen before are not even half as good as this Qin Ke!
Several professors stared like nails at Qin Ke's ever-increasing answer sheet, their expressions fascinated, and the excitement and joy in their eyes couldn't be concealed. At this point, these professors no longer doubted whether Qin Ke could The "impossible" goal has been accomplished.

Because only 23 minutes had passed by this time, Qin Ke had completed seven major questions, and the first three most difficult questions were included!
The following eight big questions are relatively easy, and it is impossible for Qin Ke to be difficult!
It is very possible that Qin Ke could really finish this paper within an hour, and he used two methods!

My God, the historical record of the national competition for a century is about to be broken!

If there is no one before, I am afraid there will be no new records for those who come later!
A new record enough to go down in history!

And they are the witnesses to the birth of this new record!
The excitement and joy in the hearts of the professors is beyond words, and there is a sense of shock and satisfaction in witnessing history!

But in complete contrast to them, Yue Wendao blushed almost to the point of bleeding.

Looking at Proof [-] written by Qin Ke, it is a new proof method that even he did not expect. It is simpler and more efficient than the standard Proof [-]. It creatively combines the construction method and uses the extremely rare Carnot Theorem and Simson line, and complete quadrilateral fold quadrilateral inscribed in a circle theorem!

However, in about 2 minutes, the second method of proof is completed!
And there is no flaw!

My most proud work was dismembered like this to pieces!

Yue Wendao didn't have the face to stay any longer, he turned around and walked away quickly, because his movements were too hasty, he almost bumped into a candidate's desk.

For a face-saving person like him, being easily relieved of his pride in public is equivalent to slapping him face to face!

Many candidates were surprised to see the usually high-spirited, calm and pretending vice president leave hurriedly with a blushing face, and their hearts were full of small question marks.

Professor Wei, Professor Han and other professors didn't notice Yue Wendao's departure at all. They were excitedly watching Qin Ke solve one problem after another with unrivaled momentum!

After an unknown amount of time, Qin Ke finally put down his pen with a "snap".

The professors who watched him do the questions were shocked, and then regained consciousness as if waking up from a dream.

They watched Qin Ke conduct the final review, and saw a six-page paper, all the answer areas were filled with answers!
God!What happened just now was really not a dream!The miracle was born in our eyes!
Qin Ke didn't have time to care about the excited professors behind him. He rubbed his sore wrists and raised his hands, "Teacher!"

The invigilator came over hurriedly. He didn't dare to neglect the student who was being watched by several old professors all the time. He hurriedly came over and asked, "Student, what's the matter?"

"Excuse me, how long has it been since the exam started?"

The invigilator first glanced at his filled answer sheet in disbelief, then looked up at the hanging clock, and said in a trembling voice, "It's 9:56 now, 56 minutes before the exam starts..."

His voice was not loud, but it could be heard clearly in the whole examination room.

For a moment, countless horrified eyes cast over, all falling on Qin Ke.

"Teacher, I know it's not the shortest time to turn in the paper and leave, but I wrote in the "Challenge Letter" that I have to complete the paper in two ways within one hour. In order to avoid people questioning my ability to finish the paper Time, I would like to invite you and the teachers behind to make a proof together. I will hand in the answer sheet to the podium now and will not touch it anymore, but my people will still stay in the examination room and leave the room after an hour and a half , can you?"

There was an uproar in the audience.

My God, could it be that Qin Ke really completed the entire test within an hour as the challenge book said?
Yang Ye's face was ashamed, and he finally solved most of the problems and couldn't solve them anymore.

Probably... I'm really just an ordinary person... Probably, mathematics is not something that ordinary people like me can handle...

Yang Ye sat on the seat with blank eyes and lost all confidence.

But none of the crowd noticed him, and all eyes fell on the invigilator and Qin Ke.

The invigilator hesitated a little. Qin Ke's proposal had never appeared in the national competition, and the rules of the examination room did not say whether it was possible. If he agreed without asking for instructions, he might bear a certain risk. During the period of hesitation, Professor Wei and Professor Han have already spoken out.

They looked happy and said: "Yes, we are happy to be this witness!"

The other professors also happily said: "We are also happy!"

Seeing that several professors looked at Qin Ke with admiration, and they all spoke together, the invigilator felt relieved. This was equivalent to being guaranteed, and he hurriedly said: "Yes."

Qin Ke smiled at the professors and said, "Thank you!"

He got up and put the paper on the podium, then returned to his seat and yawned:
"I'll sleep for a while, trouble teacher, wake me up when the shortest time for submitting papers is reached, I'm afraid I'll leave the venue late, the milk tea bought for me by the two teachers waiting for me outside the venue is cold, and it won't taste good by then gone."

Under countless astonished and shocked gazes, Qin Ke just lay down like this, and soon let out a soft snoring sound.

He is really tired.

Concentrating all your energy for an hour, pushed your science ability to the extreme, and solved [-] extremely difficult national competition problems with two solutions in succession, and even one of them was considered difficult even if IMO Even with Qin Ke's strength, he couldn't hide his tiredness about the geometry problem.

His current state is very similar to that of an online writer. After working hard every day for ten consecutive days, he insists on updating more than [-] words, which consumes a lot of energy and is in a state where he can fall asleep at any time.

So after he finished speaking, he fell asleep on the table in a blink of an eye.

Professor Wei and the others froze for a while, then smiled and shook their heads. He also whispered to the three invigilators present: "You have to look at this candidate's paper, and no one is allowed to move it, even if it is dirty. A little bit, we will settle accounts with you!"

The three invigilators nodded hurriedly, and one teacher hurriedly took out a tissue and wiped the podium clean before putting Qin Ke's paper down again...

(End of this chapter)