x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz求满足x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz,的所有正实数解

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x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz求满足x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz,的所有正实数解
x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz
求满足x^3+y^3+z^3=x+y+z
且x^2+y^2+z^2=xyz,的所有正实数解

x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz求满足x^3+y^3+z^3=x+y+z且x^2+y^2+z^2=xyz,的所有正实数解
由基本不等式:
x+y+z=x^3+y^3+z^3≥3xyz=3(x^2+y^2+z^2)≥(x+y+z)^2
得:
x+y+z ≤ 1
因此
x^2+y^2+z^2=xyz≤1/27*(x+y+z)^3≤1/27
所以
0又因为x^3+y^3+z^3=x+y+z
所以
x(x^2-1)+y(y^2-1)+z(z^2-1)=0
因x^2-1,y^2-1,z^2-1均<0,所以左边应<0,矛盾
因此原方程组没有正实数解.

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