I have been thinking this problem but have not yet gotten my ideas fixed.
proof that there exists a function F: N - N surjective such that F^(-1) (n) is infinte for each n.
Let $k:\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ be any bijection. So, $k(n) = (n_1,n_2)$. Let $k_1(n) = n_1$ and $k_2(n) = n_2$. Define $F:\mathbb{N} \to \mathbb{N}$ by $F(n) = k_1(n)$.
Edit: Here is an example of such a bijection $k$:
Every natural number can be written uniquely $n = (2a-1)2^{b-1}$ where $a,b\in \mathbb{N}$ (You might need to prove that, but it is not terribly difficult to do). Then define $k:\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ by $k(n) = (a,b)$ where $n = (2a-1)2^{b-1}$.