Difference between revisions of "2010 AIME I Problems/Problem 8"
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For a real number <math>a</math>, let <math>\lfloor a \rfloor</math> denominate the [[ceiling function|greatest integer]] less than or equal to <math>a</math>. Let <math>\mathcal{R}</math> denote the region in the [[coordinate plane]] consisting of points <math>(x,y)</math> such that <math>\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25</math>. The region <math>\mathcal{R}</math> is completely contained in a [[disk]] of [[radius]] <math>r</math> (a disk is the union of a [[circle]] and its interior). The minimum value of <math>r</math> can be written as <math>\frac {\sqrt {m}}{n}</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by the square of any prime. Find <math>m + n</math>. | For a real number <math>a</math>, let <math>\lfloor a \rfloor</math> denominate the [[ceiling function|greatest integer]] less than or equal to <math>a</math>. Let <math>\mathcal{R}</math> denote the region in the [[coordinate plane]] consisting of points <math>(x,y)</math> such that <math>\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25</math>. The region <math>\mathcal{R}</math> is completely contained in a [[disk]] of [[radius]] <math>r</math> (a disk is the union of a [[circle]] and its interior). The minimum value of <math>r</math> can be written as <math>\frac {\sqrt {m}}{n}</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by the square of any prime. Find <math>m + n</math>. |
Revision as of 14:45, 9 August 2018
Contents
Problem
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 an axis-box, for instance , is The distance from to the furthest point on a quadrant-box, for instance , is The latter is the larger, and is , giving an answer of .
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 |
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