Great Country Academician

Solution: First make auxiliary lines EI, FI, BI, CI.

Sufficiency: If BC=BE+CF, then a point K can be taken in the side BC, so that BK=BE, and thus CK=CF, linking KI.

Take the inner I of △ABC on the bisector AD of ∠BAC, and connect because BI bisects ∠ABC and CI bisects ACB, so △BIK and △BIE are symmetrical about BI, and △CIK and △CIF are symmetrical about CI

Therefore, ∠BEI=∠BKI=π-∠CKI=π-∠CFI=∠AFI, so the four points A, E, I, F share a circle

Combining four points B, E, F, and C to form a circle

Necessity: If the center I of △ABC is the circumcenter of △DEF, since AE≠AF (in fact, four points B, E, F, and C are in the same circle)

Therefore BC=BK+CK=BE+CF.

The necessity is proved.

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

This question is not very difficult. There are two key points. One is to use the four auxiliary lines EI, FI, BI, and CI to find the KI auxiliary line.

The other is the use of the value of π.

This is a point that is rarely used in high school geometry to solve triangles and co-circles, but as long as you master these two points, it is not a problem to solve the first question.

Half an hour later, the second round of integer calculation, which was more difficult, was also defeated.

"This year's questions don't seem to be very difficult."

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

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

This is the phenomenon he observed in the competition between the two lifetimes.

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

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

It was his first visit to Bashu, but the food he had eaten in the cafeteria of Bashu Middle School yesterday gave him a good impression of this place.

A high school can make such delicacies, so the store outside should taste more authentic.

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

Of course, the premise is that the food is in line with his taste.

Forget those dark dishes like stargazing, live maggot cheese, canned herring.

But as far as the current situation is concerned, the local food in Bashu is quite good.

After completing the three major questions, Xu Chuan checked twice, and handed in the paper ahead of time after confirming that there were no problems.

This year's Hunan Provincial Team's leading teacher is Mr. Gong Rihui from Xingcheng No. 1 Middle School. He is very familiar with Xu Chuan. Seeing that he was stuck and handed in the papers ahead of time, he knew that the questions this year might not be so difficult.

After handing over the bottle of water, Gong Rihui asked with a smile, "How did you do in the exam?"

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

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

"There are quite a lot of delicacies in the Bashu area. Bobo chicken, northern Sichuan jelly, dandan noodles, brown sugar glutinous rice cakes, swan eggs, Sanhe mud, three big cannons, and fat intestines powder are all very good."

"After leaving the school gate, there is a food street about 500 meters to the right. You can go there and have a look."

"But there is an exam tomorrow, so be careful not to spoil your stomach."

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

"Teacher, have you been here before, are you so familiar?" Xu Chuan asked in surprise. Just now, Gong Rihui pushed him more than a dozen snacks and delicacies in one breath.

Gong Rihui smiled and said, "My hometown is Bashu, so I'm naturally familiar with it."

Xu Chuan groaned, nodded, thanked him and planned to try the food street that Gong Rihui was talking about.

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 with a round face came from a distance, and he also handed in the paper ahead of time.

Actually Xu Chuan didn't really want to take him with him, because this young lady really talked too much.

But I couldn't bear the entanglement, so I had to bring this tail.

"Brother Chuan, Brother Chuan, for your second big question, what is the method to prove the set D(n)={ab∣∣n=ab,a,b∈Z+a\u003eb}?"

"Brother Chuan, what shall we eat later?"

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

"Brother Chuan, why do you think this Bashu sister is not afraid of the cold? It's already December, and you still have bare legs?"

"Brother Chuan,"

Along the way, Lu Tianrui kept asking non-stop, a living book of a hundred thousand whys.

Xu Chuan said that he just wanted to find earplugs to plug his ears up. How would he know why sister Bashu is not afraid of the cold? You can ask me if you have the ability.

After walking around the food street with a chattering young lady, 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 special food is not all spicy and spicy like the rumors, it is worth a try.

The only problem is that the tofu nao here is actually hot and sour, which to him is simply an outlier among outliers.

Tofu nao must be sweet!

Back in the training room, after sharing the content and experience of today's exam with other participating teammates, everyone went back to their dormitory to sleep.

There was nothing to say all night, and the next day's exam began soon.

After the entrance inspection and the distribution of test papers, Xu Chuan checked the test papers, and the expression on his face suddenly became extremely subtle.

"Good guy, I said yesterday's exam questions were so easy. It turned out that all the difficult ones were concentrated here."

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

Same as yesterday, there are still three questions, and it is still a geometry, a set of integers, and a function.

But the difficulty of the topic has been doubled several times.

Even he didn't have a specific problem-solving idea in his mind immediately after getting the test paper.

Compared with the simple two triangle figures of yesterday's first geometric triangle, today's geometric figure is more than ten times more complicated.

For a piece of geometry, the title text only occupies two or three lines, but the geometric figure below almost occupies more than half of the test paper.

Xu Chuan counted. There are four circles in this geometric figure, two big circles and two small circles. The circles are interlaced like the five Olympic rings.

In the circle and outside the circle, there are eight different triangles, including common triangles such as acute triangles, right triangles, and obtuse triangles, as well as complex triangles such as cut tile triangles and Leluo triangles.

In addition, there are six branch lines crossing the triangle and the circle, alternating horizontally and vertically, cutting the entire geometric figure into pieces.

Triangles intersect triangles, triangles intersect circles, circles intersect circles, straight lines cut, and Xu Chuan wants to tear up the test paper after looking at the complex graphics.

For math competition students, this is definitely a nightmare.