Question

1/x+1/y+1/z=0
xy/z^2+yz/x^2+zx/y^2=3 доказать. помогите

Answer 1

$frac{1}{x} + frac{1}{y} + frac{1}{z}=0quad to quad frac{yz+zx+xy}{xyz} =0 quad to; ; xy+yz+zx=0\\\ Dokazat:\\frac{xy}{z^2} +frac{yz}{x^2} + frac{zx}{y^2} =3; .$

$Formyla:\\a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc\\(xy)^3+(yz)^3+(zx)^3=underbrace {(xy+yz+zx)}_{0}(x^2y^2+y^2z^2+z^2x^2-xy^2z-\\-yz^2x-zx^2y)+3x^2y^2z^2$

$(xy)^3+(yz)^3+(zx)^3=3x^2y^2z^2; |:x^2y^2z^2ne 0\\ frac{(xy)^3}{x^2y^2z^2} + frac{(yz)^3}{x^2y^2z^2} + frac{(zx)^3}{x^2y^2z^2} =3\\ frac{xy}{z^2} + frac{yz}{x^2} + frac{zx}{y^2}=3$