Question

设abc为正整数 且a+b=c 求证a^2/3+b^2/3>c^2/3

Answer 1

证明：设a=x^3,b=y^3,c=z^3

则问题转化为已知x^3+y^3=z^3,求证：x^2+y^2>z^2

只要证明(x^2+y^2)^3>z^6即可

(x^2+y^2)^3=x^6+y^6+3x^2y^2(x^2+y^2)

≥x^6+y^6+(3x^2y^2)×2xy

=x^6+y^6+6x^3y^3

=(x^3+y^3)^2+4x^3y^3

=z^6+4x^3y^3

>z^6