Question

[Big City Boy]

Cho a+b+c=0, x+y+z=0, a/x+b/y+c/z=0. CMR: \(ax^2+by^2+cz^2=0\)

Answer 1

[Trần Minh Hoàng]

Ta có \(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0\Leftrightarrow ayz+bzx+cxy=0\).

Do đó: \(ax^2+by^2+cz^2=\left(ax+by+cz\right)\left(x+y+z\right)-axy-axz-byz-byx-czx-czy=0-xy\left(a+b\right)-yz\left(b+c\right)-zx\left(c+a\right)=0+xyc+yza+zxb=0\).