Question 23
Let $S$ be a square of side length $1.$ Two points are chosen independently at random on the sides of $S.$ The probability that the straight-line distance between the points is at least $\dfrac12$ is $\dfrac{a - b\pi}{c},$ where $a,$ $b,$ and $c$ are positive integers and $\gcd(a, b, c) = 1.$ What is $a + b + c?$