Xueba, I beg you to go and escort him quickly!

……

Thus n has property P.

Obviously p<n<p2.

Taking a prime number greater than p2, another composite number with property P can be obtained.

Therefore, there are infinitely many composite numbers n with property p.

Easy to do.

Xu Heng didn't even frown.

Then continue to turn the page and look at the third question.

An understatement!

Don't take it seriously at all!

But these three are not calm at all!

Another 5 minutes!

So... the first two questions took him less than 10 minutes! ! !

Even buy Karma!

"Thanks……"

Shet!

One examiner couldn't help but want to swear, and he was in the examination room!

He really couldn't hold back!

!!!

So perverted!

at the same time……

He also wanted to complain, are the experts and professors in the examination group all out of their minds?Such a simple question? !

!!!

But when he looked around, some of the other five students hadn't written yet!

The person who started the pen only wrote one or two lines...

The examiner looked at Xu Heng, and looked at the other five candidates to the right.

This gap is not a star and a half!

He couldn't help complaining to 010, "What's the matter with these guys? What are they waiting for? Why don't they answer? Do they find it difficult?"

Still waiting!

He even saw some students with blank faces, or frowning.

Looking at Xu Heng again, still indifferent...

He calmly examined the third question.

The "fraud guessing game" is played between two players A and B, and the game relies on two positive integers k and n that both A and B know.

At the beginning of the game, A selects two integers x and N, 0≤x≤N.

A truthfully tells B the value of N, but keeps his mouth shut about x.

B is now trying to obtain information about x by asking questions in the following way: each time a question is asked, B chooses a set S composed of several positive integers (the set used in previous questions can be reused), and asks A "Does x belong to S?".

B can ask any number of questions.After each question from B, A must immediately answer "yes" or "no" to B's question. A can lie, and there is no limit to the number of times of lying. The only limit is that A must answer at least Once the answer was the truth.

After B asks all the questions he wants to ask, B must point out a set X containing at most n positive integers. If x belongs to X, B wins; otherwise, A wins.

(1) If n2 (k power), then B can guarantee to win;

(2) For all sufficiently large integers k, there is a positive integer n (k power), so that B cannot guarantee to win.

That is, when looking at such a long topic, Xu Heng put down his pen.

Read the question!

Understand the topic!

It all takes time...

The three examiners breathed a sigh of relief.

But they didn't dare to take half a step away from Xu Heng, because they were afraid that if they were not paying attention, Xu Heng would have solved the third question!

At the same time, regardless of whether they are right or not, they really want to know how much time Xu Heng will spend on the third question!

After all, this is the final question of the day of this Mathematical Olympiad!

Even if Xu Heng thinks the first two questions are easy...

But what about this one?

Could it be any simpler? !

Even if they wanted to scold those who made the test group, at this moment, they didn't have that kind of thought. They still had confidence in the test group on the third question!

How to score in the International Mathematical Olympiad?

This is the finale question!

The three examiners couldn't understand Chinese at all, and one of them looked sideways at the test paper next to him.

Because this classmate's test paper is in English.

He watched with gusto!

He and Xu Heng read the topic almost at the same time...

Can--

Before he finished reading this question, Xu Heng slowly picked up his pen and began to answer this question!

PS: Thank you [your IQ is above the line] [zero]!

Kneel for all support! .

Chapter 209

The moment Xu Heng picked up his pen and was about to answer, he froze again.

The examiners looked up at each other, and they looked at each other.

Suddenly, the corners of their mouths rose and they smiled slightly.

It seems that this student who scored [-] points in the preliminaries has not always been invincible!

But on this question, he was stumped!

Yes!

We admit that you are extraordinary, that you are a super genius, but you can only stop there!

The examiner who read the third question couldn't help but gasp after reading all the questions.

He murmured in his heart, "Although the first two questions are not satisfactory, the third question...is very difficult!"

There are a lot of words in the topic!

You need to have a strong comprehension ability, not to mention, you also need to find a breakthrough in order to answer the questions.

In addition, in the process of answering this question, there are too many calculation rules used!

He also retracted his gaze and looked at Xu Heng.

Seeing that Xu Heng's third question was empty.

He also breathed a sigh of relief.

"call……"

But the breath is not over yet...

Xu Heng began to write.

Geek!

The three examiners suddenly pulled together.

They froze and had goosebumps all over their bodies.

Looking at the time again, only 15 minutes had passed since the entire exam!

Xu Heng wrote:

(1) First discuss the relationship between N and n:

After A selects N, B needs to reduce x from the range of N+1 numbers to the range of n numbers.Obviously when N<n, B will win without asking questions.

Therefore, A must choose N greater than or equal to n.If A chooses N=n, and B has a winning strategy at this time (that is, a certain number i from 0 to N must be excluded), then it can be proved that if A chooses any large N, B also has a certain strategy. winning strategy.

This is because for larger N, B can divide the {0,1,2,...,N} set into disjoint and non-empty subsets...

The full set will reduce at least one element, which is equivalent to reducing N by at least 1.

By analogy, since there is no limit to the number of B's ​​questions, it is inevitable that N can be reduced to n, so as to win in the end.

……

Therefore, after such an inquiry, a number must be eliminated, so B wins.

After writing the answer to the first question, what Xu Heng wrote was not (2), but still (1). At the same time, he wrote "the second solution" after this

After that is:

Consider n=2k, N=n+1.Use binary.

Write 1,2,...,2k in binary: a1, a2,...ak+1.

here……

That is to say, Si is a subset (i=1,...,k+1,2) of all elements satisfying ai=1 in T.

B adopts the following questions, which can guarantee winning: the first question, choose S1, and continue to choose S1, A's answer will appear in two situations:

Answer "No" k+1 times in a row, then...

在至多k+1次回答中,一旦出现”是”,乙接下来的k次提问,依次选取……这里a1=0,ai=0还是1取决于甲对Si的答案:若甲的回答是“是”,ai=0,否则ai=1(i=2,3,…,k+1)。

……

When B finishes asking a series of questions he wants to ask, if B can choose a set X satisfying |X|n, making x∈X, then B wins; otherwise, A wins.

……

Since pk+1N2pAj, as long as ik+1, ai=0, which makes B unable to exclude any element of S, and cannot win.

Two ways to solve the problem!

It uses a completely different way of solving problems!

Just the used answer sheet, Xu Heng has already filled three pages!

This is just the first question!

Although the three examiners don't know Chinese, they can see (1) clearly!

Although at first they thought it was Xu Heng's typo!

But after Xu Heng finished writing, when he wrote (2), they suddenly realized that it was very possible that the above two (1) were Xu Heng's different methods and different ideas to answer this final question!

Swish!

At the same time, they were staring at Xu Heng's test paper, staring at the five words "Second Solution' ~ Method", trying desperately to write these words down!

They have to wait until after the exam to ask colleagues who understand Chinese.

Although they probably guessed it, but...they couldn't pass their own test!