Chapter 6 Come on, let’s do a question first

In the quiet classroom, Xu Chuan was flipping through mathematics textbooks. Some of the thick stack of books on his desk were mathematics textbooks borrowed from the library for the first and second grades of high school.

After all, so much time had passed that he had forgotten what the textbook looked like. If he wanted to read it, he had to read it from the beginning, so as not to miss any knowledge points and lose points in the exam, which would be embarrassing and embarrassing.

If a Fields Medal winner cannot get full marks in an ordinary high school competition, he might as well be reborn, smash his medals, and go home to grow sweet potatoes and raise pigs.

After more than an hour, the lunch break bell rang, and Teacher Gong, whom Song Kaiji said would be here soon, walked into the classroom.

"Hello, Teacher Gong."

In the classroom, the classmates who were reading and doing questions all said hello. Xu Chuan looked up and saw a man wearing gray and black plaid wool. He was not tall and wore a pair of glasses.

Completely different from Song Kaiji's almost half-bald hairstyle, Gong Rihui's hair is very thick. It's so thick that it makes people wonder whether he studied mathematics. It's so thick that it doesn't fit the concept of baldness in the impression of mathematics teachers. .

"Hello, Teacher Gong."

Xu Chuan also said hello respectfully. As a contestant who often participates in competitions, although they are not in the same subject, they still know each other.

Even though this teacher Gong has thick hair and does not look like a math major, he has been leading the math training team of Xingcheng No. 1 Middle School for more than ten years. Some of his students have even won IMO gold medals. .

"Is Xu Chuan here? Come on, come up and do a question first."

Gong Rihui glanced at the classroom and saw an extra person in the classroom at a glance. He smiled and wrote a math question on the blackboard with his backhand, and then directly named Xu Chuan.

He naturally knew this talented man who studied physics. Like Song Kaiji, he also felt that it was a pity for such a good young man to study physics.

I am so good at math in college, the foundation of all subjects!

Without mathematics, you can’t even calculate how much change you’ll get for buying groceries.

"Find all pairs of positive integers (x, y) that satisfy xy=y^x-y. (^ means square)"

A mathematical problem was casually written on the blackboard by Gong Rihui. He threw the white chalk on the desk and looked at the physics genius in the audience with a smile.

Xu Chuan is no stranger to this approach of doing the questions first before entering the classroom.

Whether it was physics or mathematics, on the way to the competition, the only thing he could do was solve the problem. He had experienced this scene countless times. He was very familiar with it and was not nervous at all. He stood up and prepared to go up and do the problem.

"Brother Chuan, come on, go up and blast your ass, I support you!"

Next to him, Lu Tianrui, a little round-faced man, whispered words of encouragement.

As soon as Xu Chuan stood up, his feet suddenly staggered when he heard this, and he barely managed to hold on to the table before he could stand still.

Are you such a pervert for playing math?

In front of the blackboard, Xu Chuan casually pulled out a piece of chalk from the podium. The question asked by Teacher Gong was difficult to say. In fact, most high school students could solve part of it.

Just like the last big question in the college entrance examination mathematics, the first small question is basically free and easy to solve.

But the difficulty of the following questions increases exponentially.

The same goes for the question on the blackboard. In the college entrance examination, it belongs to the last one or two big questions. But in the competition, it is just a junior high school question after entry. It tests reasonable reasoning and analysis. The ability to solve problems is of a more comprehensive nature.

It is not difficult and easy to judge. Even a junior high school student can see that its first answer is (1, 1). In front of the blackboard, Xu Chuan glanced at the question and thought about it for a few seconds before getting started.

"Solution: It is easy to find that x=y=1 satisfies the meaning of the question, so x=y=1 is the answer and has a solution. When the equation has a solution, x≥y must exist.

If x=y, then x=y=1.

If x>y≥2, then from x^y=y^x-y, we get 1<(x/y)^y=y^x-2y, and y|x.

Assume x=ky, then k≥3, k^y=y^(k-2)y, so k=y^k-2.

Because y≥2, so y^k-2≥2^k-2, because when k≥5, y^k-2≥2^k-2>k, so, k=3,4.

When k=3, y=3, x=9; when k=4, y=2, x=8;

Therefore, all positive integer pairs (x, y) = (1, 1), (9, 3), (8, 2) are found. ”

The chalk under his hand did not pause at all. In just one minute, the complete verification process was listed on the blackboard.

Gong Rihui glanced at the blackboard, nodded with satisfaction, and said: "Yes, the idea is clear, the verification is concise, and the format is standard. It is a good idea."

"But I want to remind you that there are some differences between the math competition and the physical sciences competition. The physical sciences competition will involve some college knowledge, but there is basically no knowledge in the mathematics competition. The questions are all applicable to high school. Solved by mathematical knowledge.”

"If you have self-taught college mathematics such as advanced mathematics, try not to use it in domestic competitions, because this may lead to deductions for solving problems beyond the syllabus."

"Of course, it doesn't matter if you can break into it IMO."

Xu Chuan nodded to express his understanding.

In fact, the students participating in the competition, whether it is mathematics competition, physics competition, or even chemical and biological information, have basically read some university textbooks to some extent, but each person's degree of reading is different.

He was no exception. He had basically finished reading the textbooks on advanced mathematics, mechanics, thermal science, electromagnetism, optics, and quantum physics that were only used in universities when he was in his third year of high school.

If you use college mathematics to solve problems when participating in a competition, it will undoubtedly be a dimensionality reduction blow to other students who have learned less, which is unfair.

In the early years of the competition, whether it was domestic or international, it was generally prohibited to use university knowledge to answer questions. However, this latter rule is more liberal for other subjects except mathematics.

For example, he majored in physics. In recent years, IPHO has taken a lot of college knowledge tests. Last year, he even took a test on circuit design, which is more than sophomore level.

As for mathematics, the current IMO does not prohibit the use of university knowledge to answer questions. Instead, the question makers design the questions to prevent candidates from using university knowledge to answer questions.

Those who can give IMO questions are all masters in the field of mathematics. Their knowledge range is much better than that of the novices who participate in IMO in high school. You can only use high school knowledge to solve the questions that can be asked.

If in this case, you can still use super-category knowledge to solve the problem, it doesn't matter.

Although fairness is necessary, isn't the purpose of IMO to screen out mathematical geniuses?

Your knowledge in high school can break through the blockade of these mathematicians and mathematics professors, so there is no need to fear that you will be arrogant and forcibly deduct points from you.

Of course, this does not mean that there is no need to learn college mathematics if it is not useful. On the contrary, it is still necessary to learn some college mathematics.

Because this can broaden your thinking, make it easier for you to solve problems, and avoid doubling your brain cells.

(End of chapter)