Suppose $$f$$ is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x,y \in N$$ and $$f(1) = {1 \over 5}$$. If $$\sum\limits_{n = 1}^m {{{f(n)} \over {n(n + 1)(n + 2)}} = {1 \over {12}}} $$, then $$m$$ is equal to __________.

For some a, b, c $$\in\mathbb{N}$$, let $$f(x) = ax - 3$$ and $$\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$$. If $${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}$$, then $$(fog)(ac) + (gof)(b)$$ is equal to ____________.

For $$\mathrm{p}, \mathrm{q} \in \mathbf{R}$$, consider the real valued function $$f(x)=(x-\mathrm{p})^{2}-\mathrm{q}, x \in \mathbf{R}$$ and $$\mathrm{q}>0$$. Let $$\mathrm{a}_{1}$$, $$\mathrm{a}_{2^{\prime}}$$ $$\mathrm{a}_{3}$$ and $$\mathrm{a}_{4}$$ be in an arithmetic progression with mean $$\mathrm{p}$$ and positive common difference. If $$\left|f\left(\mathrm{a}_{i}\right)\right|=500$$ for all $$i=1,2,3,4$$, then the absolute difference between the roots of $$f(x)=0$$ is ___________.

The number of functions $$f$$, from the set $$\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$$ to the set $$\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}$$ such that $$f(x) \leq(x-3)^{2}+1$$, for every $$x \in \mathrm{A}$$, is ___________.