2010 AIME I Problems/Problem 8
Contents
Problem
For a real number , let denominate 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 .
Solution
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 .
Video Solution
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.