functions - Does there exist $f$ such that $f(n)=Omega(log n)$ and $(f(n))^2=O(f(n))$?

I have to prove/disprove the next 2 statements. I've succeeded with the second, not with the first.

1. There exists $f$ such that $f(n)=\Omega(\log n)$ and $(f(n))^2=O(f(n))$.




2. If $f$ and $g$ are monotonically increasing functions, such that $ f(g(n)) = O(n) $ and $f(n)=\Omega(n)$ then $g(n)=O(n)$.