I came across this problem when reviewing previous AMC tens.
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 $1/2$ is $(a-b\pi)/c$, where a, b, c are positive integers with gcd(a,b,c) = 1. What is $a+b+c$?
I know two solutions for this problem can be found on AoPS. In particular, I was looking at the second solution. I don't get how they get that the region has an area of $1/4$, or how they get the areas for the other regions.
Here is a link to the problem: http://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_25