Difference between revisions of "2016 AIME I Problems/Problem 14"
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Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>. | Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>. | ||
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+ | == See also == | ||
+ | {{AIME box|year=2016|n=I|num-b=10|num-a=12}} | ||
+ | {{MAA Notice}} | ||
== Solution == | == Solution == |
Revision as of 16:51, 4 March 2016
Problem
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length
whose sides are parallel to the coordinate axes. The line segment from
to
intersects
of the squares and
of the circles. Find
.
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.
Solution
First note that and
so every point of the form
is on the line. Then consider the line
from
to
. Translate the line
so that
is now the origin. There is one square and one circle that intersect the line around
. Then the points on
with an integral
-coordinate are, since
has the equation
:
We claim that the lower right vertex of the square centered at lies on
. Since the square has side length
, the lower right vertex of this square has coordinates
. Because
,
lies on
. Since the circle centered at
is contained inside the square, this circle does not intersect
. Similarly the upper left vertex of the square centered at
is on
. Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between
and
that intersect
. Since there are
segments from
to
, the above count is yields
circles. Since every lattice point on
is of the form
where
, there are
lattice points on
. Centered at each lattice point, there is one square and one circle, hence this counts
squares and circles. Thus
.
Solution by gundraja