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The points in the square that are closest to one particular edge are the ones contained in one of the triangles formed by the two diagonals of the square, so to calculate the probability in question we only need to focus on one of those triangles. Also, besides the two diagonals, the boundary of the region we want to consider is the locus of points that are equidistant from the center of the square and the edge of the square; those points lie on a parabola whose focus is the center of the square and whose directrix is the square's edge.

Recall that for the parabola $y=ax^2$ , the vertex is at $(0,0)$ , the focus at $(0,\frac{1}{4a})$ and the directrix is the line $y=\frac{-1}{4a}$ . Thus if we place the center of the square at $(0,1)$ and the lower edge of the square on $y=-1$ , the equation of the boundary parabola will be $y=\frac{1}{4}x^2$ .

With this in mind, consider the graph below:

The points in the lower quarter of the square that are closer to the center than the edge are in the shaded region; we can calculate the probability we need by dividing the area of this shaded region by the area of the lower triangle.

We start by finding the x-coordinate $a$ of the intersection of the boundary parabola and the lower right diagonal, which lies along the line $y=1-x$ .

$\begin{aligned} &\dfrac{1}{4}a^2 = 1 - a \\ \\ &a^2 + 4a - 4 = 0 \\ \\ &a = \dfrac{-4 \pm \sqrt{4^2 - 4(1)(-4)}}{2(1)} \qquad \small \text{[We want the positive root here.]}\\ \\ &a = 2\left( \sqrt{2} - 1 \right) \end{aligned}$

Then the area of the shaded region is

$\begin{aligned} &2 \cdot \displaystyle\int_{0}^{a} (1 - x) - \left( \dfrac{1}{4} x^2 \right) dx \\ \\ =\ &2 \cdot \left. \left( x - \dfrac{x^2}{2} - \dfrac{x^3}{12} \right) \right|_0^a \\ \\ =\ &2 \left[ 2\left( \sqrt{2} - 1 \right) - 2\left( \sqrt{2} - 1 \right)^2 - \dfrac{2\left( \sqrt{2} - 1 \right)^3}{3} \right] \\ \\ =\ &\dfrac{16 \sqrt{2} - 20}{3} \end{aligned}$

Finally, to find the probability we're seeking, we divide the above value by $4$ , the area of the lower triangle, giving us $\dfrac{4 \sqrt{2} - 5}{3}$ , and our answer is $4 + 2 + 3 = \boxed{9}$

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