Similar to AMC 10A #25

Calculus Level 5

Let $$S$$ be a square of side length $$1$$. Two points $$A$$ and $$B$$ are chosen independantly at random such that $$A$$ is on the perimeter while $$B$$ is strictly inside the square. The probability that the straight-line distance between $$A$$ and $$B$$ is at least $$\frac{1}{2}$$ is $$\frac{a-b\pi}{c}$$, where $$a$$, $$b$$, and $$c$$ are positive integers and $$\gcd (a,b,c)=1$$. What is $$a+b+c$$?

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