Super Scholar: Start with a low regulatory score!

Chapter 305 The Depth of Mathematics, Most People Can't Grasp It!
The next day!

Eight in the morning!

The second round of this IMO has begun!

Still four and a half hours long.

Most of the contestants, such as Jack Tom, Fang Wu Zhongjie, Miyazaki Mae, and Jin Chengyuan, have black faces, and they are about to drip black water.

Before the start of the exam, they used to show off their power in front of everyone in Dongyun team.

but now……

Ok!

When they saw the Dongyun team, they all walked around.

Don't even dare to fight face to face.

only……

In the first round, more than two-thirds of the people were eliminated, and in the second round, there were only two to three hundred participants, and there were only a few examination rooms.

anyway.

They couldn't avoid the fate of being in the same examination room as Team Dongyun.

So much so that their mentality collapsed.

But the members of the Dongyun team naturally had no pressure, and they came to the examination room to prepare for the battle with arrogance.

Jiang Nan finally got the second round of papers.

And the next second.

His eyes narrowed slightly.

"That's a good question!"

"It's much more interesting than round one!"

Almost subconsciously, Jiang Nan admired.

Ah!
If I remember correctly.

This is the first time since he participated in the competition, he looked at the Mathematical Olympiad problem with admiring eyes.

have to say!

It was nothing short of a miracle.

It is conceivable that the six questions on the second round of the International Olympiad paper are really level.

"perhaps!"

"This is the real International Mathematical Olympiad!"

"Finally, it didn't disappoint me too much. Although the challenge wasn't too great, it was still somewhat challenging!"

"This trip to the Olympiad is considered complete!"

After admiring!
Jiang Nan began to solve the problem excitedly.

That's right!

Just excited.

How long?
He really didn't know how long it had been since he had encountered such a slightly challenging math problem.

I have to say I miss it very much!

first question……

"Assume that n greater than or equal to 3 is a given positive integer, and C1, C2, ..., Cn are unit circles with a radius of 1 on the plane.

The corresponding circle centers are denoted as O1, 02, ..., 0n respectively, assuming that any straight line intersects or is tangent to at most two of the unit circles.

please prove...

All 1/OiOj (1 less than or equal to i less than j less than or equal to n) less than or equal to (n-1) π/4.

[ps: This question is one of the five most difficult problems in the history of IMO, but the symbols cannot be typed out, and the pictures cannot be drawn! 】”

"..."

The content of this question is not long.

But if you think about it carefully, it is indeed a bit difficult.

of course!

It's just a little difficult!

The key to the proof lies in the following lemma...

"Lemma: As shown in the figure (omitted), if the radius of the circle O is r, then there is: arc PQ+arc RS=4ar.

With this tiny lemma, 1/OiOj can be estimated, and then counted through the traversal.

The lemma proves that...

As can be seen from the figure above, Lambda λ+μ=2a.

therefore……

Arc PQ+arc RS=2λr+2μr=4ar.

Back to the original question: make a circle S with a sufficiently large radius r, and include the unit circles C1, C2, ..., Cn in the circle S.

Estimate 1/0i0j using the lemma. "

"..."

"..."

Less than 5 minute of work!

Jiang Nan solved the first question.

In fact, there is only one core point, which is to consider the integrity of circle cutting while using inequality scaling.

The subject is not difficult.

It's just very interesting. Candidates are required to have a very deep and solid foundation, otherwise they will be cool.

But for Jiangnan, so be it!

In fact, what really made him challenging was not to solve this problem, but to use multiple solutions.

In the first round.

Even the finale question.

He can see five solutions at a glance.

But this second round started with the first question, and he only saw four solutions.

This is incredible.

You must know that the system task is to let him continue to exceed the score in the second round, and it is over 6 points, 48 ​​points in the test!

That means that he has to use multiple solutions for each question in order to add one point to each question, and finally get 48 points.

But there are only four solutions to the first question, so what about the second, third, and sixth questions?
Think more!

The more excited Jiangnan is!

It took another 10 minutes to write out the other three interpretations of the first question, and then looked at the second question.

This is a mix of algebra and sets.

Nothing else to say.

As long as you have a better understanding of Lagrange's theorem, partial derivatives and deviation calculations, it is not difficult to do it.

The difficulty is to use multiple solutions.

But Jiangnan only took 10 minutes.

Four solutions are easy to get.

Then there is the third question.

It is another combination question, which can also be called a windmill question.

The breakthrough in the answer lies in the lemma or similar ideas.

By changing the sign to narrow the scope of the discussion, this discussion can be likened to the discrete intermediate value theorem.

The same two and 10 minutes, the four solutions are perfectly done.

Followed by the fourth question, the fifth question.

It only took Jiangnan more than 20 minutes to solve the four solutions, plus the first three questions, it took a total of more than 100 minutes to solve all of them, and finally came to the finale.

"what?"

"This question is really amazing!"

"It is said that there are four kinds of solutions, even if there are three, I didn't see it for a while, but there are only two solutions?"

"Double solution?"

"It may not be possible to add 1 point, right?"

"..."

Jiang Nan frowned slightly.

As early as in the domestic Olympiad finals, Lin Yu told him that the previous people had scored overscores, but they made three solutions.

That is to say...

If he wants to get 100% super points, he must use at least three solutions, and there are uncertainties in the two.

If the marking teacher is in a bad mood and doesn't give him a point, wouldn't it be a mission failure?

"Nima!"

"Although the system point reward is very important if it is important, it is not important at all if it is not important!"

"but……"

"Who wants to grab my points, I'm in a hurry with him!"

"In this exam, you have to score 48 no matter what, even if Jesus stops it, it won't work, I said it!"

"..."

Almost in an instant.

Jiang Nan's face sank, that was his serious look.

8 minutes!
The first solution to the finale problem is solved.

Another 10 minutes!

The second solution to the finale question is solved.

However the third solution...

"This way... and that... Then we can use the Veda jump, plus the infinite descending method, Euler's formula can also be used..."

After about 10 minutes.

No matter how small Jiangnan's handwriting is, he has already written densely in the blank spaces of the paper.

If you don't look carefully...

Others thought he was writing Morse code.

Each font is like a tadpole, and when combined, it becomes a series of extremely strange symbols.

But Jiang Nan let out a long breath, and finally stopped writing, "Phew, it's finally done!"

"This question is indeed a bit difficult. It took at least seven of my strength to come up with the third solution!"

"And there is no fourth solution in the back. Even if there is, it will overlap with the previous one. It doesn't make much sense!"

"But even so, it must be no problem for the marking teacher to add one point, right???"

"..."

Check it out a little bit.

After making sure nothing was missed.

Jiang Nan put away his pen, got up and handed in the paper.

have to say.

This exam was definitely the longest exam Jiangnan took. The six questions took him nearly 150 minutes, or two and a half hours.

The test time is only four and a half hours.

He actually spent more than half of it?

You must know that he used to do it in a few minutes, a few 10 minutes, and a few hours to get it done!

Gee!

Today is simply a failure.

"Mathematics is really deep! Unless you are like me, most people really can't grasp it!"

"I've never seen such a difficult question before, and I don't know who came up with it. It's quite interesting!"

"Originally I thought about the captain, Lao Yin and the others should have no problem getting full marks in the second round, but now it seems a little bit hanging!"

"Forget it, let's hope for good luck!"

"..."

ps: Shui Qun [Abandoned Umbrella Ear Wine Spirit, Baba Abandoned Ear] went yesterday, and it has been delayed for a long time, everyone come in to play!

I wanted to watch the blood moon total eclipse at night, but it was raining all the time, and the condition was average, so there are only two chapters!
(End of this chapter)