Question

Chứng minh \(1/(a-b)^2 + 1/(b-c)^2 +1/(c-a)^2\) là bình phương 1 số hữu tỉ

Answer 1

- Cái này mình tham khảo chứ bó tay rồi :)

* Đặt x=a-b ; y=b-c ; z=c-a thì x+y+=a-b+b-c+c-a=0

* \(\dfrac{1}{\left(a-b\right)^2}+\dfrac{1}{\left(b-c\right)^2}+\dfrac{1}{\left(c-a\right)^2}\)=\(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\)=\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-2.\left(\dfrac{1}{xy}+\dfrac{1}{xz}+\dfrac{1}{yz}\right)\)=\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-2.\left(\dfrac{x+y+z}{xyz}\right)\)=\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2\)=\(\left(\dfrac{1}{a-b}+\dfrac{1}{b-c}+\dfrac{1}{c-a}\right)^2\)