2010 AIME I Problems/Problem 8
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
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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 |