Reborn Xueba, you don't really think that studying is difficult, do you?

"Proof: Let the extension line of EF intersect the extension line of BC at J (the drawing is omitted)."

"Conclusion 1: JD is tangent to the circumscribed circle of △ABC and DEF."

"We divided the proof into the following small parts."

"(1) The four points B, F, E, and C share a circle. Make a circumcircle of △ABC, extend AD to intersect the circumcircle at M..."

"(2) The circumscribed circles of △DBC and △DEF are tangent, let EF intersect AD at H, and the conclusion of (1)..."

"(3) JD is the common tangent of two circles. By Monge's theorem..."

"Conclusion 2: The four points X, B, E, and T share a circle..."

"Conclusion 3: X, K, E, D and B, D, T, K share a circle..."

"Conclusion 4: JK=JD."

"The transformation with J as the inversion center and JD^2 as the inversion power changes E into F, C into B, A into T..."

The process is very complicated.

After all, it is an IMO real test candidate. The normal problem-solving time is four and a half hours for three questions, that is, one and a half hours for each question, or nine and 10 minutes.

This is also the reason why Ge Tianxing said before that Lin Bei and Zhao Qinghan could make it within 45 minutes, so he admitted that they are good. After all, this is really not an ordinary difficulty.

Even though Lin Bei and Zhao Qinghan had compressed their handwriting to the limit, they each wrote half a blackboard.

It is densely packed, just like tadpole text.

When outsiders saw it, they thought they had fallen into the maggots, and they were scared to death, or half to death.

But even after writing so much, regardless of whether it was Lin Bei or Zhao Qinghan, it took very little time.

5 minutes?
No no no, definitely not 5 minutes.

Not even 4 minutes.

It took Lin Bei about three and a half minutes, and Zhao Qinghan only took less than three minutes, which can be described as breaking the sky.

Then, no more.

Because both the process and the answer have already been revealed.

If you don’t believe it, everyone can try it. Just prove the above four conclusions, and you can get the final answer, that is, the straight line BC, EF and O1O2 must have the same point.

In fact, this question is said to be complicated, but it is quite complicated. It is best not to do it if you do not have a deep study of Mathematical Olympiad.

After all, doing it is a waste of time.

Don't say one and a half hours, even if it is doubled, or even given a day, if you can't do it, you can't do it.

But for Lin Bei and Zhao Qinghan, it is quite simple, and their thinking can be naturally divided into two parts.

"1: Exploring the nature of point D and dealing with the intersection of two limits. 2: Observing the co-circle of B, F, E, and C in the exploration of D is a key step."

It is not difficult to find tangent conclusions in the proof process.

After the conclusion of tangency is obtained, it is a common method to use the property of inversion to deal with the multi-circle problem.

However, compared with more conventional inversion questions, how to use point X in this question will cause some troubles. Although the proof is simple, it is not easy to consciously observe the co-circles related to X.

However, this naturally did not bother Lin Bei and Zhao Qinghan.

I saw that the movements of the two were done in one go, smooth and not separated, like flowing clouds and flowing water, which dazzled people.

then……

"Dig... dig grass..."

Seeing this scene, Ge Tianxing, who was the author of the question, finally couldn't bear it any longer, and swears directly.

He was so shocked by Lin Bei and Zhao Qinghan's performance that he didn't know what words to use to describe it. He could only dig the grass non-stop and grabbed the few hairs with both hands.

Nima!

That's the real question IMO!

It is difficult for others to make it in 90 minutes, even if he does not put in a lot of hard work, it is difficult for him to do it, but Lin Bei and Zhao Qinghan can make it in three minutes.

As if you don't need to think at all, as long as you have hands, is this still human?

It's simply a monster!

Peerless evildoers that are rare to see in a hundred years, but today he bumped into two of them at once.

Gee.

I don't know what luck he had.

As for the answer, don't look at it at all, it's 100% correct, and it's even more concise than the standard answer he prepared, it's not justified not to give two full marks.

If he himself is the strongest king, then Zhao Qinghan is the king of the Hundred Stars, and although Lin Bei is less than the Hundred Stars, after all, the time is a bit slower, but he is infinitely close, and he is 98 stars. kind.

Ge Tianxing just wanted to say: "You two big brothers, you should accept your supernatural powers! I can't stop shouting!"

But Lin Bei and Zhao Qinghan obviously won't let him go easily. After all, it's rare to come across a good fat sheep... Ahem, it's a good machine for making questions... Isn't it the person who made the question?
The question just now has a taste of minced meat and eggplant, which can strengthen the muscles and bones, and grow meat and strength.

Undoubtedly, this Ge Tianxing still has some skills, much better than those of the usual teachers. Although the questions he asked are not difficult, they are still of a certain level, and there is still a possibility of digging.

Regardless of whether it is Lin Bei or Zhao Qinghan, they are still looking forward to the big meal later.

If Ge Tianxing wants to leave without making a few servings of lobster, abalone/fish, pork belly and sea cucumber, it is just a dream.

I saw the two of them looked at each other with strong fighting intent in each other's eyes, and then looked at Ge Tianxing, who was swearing, "Mr. Ge, we have finished this question. Look at the answers. You can move on to the next question."

"And we all hope that your next question will be a little more difficult. After all, this question is too simple and not very chewy."

"☆..:*(hee Д`hee)..:*☆!"

Ge Tianxing: "┗─y(炸`) you"

He tightly covered his chest with his hands, and almost spat out a mouthful of reverse blood without holding back.

This is really, really shocking!
Ge Tianxing has lived most of his life, and he has never been hit like this by anyone, especially by two little dolls.

"You...you..."

Ge Tianxing wanted to speak, but he couldn't say it anyway, so he was really pissed off.

Then he simply swallowed all the words back, just picked up a chalk eraser, and "scrubbed" everything on the blackboard. After all, Lin Bei and Zhao Qinghan's answers were correct, and there was no need for him to check again. Just wipe it off.

Of course, it's not that he doesn't recognize Lin Bei and Zhao Qinghan's problem-solving process, but that he really wants to fight, put all his eggs in one basket, and fight Lin Bei and Zhao Qinghan to the end.

only see...

"Da da da da da!"

Accompanied by the melodious sound of chalk, several super-difficult Mathematical Olympiad questions jumped onto the blackboard.

That's right, it's just counting, and each one is more difficult than the one just now.

Ge Tianxing was also going all out, and he was going to throw out all the hidden goods in his belly at the bottom of the box. If Lin Bei and Zhao Qinghan still couldn't be troubled, then he would have nothing to do.

And the other side.

When Lin Bei and Zhao Qinghan saw this scene, their eyes lit up instantly, as if they saw some kind of rare treasure, they smiled openly, and even showed 24 big white teeth.

"Giggle!"

"Finally, there is a question that really catches the eye."

"It's not just one dish, but multiple dishes, and each dish is comparable to braised chicken wings in oil, and braised pork ribs in soy sauce!"

"If you kill them all, even if you can't be full, you should be barely half full, right?"

"After being hungry for so long, I can finally have a big meal. I must be very comfortable afterwards."

"Even if you can't be ecstatic and happy like a fairy, you should be able to be happy physically and mentally."

"(￣y▽￣)~* cover your mouth and laugh!"