I request everyone to please post the solution to this problem using the A.M. - G.M. Inequality. The problem is as follows -
Problem -
Let $a,b,c$ be positive real numbers smaller than $1$ such that $abc=0.5$ Prove that $$b(c^2+a^2+1)+c(a^2+b^2+1)+a(b^2+c^2+1)-3\geq 2(a^2b^2+b^2c^2+c^2a^2)$$
My progress - I tried using the A.M. - G.M. on all the brackets and then equating the value using the value of $abc$ but I was not able to get the correct answer. Ii request someone to please give me a hint to this problem.