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

If a+b+c=0 , then prove that a(b+c)^2+b(c+a)^2+c(a+b)^2=3abc

If a/b =b/c and a,b, c gt 0 , then prove that (a+b)^2/(b+c)^2 = (a^2 +b^2)/(b^2 +c^2)

If a,b,c gt 0, then (a )/(2a + b +c ) = (b)/(a + 2b +c)= (c )/(a +b + 2c) = ?

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 .

Prove that |[a,b,c] , [a^2,b^2,c^2] , [a^3,b^3,c^3]|= abc(a-b)(b-c)(c-a)

Prove that b^(2)c^(2) + c^(2)a^(2) + a^(2) b^(2) gt abc (a+b+c)

If a!=6,b,c satisfy |(a,2b,2c),(3,b,c),(4,a,b)|=0 then abc

If a, c gt 0 and ax^2 + 2bx + 3c = 0 does not have any real roots then prove that: (i) 4a-4b + 3c gt 0 , (ii) a + 6b + 27c gt 0 , (iii) a + 2b + 6c gt 0

In Delta ABC prove that a(b^2 + c^2) cosA + b(c^2 +a^2)cosB + c(a^2 + b^2) cosC = 3abc