September 14th, Rongcheng No. 3 Middle School.

This one of the top ten key high schools in Fujian Province looks extraordinary today because the 2007 National High School Mathematics League (Fujian Province Division) competition will be held here. This exam is organized by the Fujian Provincial Mathematical Society.

Thousands of students from all over the province gathered here. Under the leadership of their respective leading teachers, they came to Rongcheng No. 3 Middle School in Fujian Province and reported to the examination center carrying their admission tickets, ID cards and other materials.

Liu Yichen has experienced biology competitions and chemistry competitions, and is already familiar with it. He filled out the registration information and then searched for the test points according to the test point distribution map.

"Rongcheng No. 3 Middle School is indeed one of the top ten key high schools. This campus is so big and beautiful~~" Chen Qiang and Liu Yichen were deeply shocked.

He used to think that Qingcheng No. 1 Middle School was already very good, because the campus of Qingcheng No. 1 Middle School was much stronger than his junior high school campus, and much better than the No. 2 Middle School. Later, when he went to Minxi No. 1 Middle School, he felt that Minxi No. 1 Middle School

It’s already incredible, it’s the ceiling of high school campuses.

But when I came to Rongcheng No. 3 Middle School, I was overturned again. The campus of Rongcheng No. 3 Middle School covers an area of ​​50 acres. The campus area is not larger than that of Qingcheng No. 1 Middle School. However, there is a building in Rongcheng No. 3 Middle School with a total area of ​​4,598 square meters, and there are

Lighted stadium, standard indoor swimming pool, gymnastics room, table tennis room and modern art multimedia classroom multi-functional sports center; there is a 300-meter plastic circular track covering an area of ​​15 acres, and a library that looks significantly larger

Pavilion......

In terms of hardware facilities alone, it is probably a block or two behind Qingcheng No. 1 Middle School.

"Stop looking, what's there to see?" Liu Yichen said: "Your exam place should be in this room, mine is next to yours!"

Liu Yichen didn't care about this at all, because there was no comparison at all. One was a county-level middle school, and the other was a key middle school at the provincial capital city level. How could they compare? There was no comparison!

The entire league exam, the first and second exams, are all completed in the morning. The first exam is from 8:00 to 9:20, and the second exam is from 9:40 to 12:30. There is less than 20 minutes in between for a short rest.

It lasted for four and a half hours, which can be said to be a very intense exam.

Even the college entrance examination is not so intense.

After entering the examination room, the radio began to read out the examination room rules. It was not until 7:55 that the invigilator took out the sealed kraft paper containing the test papers and showed everyone that it was sealed and not unsealed. After three students signed to confirm, the invigilator

The teacher distributed one test paper after another.

After Liu Yichen got the test paper, he filled in the candidate information such as name, admission ticket number, and ID number, but did not remember to answer the questions. Instead, he checked the test paper from beginning to end to see if there were any certain or vague areas in the test paper. At the same time,

Also understand the question structure of the test paper.

The first test paper of the competition consists of three parts, multiple choice questions, fill-in-the-blank questions and solution questions. There are 6 multiple-choice questions with 6 points each, a total of 36 points. There are six fill-in-the-blank questions with 9 points each, a total of 54 points. There are 3 solution questions with 20 points each.

There are 60 points in total, and the full score of the test paper is 150 points.

After a preliminary look, the content of the first test and the current high school mathematics syllabus fall within the content of the college entrance examination, and the difficulty is slightly higher than this year's college entrance examination level.

Liu Yichen concentrated on solving the questions one by one. He immediately felt the difficulty of the first test of the competition. For example, the first question was about solid geometry. Given a regular four-sided vertebra p-abcd, ∠apc=60°, then the dihedral angle

The cosine value of the face angle. This question alone is as difficult as the most difficult multiple-choice question in this year's college entrance examination.

But it’s normal. Among the candidates who can take the first and second exams of the competition, who didn’t get a score of 140 or above in the college entrance examination?

For people who don't have this level, even if they participate, it's just participation.

After selecting the answer, then look at the second question. The answer to the question is an inequality. Suppose the real number a makes the inequality ∣2x-a|∣3x-2a∣≥a^2 always true for any real number x, then a satisfies the condition

The set composed of a. After pondering for a moment, I calculated on the scratch paper and found that the set composed of a is [-1/3, 1/3].

The more you do it, the more difficult it becomes.

However, it doesn’t feel like the difficulty is sky-high. It is still within the acceptable range. For each multiple-choice question, you can directly eliminate one option at most, and the others must be calculated and concluded. It is far from being easy to eliminate the questions of the college entrance examination mathematics paper.

There are two types of selections, and it is not like the many questions in ordinary schools where you can directly determine the type of selection at a glance. Basically, you have to calculate.

Having taken the math league test papers in previous years, he was still within an acceptable range for this level of difficulty. After all, this was how difficult it was for the first league test in the past.

Unknowingly, after completing the multiple-choice and fill-in-the-blank questions, Liu Yichen did not rush to answer the questions. Instead, he filled in the answers to the multiple-choice and fill-in-the-blank questions on the answer sheet first, and then started to answer the questions.

The first question is very difficult as soon as it comes up. It is the final question of the college entrance examination. The question is very short, but it is not simple. If you do not learn all the questions, you will not be able to solve the proof question of the first answer. The proof process is five elements.

There are 20 points for calculations, which is equivalent to 4 points for one line of calculations.

The second answer question is a typical intersection of a curve and a line. Find the intersection locus of the tangent line of curve c at points m and n. This type of question is also similar to the college entrance examination, but in the college entrance examination it appears as a circle, here it appears as a curve

.

However, the idea of ​​​​solving the problem is the same. Let the coordinates of points m and n be (x1, y1) and (x2, y2) respectively. The tangent lines of curve c at points m and n are respectively at l1,, l2, and their intersection point p.

The coordinates are (xp, yp). If the slope of the straight line l is k, then the equation of l is y=kx1, thus forming a system of equations. Eliminate y and obtain the equation about x.

This question filled up the entire answer page area before finally finding the trajectory.

After finishing this question, Liu Yichen breathed out a sigh of relief, moved a little, and then continued to answer the last question. This answer question is about functions, combined with trigonometric functions. This kind of question is the most difficult.

, it is also the test that best tests the level of candidates. Even the candidates who participate in the competition, in fact, most of them cannot do it. Candidates who can solve this question every year will definitely get the first prize in the league division, and those who can get

The first prize winner in the league division may not be able to solve this question.

Unlike the clear divisions between the first-tier, second-tier, and third-tier colleges in the college entrance examination, the quotas for each province in the league are different, so the grades are determined from high to low. For example, there are 45 first-class places in the Fujian Province Division.

There are 46 second prizes and 61 third prizes, so the 45 people with the highest scores will be the first prizes, and so on.

In other words, even if they are all first prizes in the competition area, their level will actually be high or low, and the level of players from different provinces will also be different.

If you can win the first prize, it means that you are at the top in this field in this province, but it does not mean that you are at the top in the country. To do this, you have to participate in the winter camp and participate in the finals to compete with 400 outstanding players from across the country.

, only then do you know your level in the country. And the top ones, after passing the national training team and being selected by the training team to enter the 6-member Chinese Mathematical Olympiad team, it means that you have reached the top of your peers in the country at this stage.

The answer to this question filled up the answer area of ​​the page before it was finally proved. Looking at the time again, there was still some time left, so he was not in a hurry to hand in the paper. He checked it again from beginning to end and thought about it again.

Check for deficiencies and fix them.

Finally, when the bell rang at the end of the exam, the invigilator asked the candidates to stop writing and answer and began to collect papers. Then they would have 15 minutes to go to the bathroom. Liu Yichen naturally went to the bathroom, and then moved around a little outside the classroom. For

He was still very satisfied with the first test. He was still confident of getting 140 points on the 150-point test paper.

Judging from the competition in previous years, the score is more than enough to get the first prize, but you have to look at the second test. Although the second test only has three questions, these three questions are 50 points each, and the total is 150 points. These three questions

Questions are what really eliminate most people and screen out those who have enough talent. Judging from past competitions, there is often not even one person who can solve these three questions in a session.

Of course, it is not ruled out that there will be one or two mathematical geniuses who have reached the level of ability and can easily solve them.

According to the rules of the competition, the three questions are a plane geometry question, an algebra or number theory question, and a combination question. If such a question appears in the college entrance examination, it will definitely belong to the college entrance examination super-outline question, and the college entrance examination proposition teacher will embarrass the candidates.

When the exam started and after getting the test paper, Liu Yichen noticed that all the candidates around him had frowned. It was obvious that in the second test, whether there were those who were truly qualified would be highlighted. Whether they were fake academic masters or real academic masters, they immediately

distinguish.

These two test papers were like a magic mirror, whether a fake academic master or a true academic master was revealed in front of him.

After concentrating, gathering his mind, and filling in the candidate information, Liu Yichen began to read the questions. The three-question exam took nearly three hours, which meant that each question took almost an hour to complete. From this time, it can also be seen that

The difficulty of the question.

The first path: As shown in the figure, in the acute angle △abc, ab

Simple questions are paired with a triangular figure and plane geometry, but they are extremely difficult. If you don't do the auxiliary lines correctly at the beginning, you won't be able to prove it no matter how hard you prove it.

Liu Yichen connected the lines of each point and expressed them with dotted lines. In order to prevent himself from not getting the auxiliary line right, he also specially used a 2b pencil to make the auxiliary line. If he found a mistake later, he could reconnect the line.

When answering step by step, he answers on the scratch paper first. This can avoid the chance of correction if he makes a mistake. Although it will increase the time of copying the answer, the time of the competition is actually enough if you can do it.

Yes, you can’t do it. Even if the exam time is doubled, you still can’t do it.
Chapter completed!