Qin Ke started writing and drawing in the answer area of ​​the test paper:

"Solution: Arrange 1, 2,...,13 in a circle according to the following rules: Arrange 1 first, put 9 next to 1 (the difference from 1 is 8), and put 4 next to 9 (the difference from 9 is 5

), continue like this, the number next to each number differs from it by 8 or 5, and finally get 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 simultaneously satisfy:”

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

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

Therefore, this question can be reduced to: on this circle, how many numbers can be selected at most so that every two numbers are not adjacent on the circle."

ok, done, the transformation is completed.

Is this post-reversion problem exactly the same as the example he gave to Ning Qingyun?

So it would be easy for Qin Ke to do what he did next. He could just write down the solution to that example.

"Draw another circle and rank 1, 2,...,13 in order. 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 how many numbers can be selected at most. We first select the number 1 arbitrarily. At this time, the adjacent 2 and 13 cannot be selected. Match the remaining 10 numbers into 5 pairs, which are: (3

,4), (5,6), (7,8), (9,10), (11,12). Among these 5 pairs of numbers, at most one number can be selected from each pair, that is to say

, including the number 1, only up to 6 numbers can be selected so that they are not adjacent to each other.

From this it can be concluded that the answer to this question is: 6."

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

He glanced out the window, wondering if Ning Qingyun had remembered this example and whether she could apply the transformation method. If she could remember it, then she would have secured the 25 points.

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 a major question in the additional 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 center i to vertices a, b, and c are m, n, and l respectively. Verification: al^

2b^2=abc”

This question seemed to be impossible to start with due to insufficient conditions, but Qin Ke thought about it briefly and came up with an idea.

He decided to use the area method to prove it.

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

First, introduce the radius r of the circumcircle of △abc. According to the sine theorem a/sina=b/sinb=c/sinc=2r,

Area of ​​triangle s=(1/2)absinc

=(1/2)ab·c/2r

=abc/4r,

So s=abc/4r.

Then divide Δabc into three quadrilaterals. The area s of Δabc is obviously equal to the sum of the areas of the three quadrilaterals s.

In this way, the area equation is established by taking the above s=abc/4r and the sum of the areas of the three quadrilaterals.

Furthermore, based on the fact that all 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. Then with the help of the radius r of the circumscribed circle of △abc, the sine of the angle can be eliminated. No surprise.

, we can easily prove the conclusion of this question.

OK, let’s get started.

"Proof: Suppose the inscribed circle of △abc is tangent to the three sides bc, ca, and ab at d, e, and f respectively, and connects ef, ei, fi, di, and ai respectively to obtain three quadrilaterals aeif, bfid, and cdie respectively...

…”

"So s△abc =saeifsbfidscdie=(al^2b^2)/4r, and because s△abc =abc/4r, it can be seen from the area method that the s△abc obtained by the two methods are equal, thus al^2b

^2=abc”

Qin Ke put down his pen with a relaxed expression. After checking for the last time to make sure that he had not missed any questions, he looked at the candidates on the left and right and saw that everyone was frowning. Quan Wenyan had returned to normal and was answering questions quickly.

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

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

Especially Quan Wenyan, when he heard Qin Ke's voice, he subconsciously got excited, and even his train of thought on the question was interrupted.

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

"I want to hand in my paper."

Hearing this sentence, all the candidates in the examination room who were struggling with the examination papers almost simultaneously came up with the idea that some people were finally overwhelmed by the many and difficult preliminary examination papers and had to give up early.

While everyone was sighing in their hearts, they also secretly encouraged themselves. They couldn't give up as easily as this guy. They had to persevere and exert the Olympic spirit. Even if they couldn't do it, they had to persevere until the last moment!

Come on, xxx, you can do it!

The students in the examination room had similar thoughts, Quan Wenyan and Chi Jiamu in the distance were even more energetic.

The two of them couldn't calm down for a long time after entering the examination room. They finally got into the state. They had just solved one or two questions when they heard that Qin Ke was about to hand in the paper. While they were relieved, they couldn't help but sneered "ha".

The ruffian is indeed very bad, and he doesn't know how he got into the participating teams, so get out of here.

Quan Wenyan, who had a "deep feud" with Qin Ke, responded with confidence.

It was so funny that I was a little afraid of this scumbag just now! This time, I was going to slap him in the face with hard scores! Let him know that in this Mathematical Olympiad exam, no scumbag like him could sneak in and act wild.

!

There is also Ning Qingyun, who is extremely arrogant just because she is beautiful. This time, he will keep her away by dozens of points, so that she will never be able to hold her head up in front of him!

Quan Wenyan here was filled with morbid revenge and pleasure from his own sexual pleasure. The female teacher who came next to Qin Ke was good-tempered and responsible. She said warmly: "Classmate, according to the regulations, it will take less than an hour and a half."

You can’t hand in the paper. The questions this time are indeed a bit difficult. Even if you don’t know how to do it, you should think harder, maybe..."

She is used to seeing students who collapse after taking the Mathematical Olympiad and knows how to comfort them.

"But I'm done."

The female teacher stopped abruptly in the middle of her words. She thought she heard wrongly: "What?"

Qin Ke waited for her words, then raised his voice and said: "I said, I have finished the test paper. This paper is too simple, not difficult at all. I have finished all the questions, and there is nothing to do if I stay.

Can’t you hand in the paper in advance?”

The originally enthusiastic and positive atmosphere in the whole place suddenly turned into a dead silence.
Chapter completed!