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
. 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