I really just want to be a scholar

Chapter 63 The provincial competition begins!Hamiltonian diagram!

The provincial competition of Mathematics Olympiad is still held in the largest test center in the province.

When Qin Ke got up, he felt something was wrong with his body, and he touched his forehead secretly. My sister, it's even hotter!

It was just a low fever in the morning, but after half an hour of eating bread and milk in the car, it turned into a high fever!
"Qin Ke, why don't you get out of the car?" Ning Qingyun asked outside the car. She, Lao Zheng, and the middle-aged female teacher had already got out of the car.

Qin Ke took a deep breath, forced himself up, rubbed his face vigorously, and jumped out of the car with vigor.

"Hey, I'm in such a hurry, I'm going into the examination room to find the restroom."

Qin Ke originally wanted to encourage Ning Qingjun with a few words, but when he spoke, he found that his throat was a little hoarse and painful, so he stopped talking. Carrying the document bag with documents and stationery, putting on the hat of the down jacket, lazily After waving his hand, he strode towards the inspection office.

His footsteps were a bit soft, he gritted his teeth and strode vigorously, relying on his usual good physical foundation, he didn't show anything unusual.

Lao Zheng shook his head behind and sighed: "Qin Ke, this boy, is in a good mood, he doesn't seem nervous at all. I hope he can perform well this time."

Ning Qingyun looked at Qin Ke's figure, but felt something was wrong.

"By the way, student Ning, remember those two additional questions and don't do it recklessly. If you don't have a clue, give up quickly and focus on the main question in front..." At this time, the middle-aged female teacher next to her was still talking about taking the exam. Skill.

Seeing that Qin Ke had gone far away, Ning Qingyun hurriedly interrupted the middle-aged female teacher: "Mr. Liang, Mr. Zheng, I am also entering the examination room. Goodbye!"

She said goodbye to the two teachers in a hurry, and hurried to Qin Ke: "Qin Ke, wait for me!"

Of course, Qin Ke heard Ning Qingyun's shout. It is really rare for this cheeky girl to shout such words on such an occasion, which proves that the relationship between the two is indeed different from before.

Qin Ke was a little relieved and a little proud, but he didn't look back and didn't stop. He just said silently in his heart: "School committee, come on, your hard work and sweat these days can't be wasted."

He quickly completed the inspection, and then ran into the examination building at high speed. He seemed to hear Ning Qingyun's voice behind him...

Seeing Qin Ke and Ning Qingyun enter the examination room one after the other, the middle-aged female teacher couldn't help but ask Old Zheng next to him: "Mr. Zheng, I heard that this provincial competition is difficult, how are the two of them reviewing? "

"do not know."

"I don't know?" The middle-aged female teacher is a math teacher in the third year of high school. Just because she is a woman, Vice Principal Wen temporarily transferred her to Ning Qingyun in case of any accidents. in the entourage.

Naturally, she didn't know Qin Ke and Ning Qingyun's preparations for the exam.

"I only know that they have worked hard." Lao Zheng picked out a cigarette from the cigarette case, put it in his mouth and picked it up: "For us teachers, their hard work is the best answer. It doesn't matter what the outcome is."

The middle-aged female teacher looked at his calm and clear appearance as a "Master of a Generation", opened her mouth, and finally could only say one sentence: "Mr. Zheng, you are so right."

Lao Zheng said pretending words, but he was more nervous than anyone else.

"Qin Ke, Qin Ke, I bet my future on you. Whether you can get the year-end award depends on you. You have to bring me back a top five award..."

……

In this rematch of the Olympiad, each city has only five to ten places to participate, and the total number of candidates in the whole province is only about [-].

However, there are many examination rooms, with an average of only 20 people in each examination room. Candidates from the same city are separated into different examination rooms. Met Chen Hanyin and Hong Xingwei, two candidates from Chengkong.

After he entered the examination room, he lay down for a while, and the examination began soon.

The three invigilators began to distribute the test papers after reading out the rules of the examination room. Qin Ke glanced at them. The three invigilators did not know each other, and he didn't know if the three invigilators before saw his name and deliberately avoided him. go.

But Qin Ke didn't have the time to think about it. His brain was buzzing and it felt as if it was rusted. His thinking ability was not as good as [-]% of usual, and his body was getting more and more chilly, and his hands were getting colder and colder.

Qin Ke tried his best to keep his mind awake, but he knew that his cold was getting worse, and his current state would not last long, and it would probably get worse with time, so he had to hurry up to answer the questions.

He flipped through the main paper and the additional paper. As Lao Zheng said, the main paper has ten major questions, each with 20 points, and the additional paper has two major questions, each with 50 points.

Qin Ke had already decided on an exam strategy when he lay down on the table before the exam, that is, while the current state was not bad, he would first solve the two additional questions of the most difficult national competition, and then go to the provincial competition. The problem, even if the state becomes worse by then, should be able to barely handle it.

He shook his head and focused on the first additional question.

"Additional question [-]: The graph formed by n points and several edges on the plane is not a Hamiltonian graph, but if any point and the edges connected to it are removed arbitrarily, the remaining graph is a Hamiltonian graph. Find the minimum value of n .”

Qin Ke took a deep breath. As expected of the difficulty of the national competition, it was the Hamilton map.

It is estimated that [-]% of high school students across the country have never noticed the name Hamilton.

Hamilton was a famous mathematician in Eagle Country in the [-]th century. At that time, he proposed a game called "Around the World", using the [-] vertices of a regular dodecahedron to represent [-] big cities, requiring that along the edges, Starting from a city, passing through each city only once, and then returning to the starting point, this is the famous "Hamiltonian problem".

Later, the mathematical community called "a cycle that passes through each vertex on the graph once and only once" as a "Hamiltonian cycle". If a graph contains a Hamiltonian cycle, then this graph can be called a "Hamiltonian graph".

On the surface, this Hamiltonian problem seems to be very similar to Euler's problem of the seven bridges of Königsberg, but there are essential differences between the two.

The so-called "Seven Bridges of Königsberg Problem" means that there are two islands in the river, and there are seven bridges on the river connecting the two islands and the two banks of the river. Is it possible to pass each bridge once and only once.It is also known as the "one stroke" problem.The Seven Bridges of Königsberg problem has already been solved by Euler himself, and thus created a new branch of mathematics - "graph theory".

However, the Hamiltonian problem has not been solved so far. For more than 100 years, countless first-class mathematicians have worked hard, but they have not found sufficient and necessary conditions for judging it. They have only proposed some proven necessary and sufficient conditions, which are applied to different occasion.

The difficulty of this question lies in not only requiring the solver to understand the characteristics of the Hamiltonian diagram and those proven necessary and sufficient conditions, but also to be able to use it flexibly.

As soon as Qin Ke saw this question, he knew that Ning Qingyun couldn't answer it—because time was limited, he only explained two sample questions to Ning Qingjun about the Hamilton diagram, which was not considered in-depth. The understanding of the picture is impossible to answer.

It's not just Ning Qingyun, it is estimated that in the entire examination room, there is no one else who can answer it except him.

Qin Ke rubbed his swollen temples, pondered for more than three minutes, and then began to write:
"Solution: Firstly, the degree of each point is at least 3, otherwise there is a point A that connects at most two sides, then after one side is removed, the remaining point A must not be on a certain circle, which is inconsistent with the condition, so we can get out, n≥3..."

"When n=4..."

"..."

"The condition is only true when n=10, so the answer to this question is 10. The specific illustration is as follows:"

Qin Ke drew a regular pentagon with a "one-stroke" pentagram in the middle, and each vertex of the pentagram is connected with the pentagon vertices surrounding it.

This is the graph that best matches the meaning of the question when n=10. A point and the edges connected to it are removed arbitrarily, and the remaining graph is a Hamiltonian graph.

During the answering process, I wrote a full half-page, almost filling the answer area.

(End of this chapter)