Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 12A Problems/Problem 5"

(→‎Solution 2 (Stars and Bars))
Line 5: Line 5:
 
<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)==
+
==Solution 1==
 +
Since the second point is the origin, this is equivalent to finding all points <math>(x, y)</math> such that <math>|x| + |y| \leq 20</math>. Due to the absolute values, the set of all such points will be symmetric about the origin meaning we can focus on the first quadrant and multiply by <math>4</math>.
 +
 
 +
To avoid overcounts, ignore points on the axes. This means <math>x, y > 0</math>. If <math>x = 1</math>, there are <math>19</math> solutions for <math>y</math> (<math>y = 1, 2, 3, ..., 19</math>). If <math>x = 2</math>, there are <math>18</math> solutions. This pattern repeats until <math>x = 19</math>, at which point there is <math>1</math> solution for <math>y</math>.
 +
 
 +
So we get <math>19 + 18 + 17 + ... + 1 = \frac{19(20)}{2} = 190</math> points in the first quadrant. Multiplying by <math>4</math> gives <math>760</math>. Now, the <math>x</math> axis has <math>y = 0</math> which gives <math>|x| \leq 20</math>, meaning there are <math>41</math> solutions. This is the same with the <math>y</math> axis, but we overcounted the origin by <math>1</math>.
 +
 
 +
Our final answer is <math>760 + 41 + 41 - 1 = \boxed{\textbf{(C)} \, 841}</math>.
 +
 
 +
==Solution 2 (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.  
 
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>. Also, the area of the square is <math>800</math>, so when we plug this information inside Pick's theorem, we get <math>800 = i + \frac{80}{2} - 1 \implies i = 761</math>. Then our answer is <math>761+80 = \boxed{\textbf{(C)} \, 841}.</math>
 
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>. Also, the area of the square is <math>800</math>, so when we plug this information inside Pick's theorem, we get <math>800 = i + \frac{80}{2} - 1 \implies i = 761</math>. Then our answer is <math>761+80 = \boxed{\textbf{(C)} \, 841}.</math>
Line 11: Line 20:
 
~ Oxymoronic15
 
~ Oxymoronic15
  
−
==Solution 2 (Stars and Bars)==
+
==Solution 3 (Stars and Bars)==
 
Instead of considering all points with integer coordinates, first consider points with nonnegative coordinates only. Then, we want <math>x + y \le 20</math> where <math>x</math> and <math>y</math> are nonnegative integers. We can introduce a third variable, say <math>z</math>, such that <math>z = 20 - (x + y)</math>. Note that counting the number of ways to have <math>x + y + z = 20</math> is the same as counting the number of ways to have <math>x + y \le 20</math>. Therefore, by stars and bars, there are <math>\dbinom{20 + 3 - 1}{3 - 1} = 231</math> solutions with nonnegative integer coordinates.
 
Instead of considering all points with integer coordinates, first consider points with nonnegative coordinates only. Then, we want <math>x + y \le 20</math> where <math>x</math> and <math>y</math> are nonnegative integers. We can introduce a third variable, say <math>z</math>, such that <math>z = 20 - (x + y)</math>. Note that counting the number of ways to have <math>x + y + z = 20</math> is the same as counting the number of ways to have <math>x + y \le 20</math>. Therefore, by stars and bars, there are <math>\dbinom{20 + 3 - 1}{3 - 1} = 231</math> solutions with nonnegative integer coordinates.
  

Revision as of 21:23, 17 November 2022

Problem

The $\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by $|x_1 - x_2| + |y_1 - y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?

$\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\textbf{(C)} \, 841 \qquad\textbf{(D)} \, 921 \qquad\textbf{(E)} \, 924$

Solution 1

Since the second point is the origin, this is equivalent to finding all points $(x, y)$ such that $|x| + |y| \leq 20$. Due to the absolute values, the set of all such points will be symmetric about the origin meaning we can focus on the first quadrant and multiply by $4$.

To avoid overcounts, ignore points on the axes. This means $x, y > 0$. If $x = 1$, there are $19$ solutions for $y$ ($y = 1, 2, 3, ..., 19$). If $x = 2$, there are $18$ solutions. This pattern repeats until $x = 19$, at which point there is $1$ solution for $y$.

So we get $19 + 18 + 17 + ... + 1 = \frac{19(20)}{2} = 190$ points in the first quadrant. Multiplying by $4$ gives $760$. Now, the $x$ axis has $y = 0$ which gives $|x| \leq 20$, meaning there are $41$ solutions. This is the same with the $y$ axis, but we overcounted the origin by $1$.

Our final answer is $760 + 41 + 41 - 1 = \boxed{\textbf{(C)} \, 841}$.

Solution 2 (Pick's Theorem)

Let $P = (x, y)$. Since the problem asks for taxicab distances from the origin, we want $|x| + |y| \le 20$. The graph of all solutions to this equation on the $xy$-plane is a square with vertices at $(0, \pm 20)$ and $(\pm 20, 0)$ (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 $(0,20)$ to $(20, 0)$, we can see that it goes through $21$ points in total. In addition, each of the vertices gets counted twice, so the total number of border points is $21\cdot4 - 4 = 80$. Also, the area of the square is $800$, so when we plug this information inside Pick's theorem, we get $800 = i + \frac{80}{2} - 1 \implies i = 761$. Then our answer is $761+80 = \boxed{\textbf{(C)} \, 841}.$

~ Oxymoronic15

Solution 3 (Stars and Bars)

Instead of considering all points with integer coordinates, first consider points with nonnegative coordinates only. Then, we want $x + y \le 20$ where $x$ and $y$ are nonnegative integers. We can introduce a third variable, say $z$, such that $z = 20 - (x + y)$. Note that counting the number of ways to have $x + y + z = 20$ is the same as counting the number of ways to have $x + y \le 20$. Therefore, by stars and bars, there are $\dbinom{20 + 3 - 1}{3 - 1} = 231$ solutions with nonnegative integer coordinates.

Then, we can copy our solutions over to the other four quadrants. First, so as not to overcount, we remove all points on the axes. There are $20 + 20 + 1 = 41$ such points with nonnegative integer coordinates. We multiply the $190$ remaining points by $4$ to get $760$ points that are not on the axes. Then, we can add back the $41$ nonnegative points on the axes, as well as the $40$ other points on the negative axes to get $760 + 41 + 40 = \boxed{\textbf{(C)} \, 841}.$

~ jamesl123456

See Also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_12A_Problems/Problem_5&oldid=182097"