Prove that `b^2c^2+c^2a^2+a^2b^2gtabc(a+b+c)`, where `a,b,c gt 0` .

Show that b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2)>abc(a+b+c), where a,b,c are different positivine integers.

prove that b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2)>=abc(a+b+c)

Prove that b^(2)c^(2)+c^(2)a^(2)+a^(2)+b^(2)>abc xx(a+b+c)(a,b,c>0)

Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)gt a+b+c , where a,b,cgt 0 .

Suppose A, B, C are defined as A = a^(2)b + ab^(2) - a^(2)c - ac^(2), B = b^(2)c + bc^(2) - a^(2)b - ab^(2) , and C = a^(2)c + ac^(2) - b^(2)c - bc^(2) , where a gt b gt c gt 0 and the equation Ax^(2) + Bx + C = 0 has equal roots, then a, b, c are in

Prove that 2b^(2)c^(2) +2c^(2)a^(2) +2a^(2)b^(2) -a^(4)-b^(4)-c^(4)= (a+b+c) (b+c-a) (c+a-b) (a+b-c)

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove that if a,b,c gt 0 then a^2(b+c)+b^2(c+a)+c^2(a+b) geq 6abc

In a ABC, prove that: (b^(2)-c^(2))cot A+(c^(2)-a^(2))cot B+(a^(2)-b^(2))cot C=0