After handing over the system, the state forced me to study

There will not be contestants of the same nationality in the examination room, and the six team members from each country will be assigned to different examination rooms.

From the moment you step on the playing field, it means that everyone will face a lonely war and fight alone.

It seems inappropriate to say that it is a lonely fight. Although the IMO governing party does not have a group award, some are just individual awards, but I can't stand the team from all over the world. In the end, an unofficial ranking of the group's total scores was created, that is to say, it's really embarrassing to be in the bottom.

After the members of the representative teams from various countries entered the corresponding examination room, the team leaders were not idle. They stood nervously in different examination rooms to invigilate the examination, and secretly prayed in their hearts that their country's ranking could rise by one liter. By the way, think about some The country can take a step back, preferably counting backwards.

This might be considered a psychological attack.

I just don’t know how many people’s wishes can be realized at IMO this year.

The IMO questions are not provided by the host country, but 6 questions are selected from the question bank provided by many participating countries.

The topic will be translated by the organizing committee into the four official languages ​​of English, Russian, French and German, and then translated into the language of the participating country by the team leaders of each country. Among them, about 26 languages ​​will be involved.

Although the content of the test only involves algebra, number theory, geometry and combinatorics, the questions are very flexible and changeable. While testing the intelligence, creativity, spatial thinking ability and logical thinking ability of contestants from various countries, it is also very important for contestants. There are extremely high requirements for reverse thinking ability and stress resistance ability.

Facing such a tense international event like IMO, and representing their country in the competition, contestants from various countries naturally dare not be sloppy.

The exam time is from 01 am to 30:[-] pm, and you can leave for a little food in the middle, but it is all under the double surveillance of humans and cameras, and the expected cheating is unlikely to happen.

Four and a half hours, three questions, the difficulty of the questions can be imagined.

A total of six questions were solved in two days, with 7 points for each question and a full score of 42 points.

The test papers were handed out. Before starting the test, Lu Xingyao skimmed through the questions first, and got a little better in his mind.

As the instruction to start the exam was issued, Lu Xingyao's heart also rose. From this moment, the invisible war kicked off.

International competitions like IMO, although there is no such thing as appetizers, the difficulty of the questions is still different.

The team members of each country are carefully selected, and there will be no zero points.

However, there is no guarantee that a certain question will be polarized, either it is impossible to write, or it is difficult for everyone to do it.

That's what happened at this year's IMO.

Three hours have passed now, and there is still one and a half hours left before the end of the exam. Given that the previous questions are still within the controllable range, most of the contestants can more or less write some answers even if they can't answer all of them. Come in small steps.

At this time, most people's eyes focused on the third question.

That is the last question of today's exam.

"A hunter and an invisible rabbit play a game on the European plane. The starting position A? of the rabbit and the starting position B? of the hunter are known. After n-1 rounds of the game, the rabbit is at point An-1, While the hunter is at point Bn-1, in the nth round, the following three things happen at once:

(1). The rabbit moves to point An in an invisible way, so that the distance between point An-1 and point An is exactly 1.

(2). A positioning device feeds back a point Pn to the hunter. The only thing that this device can hold to the hunter is that the distance between the point Pn and the point An is at most 1.

(3). The hunter moves to a point Bn in a visible way, so that the distance between point Bn-1 and point Bn is exactly 1.

Ask: No matter how the rabbit moves, and no matter which points are fed back by the positioning device, the hunter can always choose her moving mode appropriately, so that after 10? rounds, she can ensure that the distance between her and the rabbit is at most 100? "1

Lu Xingyao frowned, his mind went blank.

I stared at this question for 2 minutes, but I couldn't see why.

He understands all the words on the topic, but what do they mean when they are connected?

Can you take some questions about Yang Jian?

He even faintly heard scolding coming from the examination room.

Especially his back table, hey, this classmate, your fack sound has reached my ears, I advise you to pay attention, don't force me to sue the teacher.

But it was also this scolding that made Lu Xingyao wake up from a series of activities such as blank mind, flustered heart, and not knowing what to do.

He secretly told himself: Lu Xingyao, calm down, you can do it, you are the son of the unified election, think calmly, and you will definitely find the answer.

In order to remind himself, he took a pen and wrote four words in front of the test question: think calmly.

This is his habit. When he is flustered or restless, he writes down a few words to alert himself all the time.

After writing, Lu Xingyao inexplicably felt a little relaxed physically and mentally.

Even if it is a psychological effect, he also recognizes it.

Lu Xingyao took a deep breath, moved his shoulders, relaxed the hand holding the pen for a long time, drank some water, and then started to solve the problem.

It was during this short period of intermittent relaxation that Lu Xingyao realized that his hands, back, and forehead were covered with sweat from tension.

The first step in solving a problem is to understand the problem.

The second step in solving the problem is to find the knowledge that needs to be used in the process of solving the problem.

After completing the first step, Lu Xingyao tried to use common sense thinking, equivalent thinking and extreme thinking to explore the solution to this problem.

He grasped the meaning behind the word "question", and obtained two equivalent principles without changing the meaning of the title:

(1). Allow this invisible rabbit to be blessed with magic and can manipulate the detector.

(2). Affected by (1), hunters may have biased judgment in certain situations.

Analysis 1: Assume that after a certain round, the invisible rabbit is at point A, the hunter is at point B, and the distance between the two points is AB=0, 0<R≦100.

……

Lu Xingyao couldn't help but be thankful that his physics is not bad. After all, to solve this problem, he still needs to simulate the physical scene of the pursuit problem.

Of course, the reason why this question is a difficult one is that Lu Xingyao will not be able to easily come up with the answer.

In the process of solving the problem, I will get stuck from time to time.

But they have already started the process of solving the questions, and the emotions like panic before can't be a hindrance.

The more Lu Xingyao did it, the more interesting he found this question.

Sure enough, the exam at the National Convention is still not as good as experiencing IMO in person.

Lu Xingyao said that this experience is really great, and it would be even better if the title was not so perverted (teeth-gritting version).

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12017 IMO Question [-]: Magical Magic Invisible Rabbit

The n and n-1 in the title are both subscripts