All-round mathematician

Chapter 110 Chapter [-] Top Difficulty Topics
Teacher Zhang, the class teacher, asked the students to digest the information. In fact, the finalists already knew these rules.It's just that some people's spirits are still too high, and they have never been hit in mathematics.

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

After several league exams, the score gap was not as big as expected.Take the total score of 100 as an example, everyone has a score of 95 or above, but only a few points are due to a small mistake.

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

Teacher Zhang paused for a few seconds, looked at the seriousness on the immature face, nodded in satisfaction, and said, "In 10 minutes, it will be the first round of exams. According to the rules of the World Olympiad, there are three questions, 7 points for each question, and the exam time is four and a half hours. "

"Starting tomorrow, we will gather in the classroom at 7 o'clock, three questions a day, and the teacher will mark the papers in the afternoon. You can move freely in the campus, and you can't go out of the campus. You can also exchange math experience in this classroom, study and solve problems, etc. 5pm In the middle of the day, Teacher Qian and Teacher Zhou will explain the morning papers."

"We will not waste time on the book outline, and we will be eliminated if we can't keep up. Every time the scores are summed up, the total score will finally determine who will be shortlisted and who will be eliminated."

The beginning of this training camp is to be in line with the International Mathematical Olympiad.International Mathematical Olympiad, with a total score of 42 points, a two-day exam, and 3 questions for four and a half hours a day.

"If you want to go to WC, please hurry up, the exam will start in 5 minutes."

There are a total of 30 people in the empty classroom, arranged in six rows, with five people in each row.Each of the three teachers is in charge of a group and chins the papers.

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

Aren't you afraid to peek at the exchange?Naturally not afraid, on the top of the classroom, there is a monitoring dome camera in the four corners, which can turn 360 degrees, and any small movement cannot be hidden.

After receiving the test paper, Yang Fan immediately looked at the questions. The three questions had figures, so today we will start with geometry.

Above the blackboard in the front row, there is a rectangular electronic watch with bright red numbers beating, and it is counting down to the exam.

The first question, it is known that the center of the circumcircle of △ABC is O, the circle with the center of A intersects the line segment BC at D, E, and B, D, E, C are different points on the line segment BC...the four circles are interlaced , 7 vertical and horizontal lines

After reading the title, Yang Fan's eyelids twitched wildly: "Good guy, it's extremely difficult once you start."

It can be said that the two competitions he participated in were not as difficult as they are now.

Staring at the graph, combined with the given conditions, and frantically calculating in my mind, I didn't have any ideas until 10 minutes.

Take out the compasses and draw the same figure on the sketch.Start writing, write and draw, and add content according to inspiration. It took a full 20 minutes to analyze the content.

At this moment, looking at the graphics on the draft, he has added 3 dotted lines, converging point P, and forming two equilateral triangles with a solid line.

There is also his own red pen, adding 6 red solid lines to form more than 10 inner and outer triangles in two circles.

"Perverted enough, this is obviously a killing stick."

It took nearly 40 minutes for optical analysis and drawing, 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.Here, he relaxes and looks up at his surroundings.

There are five or six people who are still staring at the topic in a daze, and probably have no train of thought.The rest are checked on the draft.

"Difficult, really difficult."

Yang Fan no longer cares about other people, and continues to do it: connect FG and extend it, intersect with the extension line of BC at point P... X is on the perpendicular line AO ​​of FG, and the proof is completed.

After finishing, look at the countdown again, exactly 1 hour.

"It's really crazy. How many years did it take to come up with this kind of topic? Are some teachers just trying to make things difficult for others if they are full all day long?"

Yang Fan feels that his three views have been refreshed. This question has been simplified to the point that nothing can be missing, and there must be so many conditions for adding graphics. There is no other way, and the solution is also unique.

"Okay, let's move on to the second question."

Because of the difficulty and the large amount of calculations, he actually felt a sense of anticipation. This is the abnormal refreshment after overcoming a difficult problem.

Students, have had similar emotions.This is a kind of rising and flourishing of spirit, sense of accomplishment, sense of honor and so on.

The second question took about the same time and the difficulty was the same.

"Is this the trick? It's boring."

He thought that the difficulty would rise in steps, so he was a little disappointed after finishing the second question.Take this opportunity to give your brain a break and look up at other people.

The students have all entered the state, writing hard at their desks, and only they know the specific answers.

"Then continue to the third question."

Two hours have passed, and there are still two and a half hours left. Yang Fan is confident that if the last question is still at this level, he can submit the paper an hour earlier.

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

This time the topic is three circles with several triangles listed in them.

At first glance, there is simply no thinking.

"It's interesting." This question is an order of magnitude higher than the previous two questions. Yang Fan was a little confused on the spot, thinking that he couldn't start.

"It turns out that this question is the real killer. It can be said with certainty that at least half of the people sitting here can't solve it."

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

"This question has been honed over and over again, and it seems that what is finally sought is the positioning point of the missile. It's awesome."

He continued to sort out his thoughts, and there was an orthocenter and a midpoint, so he naturally remembered the nine-point circle theorem, that is, the midpoints of the three sides of a triangle, the three-side orthopedic feet, and the midpoint of the line connecting the three vertices to the orthocenter, a total of nine points are a circle.The center of the nine-point circle is the midpoint of the line connecting the circumcenter and the orthocenter of the triangle (that is, the Euler line).

According to the deduction and thinking, he continued to add solid and dashed lines, enlarged the original image, and used three pieces of draft paper, because the middle part of the drawing was cut off and he gave up.

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

He added two small circles and nine dotted lines within the big circle.I looked up at the countdown, and there was 1 hour left.

"I underestimated the World Olympiad. It seems that there are fewer questions of this magnitude in the past. It is really useless to rely on learning alone. After all, no matter how proficient in theory, ideas will not pop out without doing it. Just this question The amount of calculation is enough to be abnormal."

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

Answer: Construct a nine-point circle O', O is the circumcenter of △ABC, H is the orthocenter... Therefore, XK is not only the tangent of circle N, but also the tangent of the circumscribed circle of △MFK.Certificate completed.

He checked the steps of this question, a total of 41 steps.

In the end, he lamented: "It's amazing, there is a way."

All done, there is still half an hour, just to check the time.

"I don't know what's going on with the others. It's really hasty. It's the first time I've made such trouble."

When it was time to close the papers, everyone felt chills.

(End of this chapter)