Problem: Let $a$, $b$, $c$ be positive reals.
Prove that
\[ ( a^5-a^2+3 )( b^5-b^2+3 )( c^5-c^2+3 )
  \ge \left( a+b+c \right)^3. \]

Solution: Observe that for all real numbers $a$, the inequality
\[ a^5-a^2+3 \ge a^3+2 \]
holds.
Then the problem follows by H\"{o}lder in the form
\[ (a^3+1+1)(1+b^3+1)(1+1+c^3) \ge (a+b+c)^3. \]