Konigsberg wrote: Let $a$, $b$, $c$ be real numbers. Prove that $\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)} \ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}$. After squaring of the both sides use $\sqrt{(a^2+b^2)(a^2+c^2)}\geq a^2+bc$.

I did something like squaring both sides but I used $4\sqrt{(a^2+b^2)(a^2+c^2)}\ge(a+b)(a+c)$ (By AM-GM). I think that's also correct

Konigsberg wrote: I used $4\sqrt{(a^2+b^2)(a^2+c^2)}\ge(a+b)(a+c)$ . It should be $4\sqrt{(a^2+b^2)(a^2+c^2)}\ge2(a+b)(a+c)$

OOPS: sorry. Your solution is incorrect, isn't it (or maybe I am overlooking something...)

Konigsberg wrote: Your solution is incorrect, What is my mistake?