Fang Chao began to review the question.

Generally speaking, the three questions per day in IMO events are also divided into three modes: Normal, Hard, and Hell, and sometimes IMO events are more perverted, and their modes will rise by one level of difficulty, such as Hard, Hell, and Purgatory. It's just perverted, don't want it.

Of course, it all depends on the idea of the main test committee.

The first question is sauce purple.

Let the set of integers be Z, and find all functions f:Z→Z such that for any integer a, b there is f(2a)+2f(b)=f(f(a+b)).

This question!

This question......

Cool!!

In the last half month of the training camp, Professor Lin specially emphasized the function again, and at the same time did six or seven mock questions on top of such a question type, which directly raised the understanding of the six people to a higher level.

In more than 100 countries around the world, each country produces three to five questions, and the main examination committee selects them, and the possibility of a country being able to choose a question is very small, but the Korean professor has been selected several times, so he has also become famous.

This question obviously didn't come from Professor Lin, if it was Professor Lin, it would definitely come up with a similar question, just convert a few data, but even so, Fang Chao was also happy in his heart.

Because everything is the same.

Gee......

The organizing committee!

You still underestimate the strength of our national team!

What is the purpose of our national team doing so many practice questions every day? If you want to talk about the basic level, our country is called the second, which one dares to call it the first?

Do you think we're so good just because we do so many practice problems every day?

Of course not, in addition to a lot of practice every day, we are inseparable from 100 push-ups, 100 sit-ups, 100 squats and 10 kilometers of running every day!

We perceive our changes, we become stronger, because we are bald!

For the sake of the country, throw your head and spill your blood, and you should not give in.

We've got it!

If you miss this question, then whoever is wrong will go home and farm.

I can't even solve a question of such difficulty, what's the use of choosing you?

Fang Chao's blood is boiling!

The blood of battle in the body is soaring!

At this moment, at this moment, at this second, the war spirit of the nation has awakened.

Untie:

From f(2a)+2f(b)=f(f(a+b)), take a=0.

This gives f(0)+2f(b)=f(f(b)) for all integers b.

At the same time, in the original formula, a + b is used instead of b to obtain f(f(a+b)) = 2f(a+b)+f(0) and 2f(a+b) + f(0) = f(2a) + 2f(b) are brought back to the original formula to obtain 2f(a+b)+f(f(a+b)) = f(2a)+2f(b).

Take a=b+1, then f(0)+2f(b+1)=f(2)=2f(b)

That is, f(b+1)-f(b)=1/2(f(2)-f(0))=C is a constant.

The content of the IMO test is nothing more than the classification of algebra, number theory, geometry, combinatorics mathematics, combinatorial geometry, etc.

Similarly, for Fang Chao, the level of mathematics LV3 is not to improve Fang Chao's ability to solve problems, but to strengthen his ability in mathematics, such as analysis, application, thinking, derivation, imagination, and space.

If Fang Chao's math level was suddenly raised to LV10, then if he didn't study it systematically, he wouldn't even be able to figure out the Riemann conjecture, and he wouldn't even know what it was.

Only after a lot of exercises and strengthening his own knowledge can his mathematical level be truly improved.

The so-called rank is nothing more than your talent in this area.

You have an amazing talent, but if you never touch mathematics, then no matter how good your talent is, you will not be able to achieve anything in mathematics.

Fang Chao spent a lot of time in mathematics in junior high school and high school, and did a lot of math problems in his spare time, and these knowledge have now become his magic weapon to win.

Kill!

After using the formula calculated on it, Fang Chao quickly obtained that f(n) is a primary function or constant.

Let f(n)=pn+q, then 2ap+q+2bp+2q=ap?? +bp?? +pq+q。

by the arbitrariness of a, b.

So, 2p=p??,2q=pq+q.

In this case, there are only two solutions left for this problem.

(p,q) = (0,0) or (2,k).

where k is an arbitrary constant, then the function satisfying the question is a zero function or a linear function f(n)=2n+k, where k is an arbitrary constant.

Get!

7 points in hand!

On Fang Chao's body, there is obviously already an honorary award.

Since 1988, the IMO has established an honorary award for the first time.

The so-called honorable mention is for those who have participated in the competition, but have not won gold, silver or bronze medals, but have at least one question with a perfect score, which is the so-called 7 points.

This measure has greatly mobilized the enthusiasm of the participating countries and contestants.

Because if you participate in the competition and you don't get any prizes in the end, it will be a real shame.

The IMO spirit is the Olympic spirit.

Because it is the International Mathematical Olympiad itself.

The spiritual significance of the Olympics lies in the fact that the most important thing in the Olympic Games is not to win but to take part.

Translated into Chinese, the most important thing in the Olympics is not victory, but participation.

"God's special thing is not to win, but to participate, since he has participated, it must be to win, otherwise what's the point of participating?"

Fang Chao definitely didn't dare to agree with such an idea, he was going out on behalf of the country, that is, fighting for the country.

What flows in his bones is the blood of the motherland and the blood of the soul of war.

So just if you participate, he has won an award now, or an honorable mention, which is enough.

But is it really enough?

Not enough!

The little red guy on the American side deceives people too much!

After winning the IMO national team championship for a few years, you thought you could be lawless? Since the official start of the competition, China has won nearly 20 team championships so far.

In comparison, even if your American team wins another ten years, it will not be able to catch up with our national team!

In the past, the national team could hold its head high in this regard, but in recent years, it has been suppressed and scared...... Even on top of the IMO events, I don't dare to be as crazy as I have been in the past few years.

Fang Chao wants to teach the other party a lesson, and by the way, he will win the honor that belongs to the motherland.

Keep fighting!!

Soon, Fang Chao's gaze was on the second question.

On this second question, Fang Chao was first surprised, and then stunned...... It's so interesting.

It's not a regular question.

The professor who made the question, you are a little cutie, right? Could it be that he grew up eating cuteness?

Do you all play LOL with the big guys of the main test committee?

It's a god-like test.

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