The head teacher, Mr. Zhang, asked the students to digest the information, and the finalists actually understood these rules for a long time. It's just that some people are still too high-spirited and have never been hit in mathematics.

The top 30 people in the country, the content of the high school mathematics syllabus, must have already understood, it can be said that everyone is a top student.

In several league exams, the score is not as big as expected. For example, if you take a total score of 100 points, everyone is above 95 points, but it is only because of a small mistake that you can say a few points.

"Don't be in a hurry, after today, you will know what a problem is, and you must know how to be in awe. ”

Teacher Zhang paused for a few seconds, looked at the seriousness on his immature face, nodded in satisfaction, and said, "Ten minutes later, it will be the first round of exams." According to the rules of the World Mathematics Olympiad, there are three questions, each question is worth 7 points, and the exam time is four and a half hours. ”

"Starting tomorrow, you will gather in the classroom at 7 o'clock, three questions a day, and the teacher will mark the papers in the afternoon, so you can move freely on campus and not go out of the campus. You can also exchange math experiences, learn to solve problems, and so on. At half past five in the evening, Mr. Qian and Mr. Zhou explained the morning papers. ”

"We don't waste time on book outlines and knock them out if we can't keep up. Each time the scores are summarized, and the total score is used to determine who is shortlisted and who is eliminated. ”

At the beginning of this training camp, it was in line with the International Olympiad. International Mathematics Olympiad, with a total score of 42 points, two days of exams, four and a half hours a day to do 3 questions.

"If you want to go to WC, hurry up, the exam will start in five minutes. ”

There are 30 people in the empty classroom, in six rows, with five students in each row. Each of the three teachers is responsible for one group, putting the roll on the chin.

A few minutes later, the staff arrived, and the three teachers left generously.

Aren't you afraid to peek at the exchange? Naturally, at the top of the classroom, a monitoring ball machine in the four corners can pay attention, and you can turn 360 degrees, and you can't hide any small movement.

After receiving the paper, Yang Fan immediately looked at the questions, and the three questions all had graphics, so let's talk about geometry first.

Above the blackboard in the front row, there is a rectangular electronic watch, with bright red numbers jumping, entering the countdown to the exam.

In the first question, it is known that the center of the circumscribed circle of △ABC is O, and the circle segment BC with the center of the circle is A at D and E, and B, D, E, and C are the different points on the line BC...... Four circles are staggered, and 7 straight lines are vertical and horizontal

After reading the question, Yang Fan's eyelids jumped wildly: "Good guy, it's difficult as soon as you come up." ”

It can be said that the two competitions he participated in were not as difficult as the current ones.

Staring at the graphics, combined with the given conditions, the mind was crazy to calculate, until 10 minutes, there was some idea.

Take out the compass and draw the same figure on the draft. Write, write, draw, add content according to inspiration, a full 20 minutes, just finished analyzing the content.

At this moment, looking at the figure on the draft, he has added 3 dashed lines, which converge the P points and form two equilateral triangles with a solid line.

He also added his own red pen, adding 6 solid red lines, and combining them into more than 10 inner and outer triangles in two circles.

"Perverted enough, this is obviously a killer. ”

It took nearly 40 minutes for light analysis and mapping, and the confident Yang Fan almost couldn't hold on.

All the ideas are in place, and now we are just waiting for the final answer. At this point, he relaxed and looked up at his surroundings.

There are five or six people who are still in a daze with the topic, and it is estimated that they have no ideas. The rest are checked on the draft.

"It's hard, it's hard. ”

Yang Fan didn't care about the others anymore and continued to do it: connect FG and extend it, and intersect with the extension line of BC at point P...... X is on the AO of the mid-perpendicular line of FG, and the certificate is completed.

When you're done, look at the countdown again, and it's exactly 1 hour.

"It's really a bombardment, how many years has it taken for this kind of topic to be figured out, and some teachers are just trying to make things difficult when they are full all day long?"

Yang Fan felt that the three views had been refreshed, this question had been simplified to the point that it could not be any less, and the conditions for adding graphics must also be so much, there was no other way, and the solution was also unique.

"Okay, move on to the second question. ”

Because of the difficulty and the amount of calculation, he actually had a sense of anticipation, which was the refreshing of the pervert after overcoming the problem.

Students, all of whom have had similar emotions. This is a kind of spirit and upliftment, a sense of achievement, a sense of honor, and so on.

The second question takes about the same amount of time and is of the same difficulty.

"Is that all there is to it? ”

He also thought that the difficulty would rise in a cascade, and after completing the second question, he was a little disappointed. Take this opportunity to give your brain a break and look up at the others.

The students are all in the mood, writing hard at their desks, and only they know what the specific answers are.

Then move on to the third question. ”

Two hours have passed, and there are still two and a half hours left, Yang Fan is confident, if the last question is still at this level, he can hand in the paper 1 hour early.

In the acute triangle ABC, AB > AC, let the circle O be its circumscribed circle and H be its vertical center......

This time, the title is three circles, and several triangles are listed in them.

At first glance, there is no idea.

"It's kind of interesting. This question is another order of magnitude compared to the first two questions, Yang Fan was a little confused on the spot, and he pondered and didn't know where to start.

"It turns out that this question is the real big killer, and it is safe to say that at least half of the people here can't do it. ”

Yang Fan believes that he has reached a certain level of high school mathematics, and of course he can judge the level of difficulty.

"This question has been tempered for a long time, and it seems that the final goal is the positioning point of the missile, which is awesome. ”

He continued to sort out his thoughts, and naturally remembered the nine-point circle theorem, that is, the midpoint of the triangle on three sides, the vertical foot on three sides, and the midpoint of the three vertices to the vertical center. The center of the nine-point circle is the midpoint of the outer center of the triangle and the vertical line (i.e., the Euler line).

According to the inference and ideas, he continued to add solid dotted lines, enlarged the original image, and used three pieces of scratch paper, but gave up because the idea was cut off in the middle of the drawing.

The final graphics are complete, and the viewers are dazzled.

He added two small circles and nine dotted lines to the great circle. Looking up at the countdown, there was still 1 hour.

"Underestimate the World Olympiad, this magnitude of the topic, it seems to have done less before, and it is really useless to just rely on learning, after all, no matter how proficient the theory is, the ideas will not pop up all at once if you haven't done it. The amount of calculation on this question alone is perverted enough. ”

All the details are sorted out in place, as long as the final answer is made, there is no big problem here.

Answer: Make a nine-point circle O', O is the outer heart of △ABC, and H is the vertical center...... Therefore, XK is both the tangent of the circle N and the tangent of the circumscribed circle of △MFK. Certification.

He checked the steps of the question, a full 41 steps.

In the end, he lamented: "Awesome, there is a doorway." ”

When everything is done, there is still half an hour, just enough time to check.

"I don't know what happened to the others, it's really hasty, it's the first time I've done it so troublesomely. ”

When the time for rewinding came, everyone felt cold.