Looking at the time, it actually took almost fifteen minutes. Qin Ke rubbed his cold and stiff fingers, and then pinched his thigh hard to keep his brain awake, but the chill in his body became stronger and stronger, and he exhaled
My breath is getting hotter and hotter, my temples are getting more and more swollen and painful, and I have a faint feeling of dizziness.
Qin Keneng clearly felt that his thinking speed had 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 2: It is known that there are points d, e, and f on each of the three sides bc, ca, and ab of △abc, and it satisfies that ad, be, and cf intersect at a point g. If the areas of △age, △cgd, and △bgf
Equality, verify: g is the center of gravity of △abc."
Qin Ke breathed a sigh of relief. This question seemed to be relatively easier than the first question just now. The main knowledge point involves the "center of gravity" of the five centers of the triangle, which is the point where the three midlines of the triangle intersect.
This is a knowledge point that high school students will all know. If you want to prove that g is the center of gravity of △abc, you only need to prove that d, e, and f are the midpoints of △abc.
It seems simple, but it is not easy to prove this, because the question only gives the condition that the areas are equal.
The area...
Qin Ke immediately tried to prove it using the construction method and the area method, which he was best at, but after thinking about it for a while, he realized something was 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 fills a whole page, it may not be able to prove it! It is really a waste of time and energy!
This is the question maker’s trap!
Damn it, the guy who asked the question this time is pretty good... but he is not in the best condition.
Qin Ke pinched his thigh twice hard again. The severe pain finally cleared his mind for more than ten seconds. He immediately caught the flash of inspiration. Oh, yes, isn't there Ceva's theorem in plane geometry? Menie
Rolls' theorem?
Especially to solve the relationship between a point in a triangle and the points in the three sides of a triangle, the most suitable ones to use are the edge element Ceva's theorem and the corner element Ceva's theorem!
Although these two theorems are a bit unfamiliar, didn't I just explain them to Ning Qingyun the day before yesterday?
Qin Ke quickly thought of the idea of ​​​​proof, drew a picture, and then wrote:
"Proof: As shown in the figure, assuming af/fb=x, bd/cd=y, ec/ea=z, we can get xyz=1 from the edge element Ceva theorem.
For △bfc and straight line agd, using Menelaus’ theorem, we can get fe/c****/db*ba/af=1.
…
From the above formula, we can get x=y=z, from xyz=1, we can get, x=y=z=1, so we can draw the conclusion that d, e, f are the midpoints of △abc, so g is the midpoint of △abc
The focus. The original question is proved."
After writing more than thirty lines of proof process, Qin Ke let out a long sigh of relief. This question writer was obviously digging a hole, specifically targeting candidates like himself who are familiar with using various problem-solving techniques and strategies. If he is not careful, he will
Go astray.
Even a veteran like him almost got lost. He was confused by the usual problem-solving techniques and took a big detour, making the solution to this problem extremely complicated and difficult. If he really wanted to prove it, it would probably take him an hour.
Fortunately, I quickly discovered the conspiracy and directly used Ceva's theorem and Menelaus' theorem to solve the problem, which greatly shortened the proof process and the time consumed.
——Study Committee, ah, School Committee, 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! It's a pity not to get it!
However, Qin Ke no longer had the energy to think about Ning Qingyun. The dizziness in his head was getting more and more serious. He mustered up energy and hurriedly looked at the answers to the two additional questions. Seeing that there were no mistakes or omissions, he
Fold it up, use it as a pad of paper and press it under the main volume and draft paper, and start solving the ten provincial-level problems in the main volume.
The difficulty level of these ten major questions also has nothing to do with the serial number. Qin Ke has no time or thought to review the questions one by one. Anyway, the goal is to get a perfect score, so he must conquer all the questions.
He started solving it directly from the first line.
"Question 1: It is known that 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 to a question like this at a glance. But at this time, his state became extremely bad. When he looked at the words of the question, he could see some ghost images.
The brain is more like a mirror covered with a layer of water mist, blurry.
He thought hard for nearly three minutes before he figured out how to solve this problem by constructing a recursive sequence.
With great effort, he squeezed the pen tightly with his weak hands and wrote the answer one stroke at a time. It actually took Qin Ke nearly ten minutes, and he was even sweating unconsciously.
Without looking in the mirror, Qin Ke could guess that something was wrong with his face now, his hands were frighteningly cold, his lips were dry, and he even felt a little tight in his chest and felt like vomiting, which was extremely uncomfortable.
Looking at the time, nearly an hour has passed since the exam started.
This is the first time in recent months that Qin Ke has spent so long on a math test paper. Under normal circumstances, he would probably have finished all the test papers.
However, Qin Ke didn't have the energy to think about this anymore. He wrapped his down jacket tightly and took several deep breaths in succession. The cold air penetrated his chest, allowing him to quickly force himself to concentrate, and then continued to answer questions one by one.
But his physical condition became worse and worse, he began to feel sleepy and tired, he thought for longer and longer, and it took longer and longer to solve problems.
The next question is no longer a question of intelligence and skills, but a double test of perseverance and physical strength!
…
Deng Hongguo arrived at the Provincial Cultural and Sports Center just after the provincial competition started. He had an important meeting yesterday and could not miss it until last night. Then he booked the latest flight and set off in the early morning. He arrived far away in the early morning.
State Airport.
After getting off the plane, he didn't even bother to eat breakfast, so he got into Shi Cunyuan's car and drove over.
Naturally, he traveled thousands of miles without any hesitation because he wanted to see with his own eyes the rumored to be extremely powerful genius student named Qin Ke.
Now the national training team is really short of such talented young people. Even if there is a 50% chance, Deng Hongguo is not willing to miss it.
When several leaders of the Provincial Olympic Organizing Committee heard that such a big shot was coming, how could they not come over to accompany them? Deng Hongguo repeatedly waved his hand, indicating that Shi Cunyuan could accompany him. However, it was difficult to refuse the hospitality, so he had to exchange pleasantries with these leaders.
It was delayed a lot of time.
After finally putting down their teacups and leaving the office, Deng Hongguo and Shi Cunyuan braved the biting north wind and walked towards the examination building.
Deng Hongguo rubbed his brows, using the cold wind to clear his mind and dispel the fatigue from the long business trip. He turned to ask Shi Cunyuan next to him: "Brother Cunyuan, the two additional questions in this provincial competition cost me a lot of effort.
what do you think?"
Shi Cunyuan frowned and said: "It's difficult. The first question involves Hamilton's problem, which is relatively unpopular. I guess not many candidates can do it. The second question, are you specifically targeting Qin Ke?"
Hearing that his old classmate discovered the beauty of his question, Deng Hongguo's bloodshot eyes suddenly showed a look of pride:
"That's right, the second question is used to dig holes for Qin Ke. This kid has an excellent grasp of various Mathematical Olympiad problem-solving strategies. It has become his instinct to use these skills to solve problems. I also specially gave him
Chapter completed!