Difference between revisions of "2022 AMC 12A Problems/Problem 5"
Oxymoronic15 (talk | contribs) (Created page with "==Problem== The <math>\textit{taxicab distance}</math> between points <math>(x_1, y_1)</math> and <math>(x_2, y_2)</math> in the coordinate plane is given by <math>|x_1 - x_2...") |
Oxymoronic15 (talk | contribs) |
||
| Line 4: | Line 4: | ||
<math>\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\textbf{(C)} \, 841 \qquad\textbf{(D)} \, 921 \qquad\textbf{(E)} \, 924 </math> | <math>\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\textbf{(C)} \, 841 \qquad\textbf{(D)} \, 921 \qquad\textbf{(E)} \, 924 </math> | ||
| + | |||
| + | ==Solution 1 (Pick's Theorem)== | ||
| + | |||
| + | Let <math>P = (x, y)</math>. Since the problem asks for taxicab distances from the origin, we want <math>|x| + |y| \le 20</math>. The graph of all solutions to this equation on the <math>xy</math>-plane is a square with vertices at <math>(0, \pm 20)</math> and <math>(\pm 20, 0)</math> (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 <math>(0,20)</math> to <math>(20, 0)</math>, we can see that it goes through <math>21</math> points in total. In addition, each of the vertices gets counted twice, so the total number of border points is <math>21\cdot4 - 4 = 80</math>. | ||
| + | |||
| + | ~ Oxymoronic15 | ||
Revision as of 11:07, 12 November 2022
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
.
. 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
