TITLE: Discrete Math Proofs Involving Real Numbers
QUESTION [6 upvotes]: I am stuck on these two problems. 
$1$. Prove that for every three positive real numbers a, b, and c that 
$(a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9$. 
$2$. Prove that for every three positive real numbers a, b, and c that 
$a^2 + b^2 + c^2  \ge ab + bc + ac$.
I have tried direct proof and have not gotten anywhere significant. I won't put the work on there since it is way too long and I don't think it will help. There must be some sort of trick involved, but for the life of me, I cannot figure it out.

REPLY [3 votes]: For $(1)$ $$a+b+c\ge 3(abc)^{\frac13}$$ and $$\frac1a+\frac1b+\frac1c\ge3 \frac1{(abc)^{\frac13}}$$