2010 AIME I Problems/Problem 8
For a real number , let denote the greatest integer less than or equal to . Let denote the region in the coordinate plane consisting of points such that . The region is completely contained in a disk of radius (a disk is the union of a circle and its interior). The minimum value of can be written as , where and are integers and is not divisible by the square of any prime. Find .
The desired region consists of 12 boxes, whose lower-left corners are integers solutions of , namely Since the points themselves are symmetric about , the boxes are symmetric about . The distance from to the furthest point on a box that lays on an axis, for instance , is The distance from to the furthest point on a box in the middle of a quadrant, for instance , is The latter is the larger, and is , giving an answer of .
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