Ta có: \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\Leftrightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\cdot\frac{xyc+yza+zxb}{abc}=1\)

Mà \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\Leftrightarrow\frac{yza+zxb+xyc}{xyz}=0\)

\(\Rightarrow yza+zxb+xyc=0\)

\(\Rightarrow A=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)

Ta có: \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\Rightarrow ayz+bxz+cxy=0\)

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) (\(a;b;c\ne0\) )

\(\Rightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+\frac{2xy}{ab}+\frac{2yz}{bc}+\frac{2xz}{ac}=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1-2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1-2\left(\frac{ayz+bxz+cxy}{abc}\right)=1-2.0=1\)

=> đpcm

á em đổi biến lộn ạ. Em định viết H;U;Y  cho đúng tên mình mà quen tay lộn vào Y;Z ạ

Đặt \(\left(\frac{x}{a};\frac{y}{b};\frac{z}{c}\right)\rightarrow\left(H;U;Y\right)\)

Khi đó ta có:

\(H+U+Y=1;\frac{1}{H}+\frac{1}{U}+\frac{1}{Y}=0\Rightarrow HU+UY+YH=0\)

Thay vào thì :

\(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\left(H+U+Y\right)^2-2\left(HU+UY+YH\right)=1\)

Vậy ta có đpcm

với x=y=z khác 0 và a,b,c khác nhau là 1 số bất kỳ khác 0 thì (1) thỏa mãn và (2) không thỏa mãn

=> Không thể CM

ta có: \(\frac{x^2-yz}{a}=\frac{y^2-zx}{b}=\frac{z^2-xy}{c}\)

\(\Rightarrow\frac{a}{x^2-yz}=\frac{b}{y^2-zx}=\frac{c}{z^2-xy}\) (*)

\(\Rightarrow\frac{a^2}{\left(x^2-yz\right)^2}=\frac{bc}{\left(y^2-zx\right).\left(z^2-xy\right)}=\frac{a^2-bc}{\left(x^2-yz\right)^2-\left(y^2-zx\right).\left(z^2-xy\right)}\)

\(=\frac{a^2-bc}{x^4-3x^2yz+xy^3+xz^3}=\frac{a^2-bc}{x.\left(x^3-3xyz+y^3+z^3\right)}\)

\(\Rightarrow\frac{a^2-bc}{x}=\frac{a^2}{\left(x^2-yz\right)^2}.\left(x^3-3xyz+y^3+z^3\right)\)

Làm tương tự như trên. ta có:

\(\frac{b^2-ca}{y}=\frac{b^2}{\left(y^2-zx\right)^2}.\left(x^3-3xyz+y^3+z^3\right)\)

\(\frac{c^2-ab}{z}=\frac{c^2}{\left(z^2-xy\right)^2}.\left(x^3-3xyz+y^3+z^3\right)\)

Từ (*) \(\Rightarrow\frac{a^2-bc}{x}=\frac{b^2-ca}{y}=\frac{c^2-ab}{z}\left(đpcm\right)\)

Ờm thì đại khái như vầy , dùng thêm hằng cao cấp mới chơi được =))

Link : Bảy hằng đẳng thức đáng nhớ – Wikipedia tiếng Việt 

Dùng hằng mở rộng số 4

Ta có :

\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\)

\(\Leftrightarrow ayz+bxz+cxy=0\) (1)

Lại có :

\(\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\right)^2=\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{c^2}{z^2}+2.\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1^2=1\) (chỗ này dùng cái skill mở rộng) 

<=> \(\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{c^2}{z^2}+2.\left(\frac{xyc}{abc}+\frac{ayz}{abc}+\frac{bzx}{abc}\right)=1\)

<=> \(\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{c^2}{z^2}+2.\frac{ayz+bxz+cxy}{abc}=1\)

Thay 1 vào 

=> \(\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{c^2}{z^2}=1\)

mình giải hơi khác 1 chút, nhưng thôi cx đc

Sửa lại :

Lại có :

\(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{zx}{ca}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xyc}{abc}+\frac{yza}{abc}+\frac{zxb}{cba}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2.\frac{ayz+bxz+cxy}{abc}=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)

2. Có : 1/x + 1/y + 1/z = 0

=> 1 + x/y + x/z = 0 => x/y + x/z = -1

Tương tự : y/x + y/z = -1 ; z/x + z/y = -1

=> x/y + x/z + y/x + y/z + z/x + z/y = -3

Lại có : 1/x+1/y+1/z = 0

<=> xy+yz+zx/xyz = 0

<=> xy+yz+zx = 0

Xét : 0 = (xy+yz+zx).(1/x^2+1/y^2+1/z^2)

           = xy/z^2+xz/y^2+xy/z^2+x/y+y/x+y/z+z/y+z/x+x/z

           = xy/z^2+xz/y^2+xy/z^2-3

=> xy/z^2+xz/y^2+xy/z^2 = 3

=> ĐPCM

Tk mk nha

Áp dụng BĐT Cô si ta có: 

\(1=\left(a+b+c\right)^2\ge4a\left(b+c\right)\)

\(\Leftrightarrow b+c\ge4a\left(b+c\right)^2\)

Mà \(\left(b+c\right)^2\ge4bc\)

\(\Rightarrow b+c\ge4a.4bc=16abc\)

\(\frac{a^2}{x}+\frac{b^2}{y}=\frac{\left(a+b\right)^2}{x+y}\)  C/M thế này cho ít số dễ nhìn 

Quy đồng ta được

\(a^2y\left(x+y\right)+b^2x\left(x+y\right)=xy\left(a^2+2ab+b^2\right)\)

\(a^2yx+a^2y^2+b^2x^2+b^2xy=a^2xy+2abxy+b^2xy\)

rút gọn

\(a^2y^2+b^2x^2=2abxy\)

\(a^2y^2+b^2x^2-2abxy=0\) hằng đẳng thức số 2

\(\left(ay+bx\right)^2=0\) 

\(ay+bx=0\Leftrightarrow ax=-bx\)

vậy \(-bx+bx=0\) đúng 

\(\frac{a^2}{x}+\frac{b^2}{y}=\frac{\left(a+b\right)^2}{x+y}\Leftrightarrow\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}=\frac{\left(a+b+c\right)^2}{x+y+z}\)

Đặt \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=k\)\(\Rightarrow\hept{\begin{cases}x=ak\\y=bk\\z=ck\end{cases}}\)

Ta có \(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}=\frac{a^2}{ak}+\frac{b^2}{bk}+\frac{c^2}{ck}=\frac{a}{k}+\frac{b}{k}+\frac{c}{k}=\frac{a+b+c}{k}\)(1)

\(\frac{\left(a+b+c\right)^2}{x+y+z}=\frac{\left(a+b+c\right)^2}{ak+bk+ck}=\frac{\left(a+b+c\right)^2}{\left(a+b+c\right)k}=\frac{a+b+c}{k}\)(2)

Từ (1); (2) => \(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}=\frac{\left(a+b+c\right)^2}{x+y+z}\)

Từ \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)

\(\Rightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1^2\)

   \(\left(\frac{x}{a}+\frac{y}{b}\right)^2+2\left(\frac{x}{a}+\frac{y}{b}\right)\frac{z}{c}+\left(\frac{z}{c}\right)^2=1\)

\(\left(\frac{x}{a}\right)^2+2\frac{x}{a}\frac{y}{b}+\left(\frac{y}{b}\right)^2+\left(2\frac{x}{a}+2\frac{y}{b}\right)\frac{z}{c}+\left(\frac{z}{c}\right)^2=1\)

\(\frac{x^2}{a^2}+\frac{2xy}{ab}+\frac{y^2}{b^2}+\frac{2xz}{ac}+\frac{2yz}{bc}+\frac{z^2}{c^2}=1\)

\(\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)+\left(\frac{2xy}{ab}+\frac{2xz}{ac}+\frac{2yz}{bc}\right)=1\)

\(\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)+\frac{2xyz}{abc}\left(\frac{c}{z}+\frac{b}{y}+\frac{a}{x}\right)=1\)

\(\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)+\frac{2xyz}{abc}.0=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\left(ĐPCM\right)\)

\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\Rightarrow ayz+bxz+cxy=0\)

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\Leftrightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{xz}{ac}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1-2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{xz}{ac}\right)\)

\(=1-2.\frac{cxy+bxz+ayz}{abc}=1-2.0=1\)