The mathematics competition team of Yunze Province began to return under the leadership of Lao Meng.

On the way, I met a group of players from other provinces.

"Woo woo, Mr. Guo, I don't deserve to go to Qingbei......"

"Lao Guo, you're right, I'm only worthy of a second-rate garbage school like Jiangcheng, and I'll change my mind when I go back."

……

This is a familiar conversation.

How do you say it?

It can only be said that the Bosuk-Ullam theorem states that any, well, any continuous function from the n-dimensional sphere to the Euclidean n-dimensional space, must map a certain pair of pairs to the same point......

This mapping theorem is the same when applied to life!

Yi Cheng let out a sigh in his heart.

In other words, a happy life has its own happiness.

Unfortunate lives are always alike.

……

After returning to the hotel, Mr. Meng recorded the questions according to the memories of the contestants and reviewed them for everyone.

……

The next day, the second test began.

From half past eight to half past twelve.

The time was still four and a half hours.

Each question is still worth 21 points.

Pen and paper rustle in the examination room.

It's like it's raining.

It's just that this kind of moisturizing and silent silence is more terrifying than the real battlefield.

In the examination room of Yicheng, 40 top brains entered flow mode.

The first question is divided into points:

Proof that a^4-1 is divisible by 240 when the prime number a is greater than or equal to 7.

The question is very simple.

It is a student who participates in the Olympiad math competition.

In general, the self-esteem of the contestants is taken care of, so the questions are not too difficult.

This question is indeed a scoring question.

There are so many number theory theories related to diversion.

Yi Cheng only glanced at it and knew that Fermat's theorem should be used for this problem.

It is impossible for others not to know.

Yi Cheng doesn't expect to rely on it to score points, but only hopes that the last two questions can be more difficult.

At the very least, don't cut the cake below the level of yesterday.

Fermat is a man who is famous all over the world because when he read the book Diophantine wrote, next to the 8th proposition in Book 11: "It is impossible to divide a cubic number into the sum of two cubic numbers, or a power of four into the sum of two powers of four, or in general to divide a power higher than a quadratic into the sum of two powers of the same power." I'm sure I've found a wonderful way to prove this, but unfortunately the blank space here is too small to write. ”

This is the very famous Fermat's theorem, which began in 1637 and was not finally proved by the British mathematician Andrew Wiles until 1986.

It is also because of Fermat's skin that there will be a blank page at the back of the math book published after that, so as to prevent others from having excuses that they can't write.

Fermat was a man who changed the history of mathematics and the production of mathematics textbooks.

However, many people are not very familiar with Fermat's theorem.

Or people who are not majoring in mathematics have rarely heard of Fermat's theorem.

This thing is a terrible existence that has become the four major theorems of number theory along with Euler's theorem, Sun Tzu's theorem and Wilson's theorem in China.

So, what does Fermat's theorem tell about?

It says:

If p is a prime number and the integer a is not a multiple of p, then there are a^(p-1)≡1(mod p)

……

Then the proof of this problem is very simple.

Without thinking, Yi Cheng put pen to paper and wrote-

Proof:

The prime number a is greater than or equal to 7, and a is an odd number.

a^4-1=(a-1)(a+1)(a^2+1)

Moreover......

By Fermat's theorem there are:

（3，a）=1

（5，a）=1

So......

Finally, it was proved that:

240|(a^4-1)

……

It took 10 minutes for Yi Cheng to prove the first question and start to tackle the second question.

There are two questions to this question:

[Let's say you live in 13th-century Rome and you have 10 integer grams of weight and a scale.]

Now the king wants you to measure something on him.

The weight of this item is between 1 and 88 grams.

1. Can you do it? Can you do this without even a single weight?

2. Increase the number of weights to 12, among which there can be weights of the same weight, and use the balance to measure an item that the king gives you.

This item is between 1-59 grams.

Can you do it, even without any two weights? 】

After reading the question, Yi Cheng had at least 4 different ways of proving it in his mind.

But what's a little strange about this question is that-

It sets out the context of the times.

You live in the 13th century, and it's Europe.

European mathematics was still relatively backward at this time, and it was just beginning to recover from its decline.

Therefore, the method that Yi Cheng can use to prove the problem can only be before this period.

He first tried to disassemble the topic -

Take n weights, and record the weight of the ith weight as Fi

For an object weighing w, its weight can be measured with n weights.

When n=1, F3=F2+F1=2

Therefore, when F3-1 = 1 and w = 1, it can obviously be measured.

Then we will discuss the situation in the case of n and n+1......

By inductive hypothesis......

You can get the proof of the first question.

Here, after many enumerations, Yi Cheng found some patterns -

What a beautiful digital relationship.

There is only one thing that explains such a beautiful number relationship:

Fibonacci sequence.

Fibonacci was a mathematician of the early 13th century, and the application of his theories would not contradict the principles of the context of the time.

So, when the law is found to be a Fibonacci sequence, it is much easier for the second question.

Yi Cheng put pen to paper and wrote——

Constructing a generalized Fibonacci sequence:

g(n)=g(n-1)+g(n-3) (n is greater than or equal to 4).

g(1)=g(2)=g(3)=1.

Using the inductive hypothesis, it can be shown that for such n weights, even if two of them are arbitrarily removed, an object weighing 1 to g(n+1)-1 can still be weighed.

and g(13)=60.

So the second question is proven.

12 weights that meet the question can be found to weigh objects in the range of 1-59.

Finish the question.

Yi Cheng closed his eyes and savored it.

I have to say that the person who wrote the question is really great.

At least he gave people a taste of what the beauty of mathematics is in this problem.

Not only because the Fibonacci sequence is the golden section, but also because it is artistically beautiful.

More importantly, this question reflects the beauty of mathematics from exploration to conjecture to proof.

Gee.

Yi Cheng smashed his lips, and after reveling in it, he continued to tackle the last big question.

Only a third of the time has passed.

The last question is a proof question:

Let S be the parabola z=(x^2+y^2)/2 in R^3, and P(a,b,c) be a fixed point outside S, satisfying a^2+b^2 is greater than 2C, and passing P points is all tangents of S.

Proof: The tangents of these tangents fall on the same plane.

Originally, I thought it was the finale question, which should be a bit difficult, but Yi Cheng thought about it for a while and found that this question was not difficult.

In Geometry, there is a very powerful king curry stick.

It's a vector.

Just use the vector curry stick to cut everything invisible.

Yi Cheng thought about it for a while and used vectors to prove the problem.

When he was done, he discovered a miraculous thing-

This problem is not only provable on a two-dimensional plane, but can even be generalized to a quadric surface.

So Yi Cheng used vectors to prove the generalized proposition of quadrics.

After doing this, Yi Cheng thought, since the quadric is also feasible, is it possible to generalize to 3 times?

When he got carried away and carried out a higher-dimensional promotion on scratch paper-

Exam time is over.

According to the requirements of the competition, the examiner will seal the exam paper together with the scratch paper for assessment.

Yi Cheng looked dazed, and he was worried that the final step was not completed.

"It's not like you this time!"

At the gate of the arena, Li Anruo clasped his hands and mocked.

"Aren't you the first to turn in your papers every time?"