I really just want to be a scholar

Chapter 64 Ceva's Theorem Menelaus' Theorem
Looking at the time, it actually took about 15 minutes. Qin Ke rubbed his cold and stiff fingers, then pinched his thigh vigorously to keep his brain awake, but the chill on his body was getting stronger and stronger, and the breath he exhaled It is getting hotter and hotter, and the temples are getting more and more painful, and there is a faint feeling of dizziness.

Qin Ke can clearly feel that his thinking speed has dropped a lot compared to when the exam started.

No, you have to speed up!

Qin Ke took a deep breath, forced himself to concentrate, and continued to look at the second additional question.

"Additional question [-]: It is known that there are points D, E, and F on the three sides BC, CA, and AB of △ABC, and AD, BE, and CF intersect at a point G. If the areas of △AGE, △CGD, and △BGF Equal, verify: G is the center of gravity of △ABC."

Qin Ke heaved a sigh of relief. This question seems to be relatively easier than the first question just now. The main knowledge point involves the "center of gravity" in the five centers of a triangle, that is, the point where the three midlines of the triangle intersect.

This is the knowledge point of high school students. If you want to prove that G is the center of △ABC, you only need to prove that D, E, and F are the midpoints of △ABC.

It seems simple, but it is extremely difficult to prove this point, because only the condition of equal area is given in the title.

Area...

Qin Ke immediately tried to use the best structure method plus the area method to prove it, but after thinking about it in his mind for a while, he found something wrong.

Under the current conditions, no matter how it is constructed, combined with the area method, it will only make the problem more complicated. Even if it is written on a whole page, it may not be able to prove it!That would be a complete waste of time and energy!

This is a trap for the questioner!

Damn, the guy who made the question this time is a bit of a level... but he is not in a good state.

Qin Ke pinched his thigh twice again. The severe pain finally made his brain clear for more than ten seconds. He immediately caught the flash of inspiration. Yes, there are Ceva's theorem and Mene in plane geometry. Routh's theorem?

Especially to solve a point in a triangle and the relationship between points in the three sides of a triangle, the most suitable ones are the edge element Ceva theorem and the angle element Ceva theorem!

Although these two theorems are a bit out of the ordinary, didn't I just explain them to Ning Qingyun the day before yesterday?

Qin Ke quickly thought of the idea of ​​proof, picked up a pen and drew a picture, and wrote:
"Proof: As shown in the figure, assuming AF/FB=x, BD/CD=y, EC/EA=z, xyz=1 can be obtained by edge-element Ceva's theorem.

For △BFC and straight line AGD, using Menelaus theorem, FE/CG times CD/DB times BA/AF=1.

……

It can be obtained from the above formula that x=y=z, and from xyz=1, x=y=z=1, so it can be concluded that D, E, F are the midpoints of △ABC, so G is the point of △ABC center of gravity.The original title is proved. "

After writing the proof process of more than 30 lines, Qin Ke breathed a long sigh of relief. The person who wrote the question was obviously digging a hole, and it was specifically aimed at candidates like himself who are familiar with using various problem-solving skills and strategies. Go astray.

Even an old hand like him almost lost his way. He was confused by the usual problem-solving skills and took a big detour, which made the solution of this problem extremely complicated and difficult. It would take an hour to prove it.

Fortunately, I quickly discovered its conspiracy, and directly used Ceva's theorem and Menelaus' theorem to solve the problem, which greatly shortened the proof process and exhausted time.

——Academic committee, ah, I just told you about Ceva's theorem and Menelaus' theorem the day before yesterday. Don't fall for this question, but it's a full fifty points!What a pity if you can't get it!

However, Qin Ke has no energy left to think about Ning Qingyun's matter now. His dizziness is getting worse and worse. Fold it up, press it under the main volume and the draft paper as a pad, and start to attack the ten provincial-level problems in the main volume.

The difficulty level of these ten big questions is also irrelevant to the serial number. Qin Ke has no time or thought to examine the questions one by one. Anyway, the goal is to get a perfect score, so let's conquer all the questions in one fell swoop in order.

He started to solve it directly from the first one.

"Question 1: Given that the function f(x) satisfies f(x^2)-f(x)=1, find f(x)."

Normally, Qin Ke would be able to see the steps and even the final answer for such a question at a glance, but at this time his state became extremely bad, and when he looked at the words of the question, there were some double images.

The brain is more like a mirror covered with mist, blurred.

He tried his best to think for nearly 3 minutes before he figured out to solve this problem by constructing a recursive sequence.

Struggling to squeeze the pen tightly with his weak hands and finish writing the answer stroke by stroke, Qin Ke actually spent nearly 7 minutes, and his body was already sweating unconsciously.

Without looking in the mirror, Qin Ke could guess that his face must be very strange now, his hands were frighteningly cold, his lips were dry and his tongue was dry, and he even felt a little chest tight and nauseous, which was extremely uncomfortable.

Looking at the time, nearly an hour has passed since the exam started.

This is the first time Qin Ke has spent such a long time doing the math test paper in recent months. Under his usual best condition, he is afraid that he has already finished all the test papers.

But Qin Ke didn't have the strength to think about it anymore. He wrapped his down jacket tightly and took a few deep breaths. The cold air penetrated into his chest, allowing him to quickly force himself to concentrate, and then continue to work on each question.

But his physical condition was getting worse and worse, he began to feel drowsy and tired, the time of thinking became longer and longer, and the time spent on solving problems became longer and longer.

The next question is no longer a question of intelligence and skill, but a double test of perseverance and physical strength!

……

Deng Hongguo arrived at the Provincial Culture and Sports Center after the provincial competition started. He had an important meeting yesterday and couldn't be absent until last night. state airport.

After getting off the plane, he didn't even bother to eat breakfast, so he got in Shi Cunyuan's car and drove over.

The reason why he traveled so tirelessly for thousands of miles was naturally to see with his own eyes the rumored extremely powerful genius student named Qin Ke.

Now the national training team is really short of such talented seedlings. Even if there is a [-]% possibility, Deng Hongguo is unwilling to miss it.

Several leaders of the Provincial Olympic Committee heard that such a big man was coming, so how could they not come to accompany him?Deng Hongguo waved his hands again and again, indicating that Shi Cunyuan could accompany him, but the hospitality was hard to come by, and he still had to greet these leaders, which wasted a lot of time.

After finally putting down their teacups and leaving the office, Deng Hongguo and Shi Cunyuan braved the bitter north wind and walked towards the examination building.

Deng Hongguo rubbed his brows, took advantage of the cold wind to clear his brain and dispel the fatigue of long-distance business trips, then turned his head and asked Shi Cunyuan next to him: "Brother Cunyuan, the two additional questions in this provincial competition took a lot of effort. what do you think?"

Shi Cunyuan frowned and said: "It's very difficult. The first question involves the Hamiltonian problem, which is relatively unpopular. It is estimated that few candidates can solve it. The second question, you are specifically targeting Qin Ke, right?"

Hearing that an old classmate discovered the beauty of his topic, Deng Hongguo's bloodshot eyes suddenly showed complacency:
"That's right, the second question is used to dig a hole for Qin Ke. This kid has a good grasp of various Olympiad problem-solving strategies and skills. It has become his instinct to use these skills to solve problems. I also specially give it to him. Given such delicious bait, it's no wonder he didn't take the bait."

　Thanks to "Ink Retelling the Truth", "Book Friends 20200412155813055", and "Da Muze" for their rewards!Tomorrow the two chapters are big chapters, very refreshing!
　　
　
(End of this chapter)