Chapter 30 The difficult second half (I have something to do this afternoon, so I will post in advance)
Solution: First make the auxiliary lines EI, FI, BI and CI.
Sufficiency: If BC=BE+CF, then you can take a point K in side BC to make BK=BE, so CK=CF, connecting KI.
Take the inner center I of △ABC on the bisector AD of ∠BAC. Since BI bisects ∠ABC and CI bisects ACB, △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, and F are cocircular
Combine the four points B, E, F and C to form a circle
Necessity: If the incenter I of △ABC is the circumcenter of △DEF, since AE≠AF (in fact, the four points B, E, F, and C are cocircles)
Therefore BC=BK+CK=BE+CF.
Necessity is proved.
In ten minutes, Xu Chuan successfully solved the first big question.
This question is not very difficult. 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 value.
This is a point that is rarely used in high school geometry to solve triangles and cocircles, but as long as you master these two points, solving the first question is not a problem.
Half an hour later, the second and more difficult integer calculation was also defeated.
“This year’s questions don’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 lowering their heads to do the questions, which confirmed his idea.
After all, if the question is too difficult, some students will definitely look up to the sky.
This is a phenomenon he observed during his two lifetimes of competition.
"Forget it, finish the third question quickly, and then hand in the paper to try the delicious food outside."
Xu Chuan shook his head and focused on the test paper again.
It was his first time to come to Bashu, but some of the delicacies he had eaten in the Bashu Middle School cafeteria yesterday gave him a good impression of this place.
If a high school can make such delicious food, then the restaurants outside should taste more authentic.
Although he is not a foodie, he is still very keen on trying food.
Of course, the premise is that the food suits his taste.
Forget about looking up at the stars, cheese with live maggots, canned herring and other dark dishes.
But for now, the local food in Bashu is quite good.
After finishing the three major questions, Xu Chuan checked them twice, and after confirming that there were no problems, he handed in the papers in advance.
The leader of this year's Xiangnan Provincial Team is Teacher Gong Rihui from Xingcheng No. 1 Middle School. He is very familiar with Xu Chuan. When he saw that he was stuck and handed in the paper in advance, he knew that the questions this year might not be that difficult.
Handing over a bottle of water, Gong Rihui asked with a smile: "How was your exam?"
"It's okay. The questions in the first half are not very difficult. Let's watch the second half."
Xu Chuan opened the water and took a sip, then asked: "Teacher Gong, are there any delicious food nearby? I want to try it."
"There are quite a lot of delicacies in the Bashu area. Bobo chicken, Sichuan jelly, Dandan noodles, brown sugar glutinous rice cake, swan eggs, Sanhe mud, three big guns, and fat rice rolls are all very good."
"There is a food street nearby after leaving the school gate and walking 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 get upset."
Gong Rihui reminded with a smile that he was still relieved about the student in front of him. His stage was in the larger IMO, not the domestic CMO.
"Teacher, you have been here before, are you so familiar?" Xu Chuan asked in surprise. Gong Rihui had just pushed him a dozen snacks and delicacies in one go.
Gong Rihui smiled and said: "My hometown is Bashu, so I am naturally familiar with it."
Xu Chuan snorted, nodded, thanked him and prepared to try the food court Gong Rihui mentioned. But before he could take two steps, a voice behind him stopped him.
"Brother Chuan, where are you going? Wait for me."
Behind him, Lu Tianrui's voice with a small round face came from far away, and he also handed in the paper in advance.
Xu Chuan actually didn't want to take him with him because this young lady really talked too much.
But I couldn't bear the entanglement, so I had no choice but to wear this tail.
"Brother Chuan, Brother Chuan, your second big question is, what is the method to prove the set D(n)={ab∣∣n=ab, a, b∈Z+a>b}?"
"Brother Chuan, what shall we eat later?"
"Brother Chuan, you said if we wait and eat hot pot, will we get diarrhea?"
"Brother Chuan, why is this Bashu girl not afraid of the cold? It's already December and she's still bare-legged?"
"Brother Chuan,"
Along the way, Lu Tianrui kept asking questions, like a walking book of One Hundred Thousand Whys.
Xu Chuan said that he just wanted to find earplugs to plug his ears. How would he know why Bashu sister is not afraid of the cold? If you have the ability, you can ask me.
After walking around the food court with the 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 of the specialties are not spicy like the rumors, so they are worth a try.
The only problem is that the tofu noodle here is spicy and sour, which to him is an outlier among the outliers.
Things like tofu must be sweet!
After returning to the training room and sharing the content and experience of today's exam with other participating teammates, everyone went back to their dormitories to sleep.
There was nothing to say all night, and the exam started soon the next day.
After the entrance inspection and test paper distribution, Xu Chuan checked the test paper, and the expression on his face suddenly became extremely subtle.
"Good guy, I asked you why yesterday's test questions were so easy. It turns out that all the difficult questions are concentrated here."
After scanning the questions on the test paper, Xu Chuan muttered softly.
Just like yesterday, there are still three questions, and they are still geometry, a set of integers, and a function.
But the difficulty of the questions has increased several times.
Even he didn't have any specific problem-solving ideas in his mind immediately after getting the test paper.
Compared with yesterday's first geometric triangle, which consisted of two simple triangle figures, today's geometric figure is more than ten times more complicated.
For a geometry question, the text in the question only occupies two or three lines, but the geometric diagram below occupies almost half of the test paper.
Xu Chuan counted and found that there were four circles in this geometric figure, two large circles and two small circles. The circles were intertwined with each other like the five Olympic rings.
There are eight different triangles inside and outside the circle, including common triangles such as acute-angled triangles, right-angled triangles, and obtuse-angled triangles, as well as complex triangles such as the Cheval triangle and the Lelo triangle.
In addition, there are six branch lines that cross the triangle and the circle, alternating horizontally and vertically, cutting the entire geometric figure into pieces.
Triangles intersect triangles, triangles intersect circles, circles are tangent to circles, and straight lines cut each other. Xu Chuan wanted to tear up the test paper when he saw the complicated figures.
For mathematics competition students, this is definitely a nightmare.
(End of chapter)