I really just want to be a scholar

Chapter 48
Qin Ke brushed and drew while writing in the answer area of ​​the test paper:
"Solution: Arrange 1, 2, ..., 13 in a circle according to the following rules: first arrange 1, put 1 next to 9 (the difference with 1 is 8), put 9 next to 4 (the difference with 9 is 5 ), and continue down like this, the number next to each number differs from it by 8 or 5, and finally a circle as shown in Figure 1 (1, 9, 4, 12, 7, 2, 10, 5, 13, 8 , 3, 11, 6), the numbers on the circle can satisfy at the same time:”

"(1) The difference between every two adjacent numbers is either 8 or 5;

(2) The difference between two non-adjacent numbers is neither 5 nor 8.

So this question can be reduced to: On this circle, at most a few numbers can be selected so that every two numbers are not adjacent on the circle. "

OK, it's done, and the naturalization is complete.

Is this naturalized question exactly the same as the example he gave Ning Qingyun?

So there is no difficulty for Qin Ke to do it next, just write out the solution of that example.

"Draw another circle and arrange 1, 2, ..., 13 in sequence, then you can select 6 numbers that meet the non-adjacent conditions, such as 1, 3, 5, 7, 9, 11. See Figure 2.

Next, verify that you can select up to a few numbers.We first choose the number 1 arbitrarily, and at this time the adjacent 2 and 13 cannot be selected, and the remaining 10 numbers are matched into 5 pairs, which are: (3, 4), (5, 6), (7), (8), (9).Among these 10 pairs of numbers, each pair can only select at most 11 number, that is to say, together with the number 12, only 5 numbers can be selected at most so that they are not adjacent to each other.

From this we can conclude that the answer to this question is: 6. "

Qin Ke was relaxed and happy, and solved the first additional question in less than 5 minutes.

He glanced out the window, wondering if Ning Qingyun remembered the sample question and whether she could use the method of transformation. If she could remember it, then the 25 points would naturally be in her pocket.

Come on, school committee, I can only help you so far.

Qin Ke looked at the second question again. The second question was also quite difficult. No wonder it was selected as the major question in the extra paper.

"Additional question 2: Assume that in △ABC, the opposite sides of vertices A, B, and C are a, b, and c respectively, and the distances from inner I to vertices A, B, and C are respectively m, n, and l. To prove: al^ 2+bm^2+cn^2=abc"

This question seems to have insufficient conditions to start, but after thinking about it, Qin Ke has an idea.

He decided to use the area method to prove it.

The most basic idea of ​​the area method is to use two different methods to calculate the same area, and the results should be equal.

Firstly, the radius R of the circumscribed circle of △ABC is introduced, and by the sine law a/sinA=b/sinB=c/sinC=2R,

Triangle area S=(1/2)absinC
=(1/2)ab c/2R
=abc/4R,

So S=abc/4R.

Then divide △ABC into three quadrilaterals, the area S of ΔABC is obviously equal to the sum S of the areas of the three quadrilaterals.

In this way, the above S=abc/4R and the sum of the areas of the three quadrilaterals are used to establish an area equation.

According to the fact that the three quadrilaterals have circumscribed circles, and the diagonals are perpendicular to each other, it is not too difficult to express their areas with known quantities, and then the sine of the angle can be eliminated with the help of the circumscribed circle radius R of △ABC, no surprises , it is easy to prove this conclusion.

OK, let's get started.

"Proof: Let the inscribed circle of △ABC and the three sides BC, CA, AB be tangent to D, E, F respectively, connect EF, EI, FI, DI, AI respectively, and get three quadrilaterals AEIF, BFID, CDIE respectively... ..."

“所以S△ABC =SAEIF+SBFID+SCDIE=（al^2+bm^2+cn^2）/4R，又因为S△ABC =abc/4R，由面积法可知两种方法求得的S△ABC相等，由此得出al^2+bm^2+cn^2=abc”

Qin Ke put down his pen with a relaxed expression. After checking for the last time to make sure that there were no missing questions, he looked at the candidates on the left and right, and saw that everyone was frowning.

Qin Ke rolled his eyes, deliberately raised his hand and said, "Report to the teacher."

Originally, the examination room was very quiet, only the sound of the invigilator pacing and occasionally coughing, so although Qin Ke's voice was not too loud, most of the students could still hear it, and they couldn't help but look up at him curiously.

Especially Wenwen Yan, who subconsciously trembled when he heard Qin Ke's voice, and even interrupted his thinking about doing the questions.

At this time, a female invigilator teacher in her early 30s came over and asked Qin Ke, "Student, what's the matter with you?"

"I want to hand in the paper."

Hearing this sentence, all the candidates in the examination room who were struggling with the examination papers came up with an idea almost at the same time. Finally, someone was overwhelmed by the many and difficult preliminary papers and was about to give up early.

While everyone sighed in their hearts, they also secretly encouraged themselves. I can't give up easily like this guy. I have to persevere and play the Olympic spirit. Even if I can't do it, I have to persist until the last moment!
Come on, XXX, you can do it!

The students in the examination room had the same thoughts, and Wen Wenyan and Chi Jiamu in the distance were even more refreshed.

The two of them could not calm down after entering the examination room for a long time, and finally got into the mood. They had just solved one or two questions when they heard that Qin Ke was about to hand in the paper. The two of them couldn't help sneering "heh" while their hearts relaxed. The ruffian is really bad, and he doesn't know how to get into the team, so get out of here.

The full text of Qin Ke's "feud is as deep as the sea" replied confidently.

I was a little afraid of this scumbag just now, it's just ridiculous!This time, I was going to score with iron, and slapped him in the face fiercely!Let him know that this Mathematical Olympiad exam is not for a scum like him to sneak in and play wild!

There is also that Ning Qingyun, who is so arrogant just because she is beautiful, and this time, I have to get rid of her by a few points, so that she will never be able to lift her head up in front of him!
The whole text here is full of morbid revenge and the pleasure of his own sexuality. The female teacher who came to Qin Ke's side has a good temper and is also responsible. You can't hand in the paper. This time the topic is indeed a bit difficult, even if you don't know how to do it, you should think harder, maybe..."

She was used to seeing students who collapsed in the Mathematical Olympiad and knew how to appease them.

"But I'm done."

The female teacher stopped abruptly in the middle of her speech, and she thought she had heard it wrong: "What?"

Qin Ke waited for her words, then amplified his voice and said: "I said, I have finished the test paper. This paper is too simple and not difficult at all. I have finished all the questions, and there is nothing to do when I stay. Can't you hand in the papers in advance?"

The audience, which was originally a bit enthusiastic and positive, suddenly became dead silent.

All the candidates looked up at this brazen candidate in astonishment, and only one thought flashed in their minds, bragging, right?how can that be?

(End of this chapter)