2016 AIME I Problems/Problem 14
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
.
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
,