I Just Want to Be a Quiet Top Student

"Oh, I'm going, my train of thought is broken!"

Doing math problems is like writing a novel. If the thinking is smooth, it will be changed ten times a day, and if the thinking is broken, it will be changed every ten days.

At the critical moment when Shen Qi was about to solve the first geometry problem, he was interrupted by the vice president.

"This uncle is really annoying." Shen Qi had to reorganize his thoughts, which took him an extra five minutes.

The whole blank paper was filled with the verification process, and Shen Qi finally found out the value of sinψ.

The answer surprised him, sinψ is actually 1/2, which is an angle of 30 degrees.

Take a ruler and measure it, it looks like 30 degrees.

It is verified by Lobachevsky's drawing method that it is 30 degrees.

"I'm stupid, I'm really stupid..." Shen Qi realized a low-level mistake, he was confused by the complex geometric pattern, and it took nearly an hour to deduce it forward.

If the degree of ψ is directly calculated by Lobachevsky's drawing method first, and then reversely verified that the angle of ψ is 30 degrees, at least half of the time can be saved.

Of course, Lobachevsky's drawing method must not be used for drawing on examination papers, and drawing on scratch paper is no problem.

It is easier to verify a conclusion with a conclusion than to derive an unknown.

This is a game, a mathematical game between the question maker and the answerer.

War is the best training, and examination is the best review.

Although there is systematic assistance, Shen Qi also has a new understanding of mathematics in the extremely difficult semi-finals.

There must be a deeper relationship between hypothesis and proof. Truth or fallacy is not as simple and opposite as it seems on the surface. Fallacy may be a reverse of truth.

Shen Qi was distracted, he thought a lot, for a few minutes, it seemed that his math level was still too low, and he couldn't understand many problems.

"There are still two hours left in the exam, please take the time to answer the questions." An invigilator said to all the contestants.

"Two hours, there are two questions left." Shen Qi came back to his senses and started to answer the next question.

This rematch exam paper is the one with the fewest questions he has done, with only three questions.

It is also the most difficult one, and the first question with the lowest score took Shen Qi 1 hour.

The second question is an algebra question, and the title is as follows:

1

1-1

1-2-1

1-3-3-1

1-4-6-4-1

1-5-10-10-5-1

1. Please calculate the sum of all the numbers in line 1024. (5 points)

2. And prove that any of the numbers in line 4201 is a fraction or a negative number applies. (15 marks)

In fact, many high school students know this number triangle. Who doesn't know Yang Hui's triangle? Middle school students who have participated in the Mathematical Federation and Olympiad competitions all know the regularity of Yang Hui's triangle.

Of course, Shen Qi understood this digital triangle. This digital triangle is called Yang Hui's Triangle in China, and "Pascal's Triangle" in the West, named after two mathematicians in China and the West.

It is not difficult to observe the regularity of Yang Hui's triangle. Each number in the triangle array is the sum of the two numbers immediately above it.

By analogy, Shen Qi quickly calculated that the sum of all the numbers in the 1024th line is xxxxx... This is an astronomical number expressed in 2 to the 1023th power.

The first sub-question of the second question is simply a sub-question, so the score is not high, only 5 points.

The second sub-question is the most difficult, worth 15 points.

The situation where any number in line 4201 of the forward derivation is a fraction or a negative number is applicable, which is very troublesome, and there is no way to write, and there is no clue at all.

Shen Qi wanted to infer in reverse, the content required to be proved in the second question must be based on a formula, theorem or inference.

"Bernoulli's permutations or probability theory? No, not like that."

"Wedas' three special types of equation expansions? Neither."

"Fermat is a top player in this pure numbers game. That's right, it should be Fermat. He and Pascal are good friends. The two often exchange letters. This question is based on Pascal's triangle."

"Fermat put forward hundreds of hypotheses in his life, and 99% of them were proved to be true by later generations. He is known as the 'king of amateur mathematics', but I never believe that Fermat's mathematics level is as good as that of I'm as amateur as I am."

"It's a big head. Of Fermat's 273 hypotheses, I have only studied 70 at most. Which one is it? Has it touched the blind spot of my mathematical knowledge?"

Shen Qi put down the ballpoint pen, closed his eyes and rested his mind, racking his brains to think of a way.

The vice president wobbled and swayed behind Shen Qi, he showed a smug smile, and he was very satisfied with Shen Qi's current state: "Young man, even if you can solve the first geometry problem and know the laws of Yang Hui's triangle, how can you do it?" How? The difficulty of the second question I asked is beyond your imagination, right?"

At this moment, Shen Qi suddenly opened his eyes, and his eyes sparkled brightly: "I figured it out, this is Fermat's (1+a) inference! He said that this inference must be established and there is no need to prove it, but 100 years after his death, his French compatriots proved this inference."

Here comes the idea, here comes the idea!

Shen Qi's inspiration gushed out like spring water, he picked up the ballpoint pen and prepared to solve the problem.

Just when he was about to start writing, Shen Qi inexplicably felt a chill coming from behind, the same as before.

Shen Qi glanced back, very unhappy: "Fuck, it's you again! Are you passing by again?"

The vice president pretended to be calm: "The examination room is only this big, and as an invigilator, I will occasionally pass by." After speaking, he left with his hands behind his back.

"Uncle, you are really annoying!" Shen Qi calmed himself down as quickly as possible, and entered the process of proving the second question and the second sub-question.

Ten minutes later, Shen Qi wrote the entire proof process of the second sub-problem, which was verified by the inference of Feima (1+a). This proof is completely established, and any number in line 4201 is a fraction or a negative number is applicable. .

"Both the first and second major questions have been solved, and 35 points have been obtained."

Shen Qi, who broke through two questions in a row, was so confident that he started to do the last question.

The vice president is a restless person. After a stroll, he went around behind the other players to observe secretly.

"He Tao, this young man is not good. Apart from knowing Yang Hui's triangle, he can't write any other questions at all."

"Luo Ming is from Nangang Foreign Language School. After all, he is a student of a famous school. This young man is not bad. He answered the first geometry problem correctly, but the second question of the second question is obviously wrong. Young man, you are completely different. Wrong direction."

...

The vice president visually estimated that the contestant with the fastest progress and the highest accuracy in answering questions in the examination room was Shen Qi. He couldn't help it, he really couldn't help it, so he touched Shen Qi's back again and observed secretly.

Shen Qi turned around suddenly, he was about to collapse: "Don't do this, uncle, why do you insist on making things difficult for me?"

The vice president is still very innocent: "I wrote this test paper, so I'll just take a look. Can't you just say nothing?"

"Damn it, did you write this paper?" Shen Qi never expected that the master who came up with the questions in his mind was actually this wretched uncle.