Question

if a b and c are positive real numbers not all equal,prove that 6abc

Answer 1

We know that for any positive integers x and y (1/2)(x+y)^2 < or = x^2 + y^2.

Now. a+b+c =(1/2)×2×(a+b+c)

=(1/2)(2a+2b+2c) =(1/2){(a+b)+(b+c)+(c+a)}

=(1/2)[{(a+b)^2/(a+b)} +{(b+c)^2/(b+c)} +{(c+a)^2/(c+a)}]

=(1/2)(a+b)^2/(a+b) + (1/2)(b+c)^2/(b+c) +

(1/2)(c+a)^2/(c+a)

< or = (a^2+b^2)/(a+b) +(b^2+c^2)/(c+a) +

(c^2+a^2)/(c+a).