PS: Thank you for the two rewards of Zhien IU, comrades, the second watch is coming, recommend it, collect it or something, give me a reward or something, and move forward to sign a contract!

"Is he human or not?" At the end of the mental arithmetic competition, the leading professors of Korea University and Yonsei University made the same exclamation at the same time.

Unlike Korea University and Yonsei University, a group of students from Seoul University and their leading professors gave unprecedented surprises and cheers, because they really didn't expect that Lee Woo-cheol would be such a genius, multiplying thousands and thousands of digits, this guy actually took only five seconds to calculate by heart, how powerful it is.

The great demon of Goryeo was directly stunned, but in the face of the cheers and cheers around him, Li Yuzhe behaved very ordinarily, and these sounds would have made him a little excited, but now, Li Yuzhe has completely adapted to such cheers.

At this time, Li Yuzhe knew that the victory of mental arithmetic was not the final victory, and there was still that 55-point math problem in the end, and he wanted to enter Seoul University, although Lin Wanxi had promised him just now that he would let him enter Seoul University no matter whether he won or lost, but Li Yuzhe didn't want to lose, he just wanted to win, he wanted to defeat the most talented mathematics students of Korea University and Yonsei University, and give a meeting gift to the Department of Mathematics of Seoul University

"Don't panic, there is another math problem behind, that problem is 55 points, as long as we can solve it, we can still turn defeat into victory!"

At this time at the scene of the game, the leading professors of the two universities shouted at their respective students regardless of the image, Chen Guozheng was also anxious on the side of Seoul University, Chen Guozheng knew the details of Li Yuzhe, although Li Yuzhe's mental arithmetic won very beautifully, but the 55-point math problem behind it, you don't have to guess that it must be the most difficult math problem in college, you let a guy who has never been in high school solve the math problem in college, isn't that funny?

However, Chen Guozheng listened to his professor's words and also believed in luck, but unlike Lin Wanxi's belief, Lin Wanxi believed that Li Yuzhe would definitely be able to solve the math problem, while Chen Guozheng cursed that the other two universities could not solve the math problem from the referee, in this case, Li Yuzhe with 45 points would win.

While everyone was holding their breath, the referee from Sungkyunkwan University began to write out the math problems he had come up with!

It is proved that the sufficient necessary condition for the tangent of the plane Ax+By+Cz+D=0(D>0) and the quadric surface (x^2/a)+(y^2/b)+(z^2/c)=1, abc is not = 0 is aA^2+bB^2+cC^2=D^22.a1b1c1Let A=a2b2c2 be an invertible matrix, then the straight line a3b3c3x/(a1-a

1. It is proved that the sufficient and necessary condition for the tangent of the plane Ax+By+Cz+D=0(D>0) and the quadric surface (x^2/a)+(y^2/b)+(z^2/c)=1 and abc not = 0 is aA^2+bB^2+cC^2=D^2

2.a1b1c1

Let A=a2b2c2 be an invertible matrix, then a straight line

a3b3c3

The positional relationship between x/(a1-a2)=y/(b1-b2)=z/(c1-c2) and x/(a2-a3)=y/(b2-b3)=z/(c2-c3) is ____ (intersecting, parallel, coincident, different planes)

(The title was obtained on Baidu.,I don't know if it's powerful.,I don't understand it at all.,If you understand.,Let's take a look!) ）

As soon as the problem was finished, everyone present gasped, the most devastating analytic geometry proof problem in the legend, and the most tedious and difficult, when encountering such a problem, all the mathematical geniuses had to secretly shout for help.

Even several associate professors who led the team frowned slightly, and Chen Guozheng from Seoul University said speechlessly: "Why is it so difficult?" This is simply the level of the professor to prove, what is the professor of Sungkyunkwan doing? ”

Chen Guozheng's muttering made Lin Wanxi smile slightly, just now he was still praying that the other party couldn't make a question, but now people are making the topic difficult, and it is too difficult to question, this student of his own is still too young, and he is not deep in the WTO.

Just when everyone was muttering, the professor from Sungkyunkwan stood up with a smile and said: "Everyone, when I led the team to Harvard College, the other party taught us students a math problem, I am very sorry, none of the ten students we went to prove it correct, I still remember the arrogance of the other party at that time, he pointed at us and said, such a question in Korea is not a student can prove, because the IQ of Koreans is always less than half of that of Americans, that time our students were angry." , but they can't vent, because none of them have proved each other's problem, and now I am publishing this question, just hoping that the top college students in South Korea at this time will prove this problem, and tell the Americans of that group of forces, don't deceive me No one in Korea. “

The words of the Sungkyunkwan professor made the classroom that was still muttering completely silent at this time, Koreans have a strong sense of nationality, although they like to hug the thighs of Americans, but they are ridiculed, and they are still very indignant in their hearts, because no one likes their motherland to be ridiculed.

Everyone was crazy about proving there, including all the math geniuses who participated in the competition, but the proof problem was really too difficult, and the professors were also anxious in their hearts, of course they could prove it, but their own students couldn't, and they could only do it in a hurry.

As the minutes passed, the Sungkyunkwan professor whispered sadly, "Is there really no one in our country?" ”

At this moment, Li Yuzhe stood up and said lazily: "I'll try it!" ”

In a word, the focus of the audience instantly focused on Li Yuzhe's body, and soon Li Yuzhe's dragon and phoenix dance began.

It is proved that the sufficient and necessary condition for the tangent of the plane Ax+By+Cz+D=0(D>0) and the quadric surface (x^2/a)+(y^2/b)+(z^2/c)=1 and abc not = 0 is aA^2+bB^2+cC^2=D^2

If the plane is tangent to the surface, then the plane normal vector (A, B, C) is proportional to the surface normal vector at the tangent point (2x/a, 2y/b, 2z/c): A=kx/a, B=ky/b, C=kz/c. (1).

kx^2/a+ky^2/b+kz^2/c+D=0,

k(x^2/a+y^2/b+z^2/c)+D=0,k=－D.

aA^2+bB^2+cC^2

=k^2x^2/a+k^2y^2/b+k^2z^2/c)

=k^2=(－D)^2=D^2. (2)

Writing this, all the professors showed a positive smile, and the genius students of those competitions also showed a trace of enlightenment. Of course, there are still many students who don't know, but Li Yuzhe's proof has just begun! Then, Li Yuzhe continued to write without distraction:

Evidence (1): aA^2+bB^2+cC^2=D^2=(Ax+By+Cz)^2,

(a－x^2) A^2+(b－y^2)B^2+(c－z^2)C^2－2ABxy－2BCyz－2CAzx=0,

aA^2(y^2/b+z^2/c)+bB^2(z^2/c+x^2/a)+cC^2(x^2/a+y^2/b)－2ABxy－2BCyz－2CAzx=0,

(aA^2y^2/b－2ABxy+bB^2x^2/a)+(bB^2z^2/c－2BCyz+cC^2y^2/b)+(cC^2x^2/a－2CAzx+aA^2z^2/c)=0,

[√(a/b)Ay－√(b/a)Bx]^2+[√(b/c)Bz－√(c/b)Cy]^2+[√(c/a)Cx－√(a/c)Az]^2=0,

√(a/b)Ay=√(b/a)Bx,√(b/c)Bz=√(c/b)Cy,√(c/a)Cx=√(a/c)Az,

aAy=bBx,bBz=cCy,cCx=aAz,

aA/x=bB/y=cC/z,(1) Proven.

2.a1b1c1

Let A=a2b2c2 be an invertible matrix, then a straight line

a3b3c3

The positional relationship between x/(a1-a2)=y/(b1-b2)=z/(c1-c2) and x/(a2-a3)=y/(b2-b3)=z/(c2-c3) is ____ (intersecting, parallel, coincident, different planes)

Note P(a1,b1,c1),Q(a2,b2,c2),R(a3,b3,c3),

matrix A is reversible, P, Q, R are not collinear,

x/(a1-a2)=y/(b1-b2)=z/(c1-c2),

Direction vector = vector (a1-a2, b1-b2, c1-c2) = vector QP,

x/(a2-a3)=y/(b2-b3)=z/(c2-c3),

Direction vector = vector (a2-a3, b2-b3, c2-c3) = vector RQ,

The vectors QP and RQ are not parallel, so two straight lines intersect.

After the proof was written, all the students in the audience, all the professors, collectively involuntarily spit out two words from their mouths: "Genius!" ”

(The title is copied from Baidu, that's what it means, hehe!) ）

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