"It's okay, but Teacher Lin asked me to go to his place to talk about the competition test questions tonight, so I'm afraid there will be no time tonight.

Moreover, if I can’t finish speaking tonight, I might have to go there tomorrow night.” Zhao Xiancai said.

"It's okay, there's no hurry to do this. Even if you join the provincial team, there's still some time before you go to CMO.

And after you talked it from Teacher Lin and the others, you will definitely be able to speak better in the class.”

Seeing that Zhao Xiancai agreed, Fang Qiu said with a smile.

After Fang Qiu and Zhao Xiancai hadn't talked for a few words, the bell for the evening self-study was also ringing, and Zhao Xiancai also left the classroom.

...

“…It is known that the bottom edge of the positive triangular pyramid P-ABC is 1 and the height is √2. What is the radius of the incision sphere?

I don’t need to talk about this fourth question?

The first question we answered in the preliminary round is exactly the same as this question, even the side length is the same. This is the height and the answers of our several are the same.

When I was in the initial competition, I was thinking in my mind that even if this question was made on the fill-in-the-blank question, it would be fine to ask you to cut the ball inward. I didn’t expect it to be made in this league.”

During evening self-study, Zhao Xiancai, Ji Xinglei and Teacher Lin were all in the same classroom, and Zhao Xiancai was talking to them about the league test questions.

“Well, this question is exactly the same as the first question in the preliminary round.

None of you has any problem with this question, right?

There is no need to talk about this topic, you can continue to talk about it."

After listening to Zhao Xiancai finishing the question on this question, Lin Guangqiang felt that this question was so familiar.

After Zhao Xiancai finished speaking, Lin Guangqiang also remembered that this question was the original question in the preliminary round?

"The fifth question is to let a and b be a real number, and the function f(x)=ax b satisfies: for any x∈[0,1], there is 丨f(x)丨≤1, and ask what is the maximum value of ab.

I asked this question through ab=f(0)·(f(1)-f(0))=-(f(0)-f(1)/2)2 (f(1))2/4≤(f(1))2/4≤1/4. We only had the answers to the fill-in-the-blank questions before, and how did we write them before we hadn't talked about it.

Do you use this method too?”

When he said this, Zhao Xiancai looked at Ji Xinglei and the others again.

"I use the same method as yours."

"I wrote this wrong."

"Hey, I didn't use this method in this question, but the final result was calculated as 1/4 like you."

Zheng Wenyi used the same method as Zhao Xiancai, and Ji Xinglei made the problem wrong.

Finally, I said that the method used in this question is different from that of Zhao Xiancai, and it is Zhang Jingqiang.

When Zhao Xiancai heard Zhang Jingqiang say this, he became interested and asked him: "What method did you use?"

This is one of the reasons why Zhao Xiancai has finished writing all the league test questions and feels that he can get full marks, but he still agrees to tell them.

As the saying goes, three cobblers are better than Zhuge Liang and communicate with others can also bring more problem-solving ideas to Zhao Xiancai. After all, no matter how capable a person is, he is limited.

"I think so, since |f(x)|=|ax b|≤1, then there is -1≤ax b≤1. When x is equal to 1, there is -1≤a b≤1.

Then according to the basic inequality √ab≤(a b)/2, there is 0≤ab≤(a b)2/4=1/4.

So, the maximum value of ab is 1/4."

Zhang Jingqiang and Zhao Xiancai talked about their own problem-solving process.

After listening to Zhang Jingqiang's words, Zhao Xiancai felt something was wrong, but he couldn't say it for a while.

"What's wrong? Is there anything wrong with his solution?"

Zheng Wenyi, who was standing beside him, asked when he saw that Zhao Xiancai didn't speak.

"It seems to be a problem, but I can't tell you where there is a problem.

Teacher Lin, do you see what is wrong with Zhang Jingqiang’s question? Or am I thinking too much?"

After Zhao Xiancai and Zheng Wenyi talked, he asked Lin Guangqiang again.

Lin Guangqiang is worthy of being a teacher for so many years and has a lot of knowledge. He can see at once that something is wrong with Zhang Jingqiang's question.

"He has a problem with the 0≤ab≤(a b)2/4=1/4.

Zhang Jingqiang, I ask you, what is the definition of the basic inequality?"

Lin Guangqiang said to Zhang Jingqiang that he did not directly point out what went wrong with Zhang Jingqiang's problem-solving process.

“It’s √ab≤(a b)/2?

I remember the proof process, because (√a-√b)^2≥0, a-2√ab b≥0, then a b≥2√ab, and finally √ab≤(a b)/2.”

Before Zhang Jingqiang realized what he was wrong, he said confidently.

"Oh~~~ I knew something was wrong. I was just taken in by Zhang Jingqiang, but I didn't expect this problem for a moment.

Zhang Jingqiang, let me ask you, what if ab is one positive and one negative or two negative numbers?"

"One positive and one negative?

Oh, I forgot this, the premise of the basic inequality is two positive real numbers."

"The basic inequality is expressed as that the arithmetic mean of two positive real numbers is greater than or equal to their geometric mean.

I believe your math teachers have emphasized it too when they are in class. When using basic inequality, you must remember that both equations must be positive.

Zhang Jingqiang, you can't be so careless. If something happens to you because of this small problem during the college entrance examination, you may be passing by the school you want to go to."

Lin Guangqiang said to Zhang Jingqiang in a stern tone that although Zhang Jingqiang's grades are not among the top three in the third grade of Chengxian No. 1 Middle School, they are often among the top ten in the whole grade.

This time he made such a low-level mistake. Although the answer was right because of his good luck, as a teacher who taught Zhang Jingqiang for a period of time, he still had to say it.

"Remember, remember, our teacher told us at that time that when using basic inequality, we should remember the seven-character mantra of "one positive", "two fixed", and "three equality".

‘One positive’ means that both expressions are positive numbers, ‘two determination’ means that when a basic inequality is used to calculate the most value, the sum or product is a fixed value, and ‘three equality’ means that the equal sign can be taken if and only when the two expressions are equal.

But I forgot about this at that time.”

Seeing that Lin Guangqiang's tone became serious, Zhang Jingqiang also said quickly.

"But why did Zhang Jingqiang's answer come from the same answer as Zhao Xiancai's?"

Ji Xinglei asked strangely.

At this time, Zhao Xiancai also spoke up to Ji Xinglei's doubts.

"In fact, there is no problem with this method in this question, but he lacks a few steps.

Since this question is to find the maximum value of ab, then a and b must either be positive or negative, and it cannot be one positive and one negative.

Because if it is positive and negative, the product of ab is a negative number and cannot be the maximum value...

If they are all positive numbers, the process does not need to be changed.

If all are negative numbers, you can also use -a=t, -b=w to calculate..."

"Oh, so that's it."

"No wonder, you guys are so lucky."

"Okay, everyone has figured out this question, so let's get the next question.

The difficulty of these fill-in-the-blank questions has increased, right?

I remember when Zheng Wenyi talked to me at noon, she said that although the answer to her sixth question was the same as Zhao Xiancai's, her answer was guessed.

Moreover, the value range of Zheng Wenyi's question seven is considered wrong, so he skipped the eighth question directly.

So, you guys should listen more carefully after this.”
Chapter completed!