This male student with naturally curly hair and quite handsome appearance was the same as the comprehensive question just now. He only read the question once and then started to write down the steps to solve the problem. It seemed that he did not need to think at all. Of course, it is more likely that after reading the question

You will have ideas for solving the problem at the moment.

In just two minutes, he had completed the first additional question and continued to write the answer to the second question.

Xu Shichao was completely petrified and couldn't think of any other exclamation in his heart except "fuck".

Who is this boy? He can no longer be called a "student master", but a "student god", right? Is there such a strong math student in the second year of high school?

Xu Shichao asked himself that he knew all the top math students in the school. He also helped lead the team in the Mathematical Olympiad preliminary competition last year and took on the responsibility of providing life support. However, this boy clearly had never participated in the Mathematical Olympiad preliminary competition last year!

Just when Xu Shichao was stunned, he suddenly felt someone tapping his shoulder. When he looked back, it turned out that his senior, Lao Zheng, was here.

Lao Zheng made a gesture of silence, and then stood behind the boy with him, watching the boy answer the questions.

The boy is already looking at the third difficult problem that even Xu Shichao is not confident enough to solve.

"Verify: Each item in the sequence an=3^n cos(os1/3)(n=1,2...) is an integer, but none of it is a multiple of 3."

The male student finally paused for two seconds this time, and then wrote "Proof Method 1" under the gaze of the two teachers.

Xu Shichao gasped on the spot. Did this guy think of two proof methods in the two seconds of thinking just now?

"Proof method one: Suppose θ=os1/3, then cosθ=1/3, and an=3^n cosnθ,

(1) When n=1,2, a1=3cosθ=1,a2=3^2 cos2θ=9(2cos^2θ-1)=9*(-7/9)=-7

1 and -7 are both integers and not multiples of 3. The proposition can be proved.

(2) Assume that a(k-1) and ak are both integers, and neither is a multiple of 3. It can be obtained from the trigonometric formula (note: k-1 is a subscript):

a(k1)=3^(k1)cos(k1)θ=3^(k1)[2cosθcoskθ-cos(k-1)θ]=2ak-9a(k-1)

…

It can be seen from mathematical induction that the proposition is true for all positive integers."

"Proof method two: Suppose θ=os1/3, then cosθ=1/3, sinθ=2*2^(1/2)/3,

Introduce the complex number z=3(cosθisinθ)=12*2^(1/2)i

Then an is the real part of the complex number z^n=[12*2^(1/2)i]^n..."

Watching the male student easily finish writing the two proof methods, and then flipping through the paper, he finished checking it almost at a glance and ten lines, then folded the test paper and put it in the corner, covered it with a blank scratch paper, and then he typed

Yawned and started to fall asleep.

Oh my god, this student's scratch paper is actually blank!

Xu Shichao has taken math exams countless times since he was a child, and he has never had a blank scratch paper!

Xu Shichao couldn't help but secretly lift the draft paper and glanced at the name and class column on the test paper.

Qin Ke, Grade 2, Grade 3.

I don’t have much impression... Is there such a person among the top math students in the second year of high school?

Xu Shichao was struggling to remember when Lao Zheng suddenly made a gesture to the student next to him by the window, gesturing for him to pull down the curtains.

The student didn't know why, but he still obeyed.

However, Xu Shichao noticed that the sunlight coming from the window was shining on the boy named Qin Ke just now. Old Zheng was afraid that it would affect his sleep...

Xu Shichao opened his mouth to say something, but found that he couldn't say a word. Lao Zheng pulled him to the door, took out a cigarette and held it in his mouth - of course he didn't light it, it was just to satisfy his craving for cigarettes.

Xu Shichao asked in a low voice: "Team Leader Zheng, this...is this your student?"

"Well, how about it? Is your level passable?" Lao Zheng looked calm and relaxed, but his arrogant and trembling toes clearly betrayed his happy mood.

"Uh... I can only think of one word... awesome."

Lao Zheng patted Xu Shichao on the shoulder: "Calm down, he's just a genius, it's nothing. Besides, he hasn't won the first place in the Municipal Mathematical Olympiad yet, so you can't spoil him too much."

After speaking, Lao Zheng held a cigarette in his mouth, flicked the corner of his clothes, and went to inspect the examination room.

Xu Shichao's heart was filled with countless mud horses. Lao Zheng, can you show off a little more? While you were saying it was nothing, you actually booked the first place in the Municipal Mathematical Olympiad for your students early!

Xu Shichao also walked back to the examination room. When he saw other students still answering questions with sad faces, he couldn't help but sigh, why are there such big differences between people?

It seems that this mathematics test is destined to be a feast for only individuals.

…

Qin Ke was in a good mood. The math test paper was easier than expected. Although the scores for the three additional questions would not be counted into the total score, they would be counted into the ranking of the individual subjects. If nothing else, he would get 1,000 yuan.

Moreover, when sleeping during the midterm exam, the God of Study experience value was three times that of the usual classroom. Qin Ke only spent about 20 minutes doing the paper, and the remaining 130 minutes were spent sleeping, and he unexpectedly got 390 points of God of Study experience.

Value, how can the word "shuang" be so great?

The evening self-study was still in the original classroom. Qin Ke finished playing basketball with Li Xiong and Ge Zhixing as usual, took a shower and ate, and then returned to the classroom with a few boys.

During the meal in the canteen, the following conversation also took place.

Li Xiong, the bad student: "What do you think of today's math test? I don't see any difference from usual. Why do some people cry during it?"

Ge Zhixing, the second poor student: "I'm surprised too. Anyway, I'm just like before, I don't know anything. I chose C for the multiple-choice questions. I can always get a few points right, right?"

Studying student Wang Lixin frowned: "It is much more difficult than before. I was not sure about many questions and filled them in randomly. I saw a student next to me who directly greeted the teacher's family on the paper."

The quasi-student Wu Zhu, who has always been honest, was dumbfounded: "Isn't this bad? The questions this time are really difficult, and I haven't finished the big questions at the end, but scribbling on the paper like this is disrespectful to teachers."

That’s right, as a student, it’s very inappropriate to do this…”

Qin Ke did not participate in such a topic and just smiled.

Li Xiong noticed his expression and asked: "Brother Ke, did you hand in a blank paper this time?"

"No, I promised Teacher Wei, so I wrote all the papers."

Ge Zhixing patted Qin Ke's shoulder with self-understanding: "It's hard work, but bowing your head properly is to live a better life. I also filled in the questions. I copied the questions directly for the subsequent solution questions and comprehensive questions.

, I hope the teacher will give me one or two comfort points in view of my hard work in copying the questions."

By this time, they had reached the classroom. Li Xiong and others did not dare to approach Ning Qingyun, so they separated from Qin Ke.

When Qin Ke returned to his seat, Ning Qingyun was already doing the questions silently.

In fact, you don’t need to wear school uniforms for evening self-study, but Ning Qingyun still wears a blue and white school uniform. Her hair is simply tied into a single ponytail with a black ribbon. The ponytail is tied a bit high, giving off a cool and proud atmosphere.
Chapter completed!