Đặt \(A=\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=2009,B=\frac{y^2}{x+y}+\frac{z^2}{y+z}+\frac{z^2}{x+z}\)
\(=>A-B=\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{x^2}{z+x}-\frac{y^2}{x+y}-\frac{z^2}{y+z}+\frac{x^2}{z+x}\)
\(=>2009-B=\frac{x^2-y^2}{x+y}+\frac{y^2-z^2}{y-z}+\frac{z^2-x^2}{z-x}\)
\(=>2009-B=\frac{\left(x-y\right).\left(x+y\right)}{x+y}+\frac{\left(y-z\right).\left(y+z\right)}{y+z}+\frac{\left(z-x\right).\left(z+x\right)}{z+x}\)
=>2009-B=x-y+y-x+z-x
=>2009-B=(x-x)+(y-y)+(z-z)
=>2009-B=0+0+0
=>2009-B=0
=>B=2009
Vậy \(\frac{x^2}{x+y}+\frac{z^2}{y+z}+\frac{x^2}{z+x}=2009\)
thông minh đấy,mới lớp 7 mà làm được bài lớp 8
