>>515
 a^(10/3) = A,b^(10/3) = B,c^(10/3) = C とおくと本題は
 (A^3+B^3+C^3)^2 ≧ 3(A^4+B^4+C^4)

〔補題〕
 (A^3+B^3+C^3)^2 ≧ (AB+BC+CA)(A^4+B^4+C^4)

m = min{A,B,C} とおき、
{A,B,C} = {m,m+x,m+y} (x≧0,y≧0)
とする。
(左辺) - (右辺) = (A^3+B^3+C^3)^2 - (AB+BC+CA)(A^4+B^4+C^4) 
 = (m^4)(xx-xy+yy) + (2m^3)xy(x+y) + (2m^2){2xx(x-y)^2 +5xxyy +2yy(x-y)^2} + m(x+y){xx(2x-2.5y)^2 +(7/2)xxyy +yy(2.5x-2y)^2} + (x-y)(x^5-y^5) + 2(xy)^3
 ≧ 0,