While speaking, the middle-aged man who was called "Teacher Shen" by Hu Peiwei took out three pieces of paper and distributed them to everyone: "The amount of questions is not large, just 6 questions, the key is to use some brains." Because we are practicing, there is no time limit, until we are done. There is A4 paper next to it, so you have to take it yourself. If that's okay, let's get started, shall we?"

At the beginning, when Mr. Shen said that he was referring to the International Olympiad questions in previous years, Jiang Shuiyuan felt a "sigh" in his heart, because he had turned over the past International Olympiad questions and knew the treacherous mysteries in them. Hearing that there are only 6 questions, he has already considered whether to sleep here at night. Sure enough, he was a little confused when he saw the first question: "Given 100 points on the plane, there are no three points collinear." Verification: Up to 70% of the triangles formed by these points are acute triangles. ”

Let's look at the second question: "In the convex quadrilateral ABCD, AB? CD=BC? DA, the point X is inside the quadrilateral ABCD, and satisfies ∠XAB=∠XCD, ∠XBC=∠XDA. Proof: ∠BXA+∠DXC=180°. ”

Question 3: "An international community has 1,978 members from 6 countries, with 1, 2, 3, 4 ...... 1997, 1998. Prove that at least one member of the society has a sequential number equal to the sum of the serial numbers of his two siblings, or twice the sequential number of one sibling. ”

Question 4: "n is a given positive integer, S={(x,y,z)|x,y,z ∈{0,1,2... ,n},x+y+z>0} is a set of (n+1)^3-1 points in three-dimensional space. Finds the minimum number of planes whose union contains S but does not include (0,0,0). ”

Question 5......

Question 6......

The six questions cover four categories: algebra, geometry, number theory, and combinatorics, and are all of above average difficulty - note that it is the medium of the International Olympiad, not the average of the general high school monthly exam. It is important to know that mathematics is sometimes divided into three categories in secondary mathematics education: elementary mathematics, advanced mathematics, and competition mathematics represented by various Olympiad mathematics problems between elementary mathematics and advanced mathematics. In addition to the expansion of specific knowledge compared with elementary mathematics, competition mathematics is more important in thinking and methods.

To put it bluntly, the brain must be smart and flexible enough!

After sweeping the six questions, Jiang Shuiyuan said directly to Uncle Ge: "Teacher Ge, help us order lunch!"

Uncle Ge gloated a little: "No problem, do you want to book dinner for you?" ”

Jiang Shuiyuan turned his head to look sideways and found that Hu Peiwei was already writing hard, while Zhang Jin was frowning and biting the pen and meditating. He took a stack of A4 papers and prepared to answer the question.

I just read the problem once, and I have a rough idea: among the 6 questions, there are 3 problems with relatively clear ideas, such as the first problem, which seems to be a geometry problem, but is actually a combinatorial math problem, to treat a given point on the plane as a set, the triangle composed of points as a set, and the acute triangle formed as a set, and then find a way to prove the conclusion. For example, the second question must use the four-point contour and Ptolemy's theorem. Of the remaining 3 questions, 2 have a vague idea, and 1 has no clue about the fourth question.

This is the first time that Jiang Shuiyuan has encountered clueless questions in the exam since he received the blessing of the bracelet. Could it be that Hu Peiwei usually practices with this kind of topic? Or is it that Huguang Olympiad has become so powerful? In the past, when I watched martial arts, I always said that there were three kinds of people who walked the rivers and lakes that could not be provoked, and the elderly, children, and women were considered to have seen it today.

Jiang Shuiyuan shook his head, emptied all the excess emotions in his head, and began to devote himself to the great cause of solving problems.

The topic is difficult, and the changes are tortuous, I feel like a grinding little goblin, and my whole body exudes the atmosphere of a pit daddy who involuntarily falls into the pit, and I must work hard to deal with it.

As soon as you get into it, you forget about the passage of time.

As soon as the third question began, I heard Uncle Ge greet him: "Dinner is coming! Let's eat first, sharpen the knife and don't mistake the woodcutter." ”

"That's right. The emperor is not short of hungry soldiers, and only when he is full can he have the energy to do the question. Teacher Shen opened the dishes and chopsticks as he spoke, and by the way, he also taught the exam secrets: "However, you can't eat too much during the exam, eating too much, blood sugar will rise, people are easy to get sleepy, but it is easy to affect thinking." If you're worried about getting hungry in the exam room, you can bring a few chocolates with you to replenish your energy anytime, anywhere. ”

Uncle Ge glared at Jiang Shuiyuan and Zhang Jin: "Learn a little, this is all valuable experience!"

"Hmm!" Zhang Jin nodded heavily.

Jiang Shuiyuan habitually complained: Since it is a valuable experience, why didn't you say it before? You have to wait for others to mention it, and you hit the snake and go down the pole, could it be that this is the legendary "Ge Liang is the reason why Zhu (pig) is also"?

Uncle Ge saw that Jiang Shuiyuan didn't speak, but took the initiative to come to the side, picked up Jiang Shuiyuan's answer sheet and looked at it: "How many did you make?"

Jiang Shuiyuan's head was still running at high speed while eating, thinking about the fourth question that he had no clue about, and he didn't have time to fight with Uncle Ge at all, so he replied bluntly: "Two." ”

"Two questions, very good!" Teacher Shen also came over, "The International Mathematics Olympiad is a two-day test, four and a half hours a day, and three questions. Just now we started the test at ten o'clock, and it has been less than two hours now, Xiao Jiang has made two questions, this level, this speed, it is not a big problem to enter the national training team—— right, Peiwei, how many have you made?"

"It's also two. "Hu Peiwei not only eats attentively, but also eats elegantly, and the box lunch can eat the style of a Michelin restaurant.

Uncle Ge immediately retorted: "More than that, right? According to my observation, Xiao Hu has made at least three questions—even if there are no three, the third question is about to come out." Hehe, you don't want to save face for them, they don't care anyway. ”

I don't know whether it was acquiescence, or adhering to the ancient adage of "eat without saying a word", Hu Peiwei didn't speak again, which made Uncle Ge seize the opportunity to play, and the answer sheet in his hand began to foam: "Isn't it usually quite capable? I can't wait to punch Leibniz, kick Lagrange, fly into the sky and stand shoulder to shoulder with Euler and Gauss, why did I drop the chain at the critical time?"

Zhang Jin held the lunch box with a face full of shame: "I, I, I just made a ......"

Jiang Shuiyuan's cynicism towards Uncle Ge has already reached the superb realm of "the eight winds can't blow, sit firmly on the Purple Golden Lotus", and when he heard his ridicule at this time, it seemed that a lightning bolt crossed his head, as if he had grasped a key to solving the fourth question, but when he thought about it carefully, he disappeared. He frowned and subconsciously asked, "Who did you just say?"

"Say what's wrong with you?"

"Not me!"

Uncle Ge smiled: "If it's not you, who else can there be?"

"No, who do you say I punched? Kick who?——Oh, Lagrange!" Jiang Shuiyuan instantly found the key to open the fourth problem: first construct a polynomial, then use the difference method and the Lagrange median theorem to make a difference proof, and then discuss the situation to get the result.

Looking back at this problem, which is essentially an algebraic problem, it is easy to see that the minimum value is 3n, and the key is how to prove it. He had thought about using induction before, but it was too complicated, and whether it would succeed or not was still a matter of choice. Now with the difference method, not only the idea is ingenious, but also the steps are simple, it is simply a god's hand!

He immediately threw down his lunch box, took the pen A4 paper and began to prove it, lest the inspiration be lost if he delayed for a moment.

Uncle Ge saw that Jiang Shuiyuan's thoughts suddenly flowed and his pen was flying, and he explained to Mr. Shen with some weakness: "This, this, I just scolded him casually, it was purely unintentional. This shouldn't be cheating, right?"

Teacher Shen was also speechless: "It shouldn't be counted, right? After all, other people also heard it, and I didn't see any reaction from them." ”

Uncle Ge looked left and right, and found that Hu Peiwei and Zhang Jin were confused, obviously not understanding why Lagrange would be inspired by the water source of the river.

Teacher Shen continued: "And I have read all these problems, but I don't think that one needs to use the Lagrange median value theorem or the sum of four square theorem!"

"Yes, yes, yes, I remembered when you said that, it doesn't seem to be used in a place where Lagrange's theorem is used!

Their discussion did not affect the answer to the question in the slightest. Fifteen minutes later, Jiang Shuiyuan threw away the pen, picked up the fast food box and touched it, and then nodded with satisfaction: "Yes, the rice is still warm!"