Chapter 30 The difficult second half is ahead of schedule this afternoon
Chapter 30 The difficult second half (I have something to do this afternoon, so I’ll post in advance)
Solution: First make the auxiliary lines EI, FI, BI, 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 a total circle), then
Therefore BC=BK+CK=BE+CF.
Necessity is proven.
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, more difficult, set of integers was also defeated.
"Brother Chuan, why is this Bashu sister not afraid of the cold? It's already December and she still has bare legs?"
After all, if the question is too difficult, some students will definitely look up to the sky.
But the difficulty of the questions has increased several times.
Xu Chuan shook his head and focused his attention on the test paper again.
Behind him, the voice of the small round-faced Lu Tianrui came from far away, and he also handed in the paper in advance.
Although he is not a foodie, he is still very keen on trying food.
Along the way, Lu Tianrui kept asking questions, a living book of One Hundred Thousand Whys.
"But there is an exam tomorrow, so be careful and don't get upset."
Xu Chuan actually didn't want to take him with him because this young lady really talked too much.
Of course, the premise is that the food suits his taste.
If a high school can make such delicious food, then the restaurants outside should taste more authentic.
Although this little round face is a bit annoying, the food in Bashu is still very good, and some specialties are not spicy like the popular ones, so they are worth a try.
Xu Chuan said that he just wanted to find earplugs to plug his ears. How would he know why Sister Bashu is not afraid of the cold? If you have the ability, you can ask me.
"Brother Chuan,"
After scanning the questions on the test paper, Xu Chuan muttered softly.
But I couldn't bear the entanglement, so I had no choice but to wear this tail.
After the entrance inspection, the test papers were distributed, Xu Chuan checked the test papers, and the expression on his face suddenly became extremely subtle.
"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 cannons, and fat rice rolls are all very good."
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.
But before he could take two steps, a voice behind him stopped him.
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.
"This year's questions don't seem to be that difficult."
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.
"It's okay. The questions in the first half are not very difficult. Let's watch the second half."
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.
"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."
"Teacher, you have been here before, are you so familiar with it?" Xu Chuan asked in surprise. Gong Rihui had just recommended more than a dozen snacks and delicacies to him in one go.
The only problem is that the tofu noodle here is spicy and sour, which to him is simply an outlier among outliers.
"Brother Chuan, what shall we eat later?"
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 was nothing to say all night, and the exam started soon the next day.
Forget about those dark dishes like looking up at the stars, live maggot cheese, and canned herring.
For a geometry test, the text in the question only occupies two or three lines, but the geometric diagram below occupies almost half of the test paper.
Gong Rihui smiled and said: "My hometown is from Bashu, so I am naturally familiar with it."
"Brother Chuan, you said if we wait and eat hot pot, will it cause diarrhea?"
"Good guy, I asked you why the test questions yesterday were so easy. It turns out that all the difficult questions are concentrated here."
"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+ab}?"
Handing over a bottle of water, Gong Rihui asked with a smile: "How was your exam?"
After completing the three major questions, Xu Chuan checked them twice and submitted them in advance after confirming that there were no problems.
But for now, the local food in Bashu is quite good.
Xu Chuan snorted, nodded, thanked him and prepared to try the food court Gong Rihui mentioned.
"Brother Chuan, where are you going? Wait for me."
After walking around the food court with the chattering young lady, Xu Chuan returned to the training room.
Just like yesterday, there are still three questions, and they are still one geometry, one set of integers, and one function.
This is a phenomenon he observed during the competition between his two lifetimes.
Even he didn't have any specific problem-solving ideas in his mind immediately after getting the test paper.
It was his first time to come to Bashu, but some of the delicacies he had eaten in the canteen of Bashu Middle School yesterday gave him a good impression of this place.
"Forget it, finish the third question quickly, and then hand in the paper to try the delicious food outside."
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.
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 acute triangles, right-angled triangles, common triangles such as obtuse triangles, and complex triangles such as cut-off triangles and snare triangles.
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.