Super Xueba: Start With A Low Control Score!

Chapter 306 The depth of mathematics, the average person really can’t grasp it!

The next day!

Eight in the morning!

The second round of this IMO has begun!

The duration is still four and a half hours.

Most of the contestants, such as Jack Tom, Fang Wu Zhongjie, Miyazaki Mae, and Kim Seung-won, are about to drip black water.

Before they started the exams, they had all arrogantly flaunted in front of the Dongyun team.

but now……

All right!

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. In the second round, only two to three hundred people participated. There were only a few test rooms.

anyway.

They couldn’t avoid the fate of being in the same examination room with the Dongyun team.

So that their mentality is a little broken.

But the Dongyun team naturally didn’t have any pressure.

Jiang Nan finally got the second round of papers.

And the next second.

His eyes condensed slightly.

“This is a good question!”

“It’s much more interesting than a round!”

Almost subconsciously, Jiang Nan exclaimed.

Um!

If I remember correctly.

This is the first time he has looked at Mathematical Olympiad problems with appreciation since he participated in the competition.

have to say!

This is simply a miracle.

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

“perhaps!”

“This is the real International Olympiad!”

“Finally, it didn’t disappoint me too much. Although it is not very challenging, it is a bit challenging after all!”

“This trip to the Olympic Mathematics is also complete!”

After the admiration!

Jiang Nan started to solve the problem excitedly.

That’s right!

It’s just excitement.

How long?

He really didn’t know how long he hadn’t encountered this kind of math problem that made him feel a little challenging.

I have to say I miss it very much!

first question……

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

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

Please prove…

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

[Ps: This is one of the five most difficult problems in the history of IMO, but the symbols can not be typed, and the picture can not be drawn! 】”

“…”

The content of this question is not long.

But after careful consideration, it is indeed somewhat difficult.

certainly!

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 circle O is r, then there is: arc PQ + arc RS = 4ar.

With this tiny lemma, we can estimate 1/OiOj, and then count the traversal.

Proof of Lemma…

As shown in 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.

Use the lemma to estimate 1/0i0j. ”

“…”

“…”

Less than five minutes of work!

Jiang Nan solved the first question.

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

The topic is not difficult.

It’s just very interesting. The candidate must have a very solid foundation, otherwise it will be cool.

But for Jiang Nan, that’s it!

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

In the first round.

Even the finale.

He could see five solutions at a glance.

But in the second round, the first question started, and he actually saw only four solutions.

This is incredible.

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

That means that he has to use more solutions for each question, so that each question can be added by one point, and he will get 48 points in the end.

But the first question has only four solutions. What about the second question, the third question, and the sixth question?

The more I think about it!

Jiang Nan is more excited!

It took another ten minutes to write the other three solutions for the first question, and then looked at the second question.

This is a mixed problem of algebra and set.

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 make them.

The hard part is to use multiple solutions.

But Jiang Nan also spent twenty minutes.

Four solutions are easy to get.

Then there is the third question.

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

The breakthrough lies in the lemma or similar ideas.

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

In the same twenty minutes, the four solutions were perfectly completed.

Followed by the fourth question, the fifth question.

Jiang Nan only spent more than 20 minutes on each of the four solutions, plus the first three questions, which took a total of more than 100 minutes, all were done, and finally came to the final finale.

“what?”

“This question is really amazing!”

“I was told that there are four solutions, even if there are three, I will not notice it for a while, but only have double solutions?”

“Can you solve it?”

“It doesn’t necessarily add 1 point, right?”

“…”

Jiang Nan frowned slightly.

As early as in the domestic Olympics finals, Lin Yu told him that people in front of him had overscores, but that was three solutions.

In other words…

If he wants to get 100% over-score, at least three kinds of solutions must be used, and there are two kinds of uncertainty.

If the scoring teacher is upset and doesn’t give him a point, wouldn’t it be a failure of the task?

“F*ck!”

“Although the system point reward is important when it is said to be important, it is not important at all if it is said not to be important!”

“but……”

“Who wants to grab my points, I’m anxious with him!”

“In this exam, you have to score 48 points anyway. Even if Jesus blocks it, it won’t work, I said!”

“…”

Almost instantly.

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

Eight minutes!

The first solution to the finale is done.

Another ten minutes!

The second solution to the finale is done.

But the third solution…

“So…and so…and then use Veda jump, and add infinite descent method, Euler’s formula can also be used…”

About twenty minutes passed.

No matter how small the handwriting is, Jiang Nan has already written densely in the blank spaces of the paper.

If you don’t see it carefully…

Others thought he was writing the Morse code.

The fonts are like tadpoles, and when combined, they become a series of strange symbols.

But Jiang Nan let out a long breath and finally stopped the pen, “Huh, I’m finally done!”

“This question is indeed a bit difficult, at least it cost me seven successes before I made the third solution!”

“Moreover, there is no fourth solution behind it. Even if there is, it will overlap with the front. It doesn’t make much sense!”

“But even so, the scoring teacher must add one point, there should be no problem????”

“…”

Check it a little bit.

After making sure that there is nothing left out.

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

have to say.

This exam was definitely Jiang Nan’s longest time-consuming exam. The six questions combined took him nearly 150 minutes, or two and a half hours.

The exam time is only four and a half hours.

He actually spent more than half?

You should know that he used to spend a few minutes, dozens of minutes, and it took only an hour to get it done!

Gee!

Today is simply a failure.

“Mathematics is really deep! Unless you are like me, ordinary people really can’t grasp it!”

“I have never seen such a difficult question before, and I don’t know who made it. It’s quite interesting!”

“Originally thinking about the captain, it should be fine for Lao Yin and the others to get full marks in the second round, but now it seems to be a little susceptible!”

“Forget it, ask for your blessings!”

“…”