At this time, Qin Ke had already begun to examine the questions, and Yue Wendao also focused on the seventh question, which he was most proud of.
"7. As shown in the figure below, in the complete quadrilateral ABCDEF, O1, O2, O3, and O4 are the circumcenters of △ACF, △BCD, △DEF, and △ABE respectively. H1, H2, H3, and H4 are respectively △O4O2O3 and △O4O1O3.
, the vertical center of △O2O4O1, △O1O2O3, please verify: "
"(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 ≌ the quadrilateral O2O1O4O3." (PS, the author of the attached picture has put it in this chapter)
Yue Wendao is full of confidence in his proud work.
This question involves a very rare knowledge point, Mickel points, Newton's lines, and the perfect quadrilateral theorem in Mickel's theorem. Ninety-nine percent of high school students in the world have never even heard of these terms, let alone
Not to mention mastered.
Even if you know the entry point of this proof, how to draw the auxiliary lines later is the key!
If any of the nine auxiliary lines is missing, it will be impossible to prove it.
Just seeing this complicated geometric diagram will make most students dizzy. If they want to find out how to correctly draw auxiliary lines, they need to try and ponder over and over again before they can uncover the clues and gradually complete the auxiliary lines one by one.
line, finally completing the proof process.
Even if you are a mathematical genius, it is impossible to do this without spending a day.
Thinking of this, a proud smile appeared on the corner of Yue Wendao's mouth.
But as soon as his smile appeared, he froze because he clearly saw Qin Ke drawing lines directly on the picture in the answer area!
Draw vertical lines, draw rays, connect BO4, BO2, O4H'...
Qin Ke completed all the auxiliary lines almost within a few seconds of reading the question, and it was extremely correct!
This cannot be happening! The GOD must be playing a joke!
In Yue Wendao's stunned and disbelieving eyes, the proof process flowed out smoothly from Qin Ke's straight handwriting:
"Proof method one: (1) Assume M is its Mikel point, then BM is the common unity of circle O2 and circle O4, and we can get O2O4⊥BM. In the same way, we can get O2O3⊥DM..."
"∠O2O1O4=∠CAF ∠ACF=∠DFE. From this it can be proved that △O2O4O1∽△DEF. In the same way, it can be proved that △O∽△BCD4O1O3. Question (1) has now completed the proof."
"(2) The perpendicular line from O2 to O3O4 intersects BE at point H' and connects BO4..."
Yue Wendao couldn't believe his eyes. This was his proud work that he was so confident that he even thought he had a chance to be selected into the "Olympic History Museum Classic Problem"!
It was actually cracked by Qin Ke like this!
It still needs to be cracked very cleanly!
It didn't cause any trouble to Qin Ke at all. Qin Ke didn't even need time to think!
I thought about what I had said "humbly" to the question group: "My question is really not too difficult, it is just an ordinary IMO question. The top national examination candidates will spend about a day on it."
With time to study, I should still have a chance to make it..."
Yue Wendao's face was burning, as if he had been slapped loudly.
Especially when he saw Professor Wei and Professor Han next to him, their expressions went from shock to numbness to sparkling eyes, which made Yue Wendao, who always cared about face the most and loved the limelight the most, feel uncomfortable all over, as if he felt that these experts and professors were mocking him.
!
Just when Yue Wendao couldn't stay any longer, Qin Ke had completed the first proof of the most difficult task in the whole volume, and started writing the second proof again and again!
Yue Wendao was completely frozen in place!
Professor Wei and Professor Han who were watching were beaming with joy. They had completely forgotten that they came to see Qin Ke's joke in the first place.
Qin Ke's wonderful proof method completely activated their true nature as mathematicians, allowing them to regain their love for mathematics in their youth, as well as their heartfelt joy and excitement when seeing ingenious solutions to difficult problems!
Genius, real genius!
This is a truly talented student in mathematics!
The talented students I had seen before were not even half as good as this Qin Ke!
Several professors' eyes were fixed on Qin Ke's answer sheet, which was constantly adding content. Their expressions were intoxicated, and the excitement and joy in their eyes could not be concealed. At this point, these professors no longer doubted whether Qin Ke could complete the "
Impossible” goal.
Because only 23 minutes had passed at this time, Qin Ke had already completed seven major questions, including the top three most difficult questions!
The next eight big questions are relatively easy, and it is impossible to defeat Qin Ke!
Qin Keji could really finish this paper in one hour, and he could do it in two ways!
Oh my gosh, the historical record of the national competition over the past century is going to be severely refreshed!
It is also unprecedented, and I am afraid it will be difficult for anyone to set a new record in the future!
A new record worthy of being recorded in history!
And they are the witnesses to the birth of this new record!
The excitement and joy in the professors’ hearts are simply indescribable, and they even have a sense of shock and satisfaction of witnessing history!
But in complete contrast to them, Yue Wendao's face was so red that he almost bled.
Looking at the proof two written by Qin Ke, it turned out that even he had not expected it. It was a new proof method that was simpler and more efficient than the standard proof method one. It creatively combined the construction method and used the extremely uncommon Carnot theorem and
Simson lines, and the theorem that a perfect quadrilateral is inscribed in a circle!
In about two minutes, the second method of proof is completed!
And there are no flaws!
My most proud work was dismembered into pieces like this!
Yue Wendao didn't have the nerve to stay any longer. He turned around and left quickly. Because his movement was too hasty, he almost bumped into a candidate's desk.
For a person like him who is obsessed with face, being proud of being easily solved in public is equivalent to a loud slap in the face!
Many candidates were surprised to see the usually high-spirited and somewhat pretentious vice-president leave in a hurry with a red face, and their hearts were filled with little questions.
Professor Wei, Professor Han and other professors did not notice Yue Wendao leaving at all. They were excitedly watching Qin Ke solve the questions one after another with unrivaled power!
I don't know how long it took, but with a "pop" sound, Qin Ke finally put down his pen.
The professors who were watching him do the questions were shocked, and then they regained consciousness as if they were waking up from a dream.
They watched Qin Ke conduct the final review and saw that the six-page paper was full of answers in all the answer areas!
Oh my God! What just happened was really not a dream! The miracle was born right before our eyes!
Qin Ke had no time to pay attention to the excited professors behind him. He rubbed his sore wrists, raised his hands and said, "Teacher!"
The invigilator hurriedly came over. He did not dare to show any slightness to this student who had received the full attention of several old professors. He hurriedly came over and asked: "Classmate, what can I do for you?"
"Excuse me, how long has it been since the exam started?"
The invigilator first looked at his filled-in answer sheet in disbelief, then looked up at the hanging clock, and said with a trembling voice: "It's 9:56 now, 56 minutes until the exam starts..."
His voice was not loud, but the entire examination room could hear him clearly.
For a time, countless horrified eyes came over, and they all fell on Qin Ke.
"Teacher, I know it's not the shortest time to hand in and leave, but I wrote in the "Challenge" that I will complete this paper using two methods within an hour. In order to avoid anyone questioning my intention to finish the paper
Time, I would like to ask you and the teachers behind you to make a proof together. I will hand the answer sheet to the podium now and will no longer touch it, but my people will still stay in the examination room and will leave after an hour and a half.
, is that okay?”
The whole place suddenly became commotion.
Oh my god, could it be that Qin Ke really completed the entire paper in one hour as the challenge stated?
Yang Ye's face was ashen. He finally solved most of the problems and could no longer solve them.
Maybe... I'm really just an ordinary person... Maybe, mathematics is not something that ordinary people like me can do well...
Yang Ye sat in his seat with dull eyes, completely losing his self-confidence.
But no one noticed him, all eyes fell on the invigilator and Qin Ke.
The invigilator was a little hesitant. Qin Ke's suggestion had never appeared in the national competition, and the examination room regulations did not say whether this was allowed. If he agreed without asking for permission, he might bear certain risks.
During the period of hesitation, Professor Wei and Professor Han spoke out generously.
They looked delighted and said, "Yes, we are happy to be this witness!"
The other professors also happily said: "We are happy too!"
Seeing several professors looking at Qin Ke with admiration in their eyes and speaking out in unison, the invigilator felt relieved. This was equivalent to someone being a guarantee, and he hurriedly said: "Okay."
Qin Ke smiled at the professors and said, "Thank you!"
He stood up and put the paper on the podium, then returned to his seat and yawned:
"I'm going to sleep for a while. Please wake me up when the minimum handover time is up. I'm afraid I'll be leaving late. The milk tea bought for me by the two teachers waiting for me outside the venue will be cold and it won't taste good by then."
Already."
Amid countless surprised and shocked looks, Qin Ke lay down and soon began to snore softly.
He is really tired.
For an hour, I concentrated all my energy and pushed my science ability to the extreme. I used two solutions to solve fifteen extremely difficult national competition questions. There was even one question that was considered difficult even IMO.
Even with Qin Ke's strength, he was tired of geometry problems.
His current state is very much like that of an online article writer. After working hard for ten consecutive days, he insists on updating more than 10,000 words every day, 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 an instant.
Professor Wei and others were stunned for a while, then he smiled and shook his head. He also whispered to the three invigilators present: "You must take a good look at this candidate's paper. No one is allowed to touch it, even if it is soiled."
Little by little, we will settle accounts with you!"
Chapter completed!