2022 AMC 12A Problems/Problem 5
Revision as of 11:07, 12 November 2022 by Oxymoronic15 (talk | contribs)
Problem
The between points
and
in the coordinate plane is given by
. For how many points
with integer coordinates is the taxicab distance between
and the origin less than or equal to
?
Solution 1 (Pick's Theorem)
Let. Since the problem asks for taxicab distances from the origin, we want
. The graph of all solutions to this equation on the
-plane is a square with vertices at
and
(In order to prove this, one can divide the sections of this graph into casework on the four quadrants, and tie together the resulting branches.) We want the number of lattice points on the border of the square and inside the square. Each side of the square goes through an equal number of lattice points, so if we focus on one side going from
to
, we can see that it goes through
points in total. In addition, each of the vertices gets counted twice, so the total number of border points is
.
~ Oxymoronic15