Prove that `(a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c`

If a+b+c=1, then prove that (8)/(27abc)>{(1)/(a)-1}{(1)/(b)-1}{(1)/(c)-1}>8

Prove that |{:(1, 1, 1),(a, b, c),(a^(3), b^(3), c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)

If a , b , c are real and positive , prove that the inequality (1)/(a)+(1)/(b)+(1)/( c) lt (a^8+b^8+c^8)/(a^3b^3c^3) .

Prove that a^(3)+b^(3)+c^(3)-3abc=(1)/(2)(a+b+c){(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}

Prove that: a^(3)+b^(3)+c^(3)-3abc=(1)/(2)(a+b+c){a-b)^(2)+(b-c)^(2)+(c-a)^(2)}

If a,b,c are in A.P.,prove that 8b^(3)-a^(3)-c^(3)=3ac(a+c)

Prove that : |{:(1,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

Prove that: (i) |{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)

If a,b,c are in A.P.,prove that: (a-c)^(2)=4(a-b)(b-c)a^(2)+c^(2)+4ac=2(ab+bc+ca)a^(3)+c^(3)+6abc=8b^(3)