Li Qiong: "There are problems, but you don't meet the standards if you don't give up the problems!" ”

"Teacher, isn't it right for me to do this? Shouldn't that be? ”

You Mengjia wondered.

Li Qiong: "The result is right, but it doesn't mean that the process is right, you are a blind cat who has met a dead mouse!" ”

At this time, Li Qiong returned to Leng Yuan's side after walking around, reached out and picked up the roll on his desk and asked softly, "What do you usually do at home when you are fine?" Have you ever done a math Olympiad problem? ”

Leng Yuan raised his head and said with a smile: "No, you can't do anything at home, let alone the Olympiad math questions!" ”

Li Qiong nodded, didn't say anything more, she returned to the podium and kept flipping through the textbook and the thick lesson plan.

Ten minutes later.

Li Qiong raised her head and said: "Next, I will explain them one by one, you will see if these questions are difficult or not, what is wrong with you, and whether you should lose your scores!" ”

In the next one or two lessons, Li Qiong will explain the problems in detail, always be able to turn the difficult problems into simple questions, and use ingenious methods to avoid thorns and finally get the result, of course, some problems cannot be bypassed at all, you must be careful and have high calculation ability and judgment ability, mathematics is like this.

Whenever the teacher finished explaining a question, the students in the audience would let out a sigh of regret, why didn't they do it right when they should have done it? That's probably what everyone thinks.

When there are fifteen minutes left before the end of the class, all but the last question is proof.

When the teacher saw this last proof question, he smiled and said, "This question is a little interesting, it is a difficult problem, but you can't prove the second question, the first question should be fine, right?" It's not beyond your means, is it? Raise your hand and let me see, who made this question first? ”

Only ten of the sixty-four people raised their hands, the fat man Wang Di was in the column, the math class representative Li Qi was in the column, the squad leader Mo Yao was in the column, Leng Yuan was in the column, and there was a female comrade, and the rest were all gay men sitting in the front rows.

"Put your hand down, and I'll take a look at the second question!"

Hearing Li Qiong say this, the ten people who raised their hands suddenly put down nine helplessly, and only Leng Yuan showed off a flower.

"Put your hands down! There are more than 1,600 people in the whole school, and Leng Yuan in our class has done this problem completely, to be honest, I admire him very much, in just two hours, there is not a lot of time left for this problem, even if he does it in the end, there is no time left, but Leng Yuan can make such a complex problem in just ten minutes, which can only show that he is very sensitive to mathematics! ”

"For the rest of the time, give it to Leng Yuan and ask him to talk about his method, to be honest, it is much simpler than my method! Come, Leng Yuan will come to the stage to speak! ”

Leng Yuan got up immediately, two classes, enough to rest, the pain in his body was not so obvious, I really don't know how he did it, the blue and black eyes were much lighter, if you don't look carefully, you can't see it at all.

Leng Yuan came to the podium and glanced up at the compatriots below, and there was an indescribable feeling, like a free man who jumped out of the game, and the classmates in front of him couldn't escape these eyes with every move.

Suddenly, he was stunned, so what about the small actions he has done in the past few months, sleeping and reading other books have been seen by the teacher, but he didn't say it on the spot, it seems that he has to check some in the future, otherwise it will be a little disrespectful.

Leng Yuan coughed and began to talk about the topic.

"There are two questions in this question, the first question proves that the sum of the three xyz is not more than one, and the product of any two is not less than one, and the condition in the question is ......"

Leng Yuan stood on the podium, his eyes swept over the students from time to time, and at the same time analyzed the content of the question.

"At the beginning, I thought that I could solve the requirements of the problem by solving the range of each unknown quantity, but when I solved it, I found that the range of position quantities was too large to prove the requirements of the problem. Then it occurred to me that it was very likely that I was wrong, and the question didn't seem to ask me to do that, because the second question gave me a hint. ”

"In the second question, a function is given, and the range of the value of the function is required to be proved to be in the left-open and right-closed range from 0 to 1, so I thought that the proof process of the first question may require three unknown quantities in the same formula to be proved."

So the question arises again, what kind of formula should these three unknown quantities be put into order to meet the requirements of the problem? Looking at the first question, there is the sum of the three, and there is the difference between any two. ”

Speaking of this, Leng Yuan paused and continued: "So I thought of a model, that is, the triangular model!" Prove that the sum of the two sides in the triangle is greater than the third side, and the difference between the two sides is less than either side! ”

Speaking of which, all the students in the class understood the meaning, and their eyes were all lit up, and their hearts said: Sure enough, there is a li! So they nodded in agreement.

Speaking of this, Leng Yuan drew a triangle on the leftmost blackboard at random, marking xyz in order rather than randomly, which is the length of each side.

After doing this, Leng Yuan exhaled and continued.

"But the question arises again, how can we get both the sum of the three and the product of the two?"

Mo Yao, who was facing Leng Yuan's four eyes, suddenly said: "The square of the three!" ”

Leng Yuan said: "That's right, what you need is square!" ”

So Leng Yuan turned around, and wrote out the sum of xyz in a few clicks, flat and squared.

"What do you do next, so that you can solve it?"

Mo Yao continued: "Bring the range of XYZ into the formula! ”

Leng Yuan brought it into it according to his words, and went through the two-part formula change, but as a result, he wanted the formula, but he didn't get the desired result.

Leng Yuan looked at the blackboard and then looked at the people under the stand and said, "There are some problems, what's going on?" ”

At this time, no one answered, only silence.

Leng Yuan turned around and said: "The problem is here, the condition given in the question is not to furnish it, it shows that the triangle with xyz as the three sides is a right triangle, what does the right triangle mean?" Explain that the sum of squares of xy is equal to the sum of squares of z, and if you bring it into the equation, then the square direction of xyz becomes an item! ”

As he spoke, Leng Yuan drew a triangle again, this time at a right angle, and marked the relationship between their sides.

This time the relation of xyz is brought into the formula, and the result really appears, and it is two formulas.

Wang Di sighed: "It's okay to do this, it's really amazing!" I didn't even think about it! ”

Leng Yuan continued: "In this way, the first question is proven, and the next question needs to be obtained with the help of the result of the first question, which is the characteristic of this kind of multi-question proof question, and ninety-five out of a hundred questions are like this. ”

When he finished saying this, he didn't notice that Li Qiong suddenly glanced at him and thought for a while.

“… The second question is such a long string of formulas, how can we use the first question? All we need to do is deform it! ”

Leng Yuan quickly wrote the formula on the blackboard, and then transformed it into a horizontal direction of the sum, in the form of three twos and two products, but there was an extra two-fifths coefficient in front of it, and then combined them together, and the result was less than or equal to one.

Thus the right side of the proposition is proven, and the next is the left side. There is no need for this proof on the left, just find the minimum value of the function! ”

"Because the square of the sum is greater than or equal to zero, the square of the three is the smallest when taken as zero. However, it is given in the question that xyz is a positive number greater than zero, so you can't take zero at the same time! ”

"This can only be done in a limiting way, when xyz infinity is infinitely close to zero, and the value of the function is zero, but in fact it is not equal to zero!"

Therefore, the second proposition is proven, and the value range is 0 to 1 in the range of left open and right closed! That's it! …”

Looking at the blackboard again, three blackboards are already full.

Dense small print, as well as several function images, are very rich.

--Crash--

When Leng Yuan gently clapped his hands and walked off the podium, the whole class burst into warm applause, even their teacher Li Qiong.

Li Qiong smiled: "Do you understand everything?" If I could say that, maybe it wouldn't be as good as that! ”

Wang Di smiled bitterly: "I understand, I understand, but this is too complicated and difficult, right?" How could you be so nervous during the exam? ”

"It's not complicated anymore, wouldn't you cry if I said it? This method is very good, take this down, and look at it when you go back, it's going to be time! ”

"Teacher, is there any other way to solve this problem?" You Mengjia asked.

"Yes, but I don't recommend that!"

Having said that, Li Qiong still briefly talked about that method, and when he finished listening, everyone, including Leng Yuan, was stunned, there were too many windings, and he couldn't remember it if he wanted to!

Li Qiong smiled: "If there is anything that is not confused, raise your hand and let me see!" ”

None of the class raised their hands, and they all looked like they were constipated.

"Look, I know this is the result, you better remember Leng Yuan's method. What is wrong with you? ”

Li Qiong pointed at Mo Yao and asked.

"I had the same idea as he had in the first question, which was to narrow down the range of xyz as much as possible, and then add them to multiply, and proved the first question, but the second question was not proven!"

"It's not bad, you're the only one in the class who didn't get everything right except for the second question! Look at it, this paper is not difficult, and then look at your exams, all kinds of mistakes are different, this is going to teach people to be shot early on the battlefield. There are only a few 100 points, and there are only a few 113 in other classes..."

--Bell Bell Ring--

At this moment, the bell rang for the end of class, and the irritability in the class was diluted a lot.

"Let's get out of class, go back and look at this set of papers for yourself!"

"Stand up!"

"Goodbye, teacher!"

…

I took two classes in a row, which was a kind of torment, I was a little tired and needed to take a break, so the school prescribed a morning meal at this time, for the sake of the student's body and the student's mental outlook.

Between classes, Leng Yuan was just about to go out, but was stopped by You Mengjia.

Leng Yuan said: "What are you doing?" ”

You Mengjia pinched her waist and laughed softly: "Okay, I didn't expect your math to be so good?" If you have time, tell me about it, but you are not allowed to shirk, and some people who resist Hugh's strange old lady for committing domestic violence against you! ”

The corners of Leng Yuan's eyes and mouth twitched involuntarily, and when he heard this, he said: "Forget it, you should find the class leader, her old man is not much better than me, that is a full score at every turn, the guy who is the first in the grade, if you think that guy is too self-righteous, it's okay to find our great class representative, I just ask for one thing, don't do this to me anymore, you are uncomfortable, goosebumps are layered layer by layer!" ”

"Bastard pervert, how dare you refuse me? Underwhelmed beaten! ”

You Mengjia lifted one leg high and kicked Leng Yuan's back regardless of wearing her skirt.

Leng Yuan took a step forward suddenly, dodged You Mengjia's leg, and turned around to see the squad leader Mo Yao

Leng Yuan said: "If you want to be like this, I can fight back, I can't be bullied if I really be an uncle?" I'm in a hurry, I'm going to look for you all over the place! ”

However, Leng Yuan saw that the startled face of the squad leader Mo Yao was returning to normal, he naturally didn't know what was going on, and just glanced at him and went downstairs to the field.

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