\(A=\frac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\frac{y^2-xz}{\left(y+z\right)\left(y+x\right)}+\frac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)

\(=\frac{\left(x^2-yz\right)\left(y+z\right)+\left(y^2-xz\right)\left(z+x\right)+\left(z^2-xy\right)\left(x+y\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)

\(=\frac{x^2y+x^2z-y^2z-yz^2+y^2z+y^2x-xz^2-x^2z+z^2x+z^2y-x^2y-xy^2}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)

\(=\frac{0}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}=0\)

Vậy : \(A=0\)

\(\frac{(x^2-yz)(y+z)}{(x+y)(x+z)(y+z)}\) = ​​​\(\frac{(y^2-xz)(x+z)}{(x+y)(x+z)(y+z)}\)​= \(\frac{(z^2-xy)(x+y)}{(x+y)(x+z)(y+z)}\)

\(A=\frac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\frac{y^2-xz}{\left(y+z\right)\left(y+x\right)}+\frac{z^2-xy}{\left(z+x\right)\left(z+y\right)}\)

\(=\frac{\left(x^2-yz\right)\left(y+z\right)+\left(y^2-xz\right)\left(x+z\right)+\left(z^2-xy\right)\left(x+y\right)}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}\)

\(=\frac{x^2y+x^2z-y^2z-yz^2+xy^2+y^2z-x^2z-xz^2+xz^2+yz^2-x^2y-xy^2}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}\)

\(=\frac{0}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}\)

\(=0\)

Study well ! 

Xem lại cái đề đi Tuyển. Hình như giá trị nhỏ nhất của cái biểu thức dưới còn lớn hơn là 1 thì làm sao bài đó có giá trị x, y, z thỏa được mà bảo tính A.

Xét tích : \(\left[x^2\left(z-y\right)+y^2\left(x-z\right)+z^2\left(y-x\right)\right]\left(x+y+z\right)\)

=\(x^3\left(z-y\right)+x^2\left(z-y\right)\left(z+y\right)+y^3\left(x-z\right)+y^2\left(x-z\right)\left(x+z\right)\)

\(+z^3\left(y-x\right)+z^2\left(y-x\right)\left(y+x\right)\)

\(=x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)+x^2\left(z^2-y^2\right)+y^2\left(x^2-z^2\right)+z^2\left(y^2-x^2\right)\)

\(=x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)+x^2z^2-x^2y^2+y^2x^2-y^2z^2+z^2y^2-z^2x^2\)

\(=x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)\)

Như vậy:

 \(\left[x^2\left(z-y\right)+y^2\left(x-z\right)+z^2\left(y-x\right)\right]\left(x+y+z\right)\)\(=x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)\)

<=> \(\frac{x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)}{x^2\left(z-y\right)+y^2\left(x-z\right)+z^2\left(y-x\right)}=x+y+z\)

Ta có: \(\frac{\frac{x^2\left(z-y\right)}{yz}+\frac{y^2\left(x-z\right)}{xz}+\frac{z^2\left(y-x\right)}{xy}}{\frac{x\left(z-y\right)}{yz}+\frac{y\left(x-z\right)}{xz}+\frac{z\left(y-x\right)}{xy}}\)

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

\(=\frac{x^3\left(z-y\right)+y^3\left(x-z\right)+z^3\left(y-x\right)}{x^2\left(z-y\right)+y^2\left(x-z\right)+z^2\left(y-x\right)}=x+y+z\)

\(A=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(y-x\right)\left(y-z\right)}+\frac{xy}{\left(z-x\right)\left(z-y\right)}\)

=> \(\left(-A\right)=\frac{yz}{\left(x-y\right)\left(z-x\right)}+\frac{xz}{\left(x-y\right)\left(y-z\right)}+\frac{xy}{\left(z-x\right)\left(y-z\right)}\)

<=> \(\left(-A\right)=\frac{yz\left(y-z\right)+xz\left(z-x\right)+xy\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}=\frac{y^2z-yz^2+xz^2-x^2z+xy\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)

<=> \(\left(-A\right)=\frac{z^2\left(x-y\right)-z\left(x^2-y^2\right)+xy\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)=> \(\left(-A\right)=\frac{\left(x-y\right)\left(z^2-zx-zy+xy\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}=\frac{\left(x-y\right)\left[z\left(z-x\right)-y\left(z-x\right)\right]}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)

\(\left(-A\right)=\frac{\left(x-y\right)\left(z-x\right)\left(z-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}=-1\)

=> A = 1

Đáp số: A=1

Giúp mình với các bạn

\(\frac{x^2-yz}{\left(x+y\right)\left(x+z\right)}+\frac{y^2-xz}{\left(x+y\right)\left(y+z\right)}+\frac{z^2-xy}{\left(x+z\right)\left(y+z\right)}\)

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

\(=\frac{x^2y-y^2z+x^2z-yz^2+y^2x-x^2z+zy^2-xz^2+z^2x-x^2y+yz^2-xy^2}{\left(x+y\right)\left(x+z\right)\left(y+z\right)}\)

\(=\frac{0}{\left(x+y\right)\left(x+z\right)\left(y+z\right)}\)

\(=0\)\(\left(\text{Đ}K:x+y,y+z,z+x\ne0\right)\)

Tham khảo nhé~

\(1A=\frac{xy}{\left(z-x\right)\left(z-y\right)}+\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{zx}{\left(y-x\right)\left(y-z\right)}\)

\(=-1\left(\frac{xy}{\left(y-z\right)\left(z-x\right)}+\frac{yz}{\left(x-y\right)\left(z-x\right)}+\frac{zx}{\left(y-z\right)\left(x-y\right)}\right)\)

\(=-1.\left(\frac{xy\left(x-y\right)+yz\left(y-z\right)+zx\left(z-x\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\right)\)

 \(=\frac{-1\left(x-y\right)\left(z-x\right)\left(z-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}=1\)

Áp dụng BĐT Cô - si cho 3 bộ số không âm

\(\Rightarrow\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}+\frac{x\left(yz+1\right)^2}{z^2\left(xz+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}\ge3\sqrt[3]{\frac{xyz\left(xy+1\right)^2\left(yz+1\right)^2\left(xz+1\right)^2}{x^2y^2z^2\left(yz+1\right)\left(xz+1\right)\left(xy+1\right)}}=3\sqrt[3]{\frac{\left(xy+1\right)\left(yz+1\right)\left(xz+1\right)}{xyz}}\)

Xét \(3\sqrt[3]{\frac{\left(xy+1\right)\left(yz+1\right)\left(xz+1\right)}{xyz}}\)

\(=3\sqrt[3]{\left(\frac{xy+1}{x}\right)\left(\frac{yz+1}{y}\right)\left(\frac{xz+1}{z}\right)}\)

\(=3\sqrt[3]{\left(y+\frac{1}{x}\right)\left(z+\frac{1}{y}\right)\left(x+\frac{1}{z}\right)}\)

Áp dụng BĐT Cô - si

\(\Rightarrow\left\{\begin{matrix}y+\frac{1}{x}\ge2\sqrt{\frac{y}{x}}\\z+\frac{1}{y}\ge2\sqrt{\frac{z}{y}}\\x+\frac{1}{z}\ge2\sqrt{\frac{x}{z}}\end{matrix}\right.\)

\(\Rightarrow\left(y+\frac{1}{x}\right)\left(z+\frac{1}{y}\right)\left(x+\frac{1}{z}\right)\ge8\)

\(\Rightarrow3\sqrt[3]{\left(y+\frac{1}{x}\right)\left(z+\frac{1}{y}\right)\left(x+\frac{1}{z}\right)}\ge3\sqrt[3]{8}\)

\(\Rightarrow3\sqrt[3]{\left(y+\frac{1}{x}\right)\left(z+\frac{1}{y}\right)\left(x+\frac{1}{z}\right)}\ge6\)

\(\Leftrightarrow3\sqrt[3]{\frac{\left(xy+1\right)\left(yz+1\right)\left(xz+1\right)}{xyz}}\ge6\)

Mà \(\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}+\frac{x\left(yz+1\right)^2}{z^2\left(xz+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}\ge3\sqrt[3]{\frac{\left(xy+1\right)\left(yz+1\right)\left(xz+1\right)}{xyz}}\)

\(\Rightarrow\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}+\frac{x\left(yz+1\right)^2}{z^2\left(xz+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}\ge6\)

Vậy GTNN của \(\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}+\frac{x\left(yz+1\right)^2}{z^2\left(xz+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}=6\)