2010 AIME I Problem 8

For a real number a, let \lfloor a \rfloor denote the greatest integer less than or equal to a. Let \mathcal{R} denote the region in the coordinate plane consisting of points (x,y) such that \lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25. The region \mathcal{R} is completely contained in a disk of radius r (a disk is the union of a circle and its interior). The minimum value of r can be written as \frac {\sqrt {m}}{n}, where m and n are integers and m is not divisible by the square of any prime. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.