This is a good topic, if you use analytic geometric representations to transform into algebraic problems, it is quite complicated, to establish triangle representations, inscribed and circumscribed representations of two circles, it is difficult for any triangle to describe the features.

Liu Sen has been thinking for a long time, and has been sketching the equation, the more he sketches, the more complicated it becomes, and the competition thinking tells him that there must be an extremely ingenious solution to this problem.

"Student Liu, it seems that your weekend vacation plan is going to be in vain. ”

Zhang Qian said with a dark face.

The rest of the students were also watching the excitement.

"Student Zhang, the question is also a technical job, can you have a little content?"

Zhang Qian almost choked to death.

"Wait until you do it, or it will only add to the joke. ”

Liu Sen was stunned, but his thinking did not stop for a moment.

Li Jie walked in unexpectedly, and when he saw this question, his brows furrowed, and he said: "This problem is good, through analytic geometry into algebraic equations, can it be proved that it is not good, Liu Sen, can you solve it?"

How could the old fox laugh so gloatingly?

Zhang Qian pouted, this is not his question, but to ask for help from idols, the real god of mathematics, Yu Zhanbo, a genius in Beijing, won the first prize when he participated in IMO last year, and he still participates this year.

Li Jie's eyes lit up, he thought of his thoughts, and shouted: "Except for Zhang Qian, if the other students take the lead in unraveling, it will be regarded as Liu Sen's defeat." ”

MMP, don't play like this, what if Zhang Qian tells other students the solution?

The old fox's face is like a thief, Liu Sen knows that he has been tricked, and he has no intention of approving the fake at all.

Yu Hanming looked at Zhang Qian with a look, you give me the answer, and I will kill Liu Sen.

Zhang Qian just didn't see it, why should he give it to you?

The bureau that he worked so hard to set up, why let his rival pretend to be forced, anyway, Liu Sen was also stumped, and he would never give the opportunity to others.

Li Jie did this, Liu Sen was also in a hurry, he had to have a particularly ingenious angle, drew six lines on the blackboard, and suddenly opened his mind, the old god was there, and said: "so-easy." ”

Li Jie's eyes lit up, and he was the first to get the knowledge points.

Zhang Qian frowned, and before he could react, he said, "Then you can explain, it's not okay to draw six lines." ”

Liu Sen sighed: "Yu Hanming, do you understand?"

Paralyzed, Yu Hanming really wants to scold, you can solve the problem, cue what am I doing? You are going to give me a set, I said I don't understand and have no face, I understand, maybe I have the cheekiness to let me talk about it.

This is a really difficult question to answer, with logic and game theory.

It's like a guy saying, "You're stupid, you're stupid." ”

Another person asked, "Are you stupid?"

The answer yes or no is incorrect.

Li Jie also asked: "Yu Hanming, do you understand?"

If you can see the solution from these six lines alone, it definitely belongs to active thinking, and if you can't see it, it is a level lower, and I still look forward to this batch of trainees in addition to Liu Sen, and someone can impact the national team.

Yu Hanming's face twitched, and his classmates were all looking at him at this time, and it was difficult to ride a tiger.

"Teacher, I suddenly have an upset stomach, so I'll go to the toilet first. ”

What a witty boy.

Zhang Qian said, "Teacher Li, you can't just draw a few lines, right?"

When he said this, it proved that he didn't figure it out, Li Jie was a little disappointed, this was his problem, he must know a solution, he thought about it deeply, and saw that the auxiliary line could not be bypassed, and it was obviously a lower qualification.

Liu Sen said: "Classmate Zhang Qian, you really disappointed me. ”

Zhang Qian: MMP, now I'm testing you.

"△ABC, the inscribed circle is k1, the circumscribed circle is k2, and the parallel lines of the opposite sides of the vertices of △ABC intersect to form △A1B1C1, which is a very ingenious idea and the most important step in solving the problem, △ABC is the midpoint triangle of △A1B1C1, △ABC∽△A1B1C1, and the radius of the inscribed circle of △A1B1C1 is 2r. Because each side of △A1B1C1 has at least one intersection point with the circle k2, and the tangent of the circle k2 is made parallel to each side of △A1B1C1, and △A2B2C2 is crossed, which can be proved. ”

So concise.

Li Jie clapped his hands and praised: "Great solution, if this problem is out in this IMO, Liu Sen's solution will have a chance to win a special award." ”

You must know that from 1985 when China began to participate in the IMO to the present, the IMO has produced more than hundreds of gold medals with full scores, and only 4 people have won the IMO special award, one in 1986, one in 1988, one in 1995, and the most recent one was won by Iurie Boreico in 2005, which has been hanging for more than ten years.

Zhang Qian's face turned red, and he was embarrassed for a while.

This is the problem and solution given by the great god in my mind, but it is not as concise as Liu Sen's solution, which is really not a small blow.

He silently returned to his seat, picked up the "Seventeen Lectures on the Olympiad of Mathematics" and looked as if nothing had happened.

Instead of dwelling on the gains and losses of the previous moment, it is better to devote yourself to the next moment of learning.

This is the motto that his father taught him since he was a child, and he has always practiced it.

Li Jie nodded slightly, his thinking was not active enough, but he had the toughness of a rock, and his mentality adjusted very quickly, no wonder he had such an achievement.

A true mathematical researcher must have such qualities, just like the recent master Wiles who tried to crack Fermat's theorem, and he has been the same for ten years.

Li Jie thinks about Liu Sen's character again, he is a senior sister, loves to pretend to be forced, and he is in the limelight, but it's a pity, this kid is okay to compete, if he studies mathematics in the future, I'm afraid it will not be as high as Zhang Qian's achievements.

After all, academic research is completely different from competition.

With this topic finished, the time has come.

Liu Sen and Li Jie walked out of the classroom together, and Liu Sen smiled and said: "Teacher Li, you said good rules, but you temporarily changed the hexagram, and the other students solved it unfairly." ”

"I'm just trying to provoke you, isn't that going to work? ”

is shameless, Liu Sen had to suffer this stuffy loss, and was almost caught.

During the evening break, Zhang Qian sent an email to the great god Yu Zhanbo, informing him of the solution to this problem, and did not say anything else.

The great god quickly replied.

"You actually thought of such an ingenious solution, and you think that this year has made a lot of progress, this time CMO, we are competing together. ”

"This is our provincial Liu Sen's explanation, he is the first perfect score in the provincial league this time. ”

"Your province has finally produced a talent, and I'll see you in the field then. ”

Liu Sen was very happy, he could play for two days on the weekend, Tang Xiaoya called over and said angrily: "It's all your fault for agreeing blindly, Teacher Li can't give you leave." ”

"I'm sorry, Mr. Li approved me for two days off. ”

"No way. Tang Xiaoya didn't believe it, Teacher Li was notoriously cunning and harsh.

Liu Sen said the fight again, and said to himself: "Can those scraps stump me?" Hey, Mr. Li wants to trap me, but he doesn't know that he is one foot tall, and I am one foot taller. ”

"I think you're the monster. Tang Xiaoya was not angry, "Then you are free, right, go out with me in the evening." ”

has the demeanor of a big sister, let's talk about sister, can you not be so rude when you ask someone?

"Where?"

"You'll know when you get there. ”

I wiped it, and I didn't say it, Liu Sen was very curious.