Problem:
A point is chosen at random in the interior of a unit square . Let denote the distance from to the closest side of . The probability that is equal to , where and are relatively prime positive integers. Find .
Solution:
Note that because the square has area , the requested probability is equal to the area of the region determined by the given conditions. For , let denote the square concentric with which has side length . Every point inside except its center lies on the boundary of for exactly one , and for such a point, the distance is . The given inequality is satisfied if is inside but outside . This occurs with probability
and the requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions