A genius? I just love studying.

Chapter 3 This question is not difficult
On the podium, An Chengzhang put his hands behind his back, his thick eyebrows and big eyes gazing over the classroom like an eagle, looking at the students scratching their heads below, which was not surprising.

He is the head teacher of Class 7. He also teaches mathematics to a gifted class and coaches the school's math competition team.

Today's set of in-class test questions are naturally not normal exam questions, but selection questions for the mathematics competition team.

In recent years, No. 2 Middle School has not had any good talents, and the mathematics competition team has performed very poorly. This year, there are only two or three big cats and small cats, and even the entire team is not complete.

He gave the students this set of questions not because he wanted to select players for the math competition team from the parallel class Class 7, but because he wanted to pour some cold water on the students who had just been out for a month and had not yet recovered from the joyous atmosphere of the New Year, so that they could calm down quickly and devote themselves to the intense study as soon as possible.

Class 7 is not without good talents.

An Chengzhang looked at Liang Peixuan who was sitting in the fourth row in the middle of the classroom.

This kid not only has good grades in math, but also has excellent grades in other subjects.

There is naturally no hope of getting into Peking University or Tsinghua University, but those schools in the middle of the second tier may still have a chance.

Well, I've started solving the questions.

An Chengzhang nodded secretly, very satisfied with Liang Peixuan's speed in solving the questions.

If Liang Peixuan can solve the second to last question, he will be eligible to join the math competition team.

Maybe let him try?
An Chengzhang narrowed his eyes slightly, and his brain started thinking quickly.

With Liang Peixuan's strength, he certainly cannot expect to win a prize in the CMO (China Mathematical Olympiad), and he can't even make it into the provincial team. But with his guidance and with more effort, there is still a hope for him to get second or third place in the provincial team.

There is no extra score for the second or third provincial college entrance examination, but it is also included in the strong foundation plan. Perhaps Liang Peixuan can be recommended to participate in independent enrollment. This way, it should be easier to get into a second-tier prestigious university than to take the college entrance examination directly.

Wow...

The sound of flipping test papers woke An Chengzhang from his thoughts, and he subconsciously looked towards the direction where the sound came from.

It’s Chen Hui!
A hint of surprise flashed in An Chengzhang's eyes.

He was the one who made up this set of questions, so of course he knew that the first page of the test paper contained eight multiple-choice questions and one essay question.

Turning over naturally meant that Chen Hui had finished the first question.

Looking from the podium, he could see a lot of handwriting on Chen Hui's test paper, especially in the blank space of the first answer question, where the handwriting was even denser.

Mathematics is no different from other subjects. If you don’t know how to do it, you won’t know what to write even if you just write it randomly.

An Chengzhang shifted his gaze slightly and looked at Liang Peixuan who was in front of Chen Hui on the right.

Liang Peixuan was still reviewing the first question.

How can it be?

An Chengzhang's heart skipped a beat and he felt some absurd emotions.

He created this set of questions himself, so he naturally knew the difficulty level and Chen Hui's math level.

But now it seems that at least one of them is in trouble.

When An Chengzhang saw Chen Hui writing furiously on the blank space of the second question, he could no longer hold back and walked down from the podium.

He did not go to see what happened to Chen Hui immediately, but came behind Liang Peixuan. At this time, Liang Peixuan had finished the first question and began to review the second question.

10. Given a positive integer m (m≥3), suppose the positive arithmetic sequence {an} and the positive geometric sequence {bn} satisfy: the first term of {an} is equal to the common ratio of {bn}, the first term of {bn} is equal to the common difference of {an}, and am=bm. Find the minimum value of am and determine the ratio of a1 to b1 when am takes the minimum value.

This question was not difficult, and Liang Peixuan started writing quickly.

First, set the common ratio q and common difference d, then write the expression of d and q for k=am=bm. After eliminating d, we get an expression with q as the unknown variable.

To find the minimum value, in high school, one simply uses inequalities and derivatives.

Using inequalities is equivalent to standing on the shoulders of giants. Directly using the results derived by others can save some calculations and has the advantage of less computational effort.

Using derivatives requires a lot of calculations, but the advantage is that the idea is natural.

With a little processing of this problem, we can form a mean inequality. Let k be greater than or equal to an expression. Since it is greater than or equal to, it is natural that the minimum value is when it is equal.

At the same time, expressions with unknown variables are functions. Since they are continuous functions, they can naturally reach their extreme values ​​when the derivative is equal to zero.

There is no essential difference between the two, they lead to the same goal.

An Chengzhang naturally knew the answer to the question he had set himself.

Liang Peixuan looked at the expression in silence for a few minutes, and finally began to take the derivative.

Seeing this, An Chengzhang nodded. The subsequent results were no longer important. Liang Peixuan was on the right track. He believed that with Liang Peixuan's strength, the result would not be wrong.

Then, he came up behind Chen Hui, who had just stopped writing and had already finished answering the second-to-last question.

In summary, when am takes its minimum value, a1/b1=(m-1)^2.
The answer is correct!

Chen Hui used the mean inequality,

Having a poor memory does not mean that you cannot remember things. Thanks to the fact that he spent ten times more time than his classmates on rote memorization, he saved Chen Hui several minutes during the exam.

An Chengzhang's pupils shrank slightly and his heart was shocked.

Simple and elegant!

When he saw the test paper written by Chen Hui, these two adjectives unconsciously appeared in his mind.

If such an answer came from a player of the Digital Competition Team, it would be natural, but who is Chen Hui?

I only got 90 points in the math test at the beginning of the school year yesterday. In last year's final exam, I was ranked 387th among the 459 students in my grade.

Could he write such an answer?
An Chengzhang was a little confused. Could it be that diligence really make up for one's shortcomings?
But isn’t this a bit exaggerated?

At this moment, Chen Hui had already started writing again.

"???"

Several big question marks popped up in An Chengzhang's head.

Just now he was looking at the second to last question that Chen Hui had finished. This amount of time was just enough for him to read the question once, and he had already found the solution?

If this question was not set by him, even if it was him, he would have to think carefully before solving it.

However, at this time Chen Hui had stopped writing again, and saw a big word "解" (Solution) appear in the blank space of the last question.

"..."

An Chengzhang was in a complicated mood.

I can only say that the students I taught have developed good habits.

11. In a rectangular coordinate system, the hyperbola (Gamma) Γ: x^2/3 - y^2 = 1. For any point P in the plane not on Γ, let Ωp be the set of points that intersect Γ at two points through P. For any point (a't) ι∈Ωp, let M and N be the two points of intersection of ι and Γ, and define fp(ι) = |PM|·|PN|. If there exists a line ι0 that intersects Γ at two points on opposite sides of the y-axis, and for any line ι∈Ωp, ι≠ι0, fp(ι) > fp(ι0), then P is called a "good point." Find the area of ​​the region formed by all good points.

The title is long, but that’s a good thing.

After writing down the solution, Chen Hui read the question again and began to write as if possessed by God.

Since the question is about the area of ​​the region formed by point P, it is natural to construct some restrictions on the coordinates x0 and y0 of point P based on the conditions of the question.

For the time being, Chen Hui had no idea, but this was an analytic geometry problem, so Chen Hui did not try a new approach and began to solve the problem according to the conventional method of analytic geometry.

First, find an expression for the straight line passing through the hyperbola at point P, then combine the expressions for the straight line and the hyperbola. Since there are two intersection points, (delt)Δ is greater than zero. Once you get an inequality, keep it for future use.

The Δ formula is also a conclusion, and Chen Hui happened to have memorized it, so he didn't even need to derive it, he could just use it.

Then, based on the known conditions of the question, use the dot product double root method to quickly write the expression of fp(ι). According to the question, fp(ι) has a unique minimum value. Analyze the minimum value of fp(ι) and get an expression for x0 and y0. Combined with the inequality of Δ obtained previously, we finally get the inequality of the four P point coordinates (x0, y0).

Draw the figure and finally calculate that the area is equal to S = double root of 2·double root of 2·one-half=4!

All in one go!
call!
After letting out a long sigh, Chen Hui felt refreshed, as if all his Ren and Du meridians were opened up, and he felt comfortable all over.

This set of questions was of moderate difficulty, and the last question was not difficult either, except that it required a lot of calculations. After figuring out the answer, he was already sweating profusely. The intense mental exercise had drained his physical strength.

Jingle Bell!
"Fuck, this is too difficult!"

"Xiao Ming, have you finished the second question in the test?"

"What's the second question? I couldn't even finish the sixth fill-in-the-blank question!"

"Who am I, where am I, what am I doing?"

"Who will save me?"

"The classes will be divided into arts and science classes in the second half of the year. I think I'd better go to the arts class!"

The bell for the end of get out of class saved the students who were tortured to death. Freed from the test papers, many of them collapsed in their seats, groaning in pain.

"?"

Chen Hui rarely raised his head from studying, his eyes full of confusion.

The questions this time are not difficult, right?!
(End of this chapter)