Udowodnij równość, jeżeli...

[ranatharu]

Udowodnij, że jeżeli x+y+z = xyz, to

x(1 - \(\displaystyle{ y^{2}}\))(1 - \(\displaystyle{ z^{2}}\))+y(1 - \(\displaystyle{ z^{2}}\))(1 - \(\displaystyle{ x^{2}}\))+z(1 - \(\displaystyle{ x^{2}}\))(1 - \(\displaystyle{ y^{2}}\))=4xyz

[smigol]

\(\displaystyle{ L=x+y+z-xz^2-xy^2=xy^2z^2-yx^2-yz^2+yz^2x^2-zy^2-zx^2+zx^2y^2=x+y+z-xz^2-xy^2+yz(x+y+z)-yx^2-yz^2+zx(x+y+z)-zy^2-zx^2+xy(x+y+z)=x+y+z+xyz+xyz+xyz=4xyz=P}\)