Become a master from now on

Chapter 183 is not a regular question (ask for a monthly ticket subscription!)
Fang Chao began to examine the questions.

Generally speaking, the three daily questions of IMO events are also divided into three modes: normal, difficult, and hell. Sometimes the IMO events are more abnormal, and their modes will increase the difficulty of one level, such as difficult, hell, and purgatory. Three modes.It's simply perverted, don't want it.

Of course, this all depends on the thinking of the examining committee.

The first question is Jiangzi.

Let the set of integers be Z, and find all functions f: Z→Z such that for any integers a, b, 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, and at the same time, he did six or seven simulated questions on top of this question type, which directly raised the six people's understanding of functions by one level. steps.

The current question is almost certainly not written by Professor Lin. There are more than 100 countries in the world, and each country produces three to five questions, which are screened by the main examination committee. The possibility of a country being able to select one question is very small. , but several of the questions proposed by Korean professors were selected, so they also became famous.

This question obviously did not come from Professor Lin. If it was Professor Lin, he would definitely come up with a similar question, just by converting 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!

Why do our national team do so many practice questions every day?In terms of the basic level, our country is called the second, which one dares to be the first?

Do you think we are so strong just because we do so many exercises every day?

Of course not, in addition to a lot of practice every day, our strengthening is also inseparable from doing one hundred push-ups, one hundred sit-ups, one hundred squats and running ten kilometers every day!

We feel our change, we are stronger because we are bald!
He sacrificed his life and blood for the country and did his duty.

We won this question!

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

What's the use of choosing you if you can't even handle such difficult questions?

Fang Chao's blood is boiling!

The fighting blood in the body is soaring!
At this moment, at this moment, at this second, the fighting spirit of the nation is awakened.

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

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

同时，在原式中，以a+b代替b，得到f（f（a+b））=2f（a+b）+f（0）带回原式又得到2f（a+b）+f（0）=f（f（a+b））=f（2a）+2f（b）。

取a=b+1，于是f（0）+2f（b+1）=f（2）=2f（b）

即f（b+1）-f（b）=1/2（f（2）-f（0））=C是一个常数。

The content tested by IMO is nothing more than algebra, number theory, geometry, combinatorics, combinatorial geometry and other categories.

Similarly, for Fang Chao, the improvement of Mathematics LV3 is not Fang Chao's ability to solve problems, but to strengthen his ability in analysis, application, thinking, derivation, imagination, space and other aspects of mathematics.

If Fang Chao's mathematics level has been raised to LV10 all of a sudden, then without systematic study, he can't even figure out the Riemann Hypothesis, and he doesn't even know what it is.

Only after doing a lot of exercises and strengthening the amount of knowledge he has accumulated, can his mathematics level be truly improved.

The so-called level is just your talent in this area.

You have an amazing talent, but if you never touch mathematics, then no matter how talented you are, you will accomplish nothing 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. This knowledge has now become his magic weapon to win.

kill!

After using the formula calculated above, Fang Chao quickly obtained f(n) as a linear function or constant.

Let f(n)=pn+q, then get 2ap+q+2bp+2q=ap+bp+pq+q from the original formula.

By the arbitrariness of a, b.

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

If so, there are only two solutions to this problem.

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

Where k is any constant, the function that satisfies the meaning of the question is a zero function or a linear function f(n)=2n+k, where k is any constant.

Get it!

7 points in hand!
Fang Chao obviously already has an honorary award.

Since 1988, IMO has set up honor awards for the first time.

The so-called honorary award is for those contestants who have not won gold, silver, or bronze medals, but have at least one question with full marks, which is the so-called 7 points.

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

Because if you participate in the competition but don't get any prize in the end, it would be really embarrassing.

The spirit of IMO is the Olympic spirit.

Because its full spelling itself is the International Mathematical Olympiad.

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

Translated into Chinese, this sentence means that the most important thing in the Olympic Games is not victory, but participation.

"What the hell is not winning, but participating. Since participating, it must be winning. Otherwise, what's the point of participating?"

Fang Chao absolutely disagrees with such an idea. He is going to fight for the country, that is, to fight for the country.

What flows in the bones is the blood of the motherland, the blood of the fighting spirit.

So if he just participated, he has already won an award, and it is still an honorary award, which is enough.

But is it enough?
Not enough!
The little red guy in the United States is bullying people too much!
Just take the IMO national team championship for several years and think that you can be lawless?Since the official start of the competition, our country has won nearly [-] team championships so far.

In comparison, even if your U.S. team takes another ten years, it will not be able to catch up with our national team!
In the past, the national team could hold their heads high in this regard, but in recent years, they are afraid of being suppressed... Even in the IMO competitions, they dare not be as crazy as in the past few years.

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

Keep fighting! !
Soon, Fang Chao's eyes were set on the second question.

On the second question, Fang Chao was surprised first, then stunned...it's so interesting.

This is not a common question.

The professor who wrote the question, you are cute, right?Could it be that he grew up eating cute?

Do you guys from the main test committee also play LOL?
This is simply a god-like test question.

(End of this chapter)