Solution: Do the auxiliary lines EI, FI, BI, CI first.

Sufficiency: If BC=BE+CF, then you can take a little K in the edge BC to make BK=BE, so that CK=CF, and connect KI.

On the bisector AD of ∠BAC, the inner I of △ABC is taken to connect the ACB due to BI biparting ∠ABC and CI biparting, so △BIK and △BIE are symmetrical with respect to BI, and △CIK and △CIF are symmetrical with respect to CI.....

Therefore, ∠BEI=∠BKI=π-∠CKI=π-∠CFI=∠AFI, so that A, E, I, F four points are round......

Combined with B, E, F, C, four points of the circle......

Necessity: If the inner I of △ABC is the outer heart of △DEF, since AE≠AF (in fact, it is a circle of four points: B, E, F, and C..... Therefore......

Therefore BC=BK+CK=BE+CF.

The necessity is proven.

.......

In ten minutes, the first big question was successfully killed by Xu Chuan.

The difficulty of this question is not very great, and there are two key points, one is to use the four auxiliary lines of EI, FI, BI, and CI to find the KI auxiliary line.

The other is the use of π values.

This is one of the few points used in high school geometry to solve triangles and contours, but as long as you master these two points, it is not a problem to solve the first problem.

Half an hour passed, and the second integer set, which was more difficult, was also killed.

"This year's question doesn't seem to be very difficult."

Looking at the last function, Xu Chuan touched his chin and glanced at the examination room, most of the students were bowing their heads to do the questions, which confirmed his thoughts.

After all, if the difficulty of the question is too high, there must be students looking up at the sky.

This is a phenomenon observed in his two lifetimes.

"Forget it, hurry up and finish the third question, and then hand in the paper to try the food outside."

Xu Chuan shook his head and refocused his attention on the test paper.

It was the first time he had come to Bashu, but some of the delicacies he had eaten in the cafeteria of Bashu Middle School yesterday gave him a good impression of this place.

If a high school can make such a delicacy, then the restaurant outside should taste more authentic.

Although he is not a foodie, he is still keen to try food.

Provided, of course, that the food is to his taste.

Dark dishes like stargazing, live maggot cheese, canned herring, etc.

But for now, the local food in Bashu is still quite good.

......

After completing the three major questions, Xu Chuan checked it twice, and after confirming that there were no problems, he submitted the paper in advance.

The leader of this year's Xiangnan Provincial Team is Gong Rihui, a teacher from Xingcheng No. 1 Middle School, who is very familiar with Xu Chuan.

Handing over a bottle of water, Gong Rihui asked with a smile, "How was the test?" ”

"It's okay, the difficulty of the questions in the first half is not very high, watch the second half."

Xu Chuan took a sip of the water, and then asked, "Teacher Gong, is there any food around here?" I want to try it. ”

"There are quite a lot of delicacies in the Bashu area, such as bowl chicken, northern Sichuan cold noodles, dandan noodles, brown sugar glutinous rice cakes, swan eggs, Sanhe mud, three major cannons, and fat intestine powder..... These are all very good. ”

"There is a food street about 500 meters to the right after leaving the school gate in this neighborhood, you can go there and have a look."

"But there is still an exam tomorrow, so be careful, don't spoil your stomach."

Gong Rihui reminded with a smile that he was still relieved about the student in front of him, and his stage was on a bigger IMO, not a domestic CMO.

"Teacher, you've been here before, so familiar?" Xu Chuan asked with some surprise, just now Gong Rihui pushed him more than a dozen snacks and delicacies in one go.

Gong Rihui smiled and said, "My hometown is from Bashu, so I am naturally familiar with it." ”

Xu Chuan sighed, nodded, thanked him, and prepared to try the food street in Gong Rihui's mouth.

But before he could take two steps, a voice behind him stopped him.

"Brother Chuan, where are you going, wait for me."

Behind him, the voice of Lu Tianrui, the little round face, came from afar, and he also handed it in advance.

Xu Chuan didn't really want to take him, because this little Zhengtai really talked too much.

But I couldn't bear the entanglement, so I had to take this tail with me.

"Brother Chuan, Brother Chuan, your second big question, what is the method of verifying the set D(n)={a−b∣∣n=ab,a,b∈Z+a>b}?"

"Brother Chuan, what are we going to eat later?"

"Brother Chuan, do you think we will have diarrhea if we wait to eat hot pot?"

"Brother Chuan, why don't you say that this Bashu sister is not afraid of the cold? It's December, and you're bare-legged? ”

"Brother Chuan,....."

Along the way, Lu Tianrui kept asking questions, a living book of 100,000 whys.

Xu Chuan said that he just wanted to find an earbud to plug his ears up, why is Sister Bashu not afraid of the cold, how does he know, if you have the ability, you can ask me.

......

After taking a stroll around the food court with a chattering little Zhengtai, Xu Chuan returned to the training room.

Although this little round face is a bit annoying, the food in Bashu is still very good, and some of the special delicacies are not all spicy as they are rumored, so they are worth trying.

The only problem is that the tofu brain here is actually sour and spicy, which is simply an outlier among the different for him.

Tofu brain, this kind of thing, it has to be sweet!

After returning to the training room and exchanging the content and experience of today's exam with other teammates, everyone went back to their dormitories to sleep.

After a night of silence, the next day's exams soon began.

Entering the inspection, the test papers were distributed, Xu Chuan checked the test papers, and the expression on his face suddenly became extremely subtle.

"Good guy, I said how could yesterday's exam questions be so easy, it turns out that the difficult ones are concentrated here."

After scanning the questions on the test paper, Xu Chuan muttered softly.

Just like yesterday, it's still three questions, and it's still a geometry, an integer set, and a function.

But the difficulty of the questions has increased several times.

Even he, after getting the test paper, no specific solution ideas appeared in his mind for the first time.

Compared to the first two simple triangles of yesterday, today's geometry is ten times more complicated.

For a geometry, the text of the question only occupies two or three lines, but the geometric diagram below occupies almost half of the version of the test paper.

Xu Chuan counted it, and there were four circles in this geometric diagram, two large circles and two small circles, and the circles were intertwined with each other like the Olympic rings.

In the circle, outside the circle, there are eight different triangles, there are acute triangles, right triangles, obtuse triangles such as common triangles, as well as complex triangles such as Chewa triangles and strangled triangles.

In addition, there are six branch lines that cross the triangle and the circle, alternating horizontally and vertically, cutting the entire geometry into pieces.

Triangle to triangle intersects, triangle to circle intersect, circle to circle tangent, straight line cutting, complex graphics look at Xu Chuan want to tear the test paper.

This is an absolute nightmare for math competition students.

.......