Three distinct numbers are selected randomly from the set $\{1,2,3, \ldots, 40\}$. If the probability, that the selected numbers are in an increasing G.P., is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to __________ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(x) \geq 0, f^{\prime}(x)=f(x), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _e 3\right)$ is equal to __________ .

If the area of the region $\left\{(x, y):\left|4-x^2\right| \leq y \leq x^2, y \leq 4, x \geq 0\right\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right), \alpha, \beta \in \mathbf{N}$, then $\alpha+\beta$ is equal to _________.

The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ________ .