Find all integer-valued polynomials $$f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x)$$
holds for all positive integer $x$, where $\tau (x)$ is the number of positive factors of $x$

[i]Proposed By - ckliao914[/i]
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   The solutions are pairs of constant functions \( (f, g) = (k, \tau(k)) \) where \( k \in \mathbb{N} \) and \( \tau(k) \) is the number of positive divisors of \( k \).

The final answer is \( \boxed{ (f, g) = (k, \tau(k)) } \) where \( k \in \mathbb{N} \).