فرض کنید:
$$A:=xyz,B:=xy+yz+zx,C:=x^2y^2+y^2z^2+z^2x^2,D:=x^4+y^4+z^4$$
بنا به اتحاد مربع و اویلر داریم:
$$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$
$$ \Rightarrow 1^1=9+2B \Rightarrow 2B=1-9=-8 \Rightarrow B=-4$$
$$,x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$$
$$ \Rightarrow 1-3A=1(9-(-4))=13 \Rightarrow 3A=1-13=-12 \Rightarrow A=-4$$
$$,(xy+yz+zx)^2=x^2y^2+y^2z^2+z^2x^2+2(xy^2z+yz^2x+zx^2y)$$
$$=x^2y^2+y^2z^2+z^2x^2+2xyz(x+y+z)$$
$$ \Rightarrow (-4)^2=C+2(-4)(1) \Rightarrow C=16+8=24$$
$$,(x^2+y^2+z^2)^2=x^4+y^4+z^4+2(x^2y^2+y^2z^2+z^2x^2)$$
$$ \Rightarrow 9^2=D+2(24)$$
$$ \Rightarrow D=81-48=33$$
$$ \Rightarrow \frac{4}{x^4+y^4+z^4}=\frac{4}{D}=\frac{4}{33}$$
$\Box$
