ta có : \(x^2+y^2+z^2=xy+yz+zx\Leftrightarrow2x^2+2y^2+2z^2=2xy+2yz+2zx\)
\(\Leftrightarrow2x^2+2y^2+2z^2-2xy-2yz-2zx=0\)\(\Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\) \(\Leftrightarrow x=y=z\left(đpcm\right)\)
CMR: \(x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
CMR: x3+y3+z3-3xyz= (x+y+z)(x2+y2+z2- xy - yz - xz)
cho x,y,z là các số dương thỏa mãn x+y+z=3
Cmr \(\frac{1}{x^2+y^2+z^2}+\frac{2009}{xy+yz+xz}\ge670\)
Cho x2-yz =a
y2 -xz=b
z2- xy=c (x, y,z ≠0)
CMR: ax+by+cz= (x+y+z)(a+b+c)
cho x,y,z >0 thỏa mãn x ≥ z. Cmr:
\(\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{x+2z}{x+z}\ge\frac{5}{2}\)
Với mọi x,y,z là các số thực bất kì , CMR \(x^2+y^2+z^2\ge xy+yz+xz\)
103,CM:\(\frac{\frac{x^2\left(z-y\right)}{yz}+\frac{y^2\left(x-z\right)}{xz}+\frac{z^2\left(y-x\right)}{xy}}{\frac{x\left(z-y\right)}{yz}+\frac{y\left(x-z\right)}{zx}+\frac{z\left(y-x\right)}{xy}}=x+y+z\)
CMR:
a,\(x^2+y^2+z^2\ge\frac{1}{3}\left(x+y+z\right)^2\)
b,\(\left(x+y+z\right)^2\ge3\cdot\left(xy+yz+xz\right)\)
Cho x,y,z là các số thực dương. Tìm Max
Q=\(\frac{xy}{x^2+xy+yz}+\frac{yz}{y^2+yz+xz}+\frac{zx}{z^2+zx+xy}\)