Problem: Let $a$, $b$, $c$ be positive reals. Prove that
\[ \frac{a}{\sqrt{a^2+8bc}} + \frac{b}{\sqrt{b^2+8ca}} + \frac{c}{\sqrt{c^2+8ab}} \ge 1. \]

Solution: By Holder, we have
\[
  \left( \sum_{\text{cyc}} \frac{a}{\sqrt{a^2+8bc}} \right)^2
  \left( \sum_{\text{cyc}} a(a^2+8bc) \right)
  \ge (a+b+c)^3.
\]
So it suffices to show $(a+b+c)^3 \ge a^3+b^3+c^3+24abc$ which is clear by expanding.