Difference between revisions of "2011 AMC 12B Problems/Problem 23"

m (Changed "|x-3| + |y-2| is the shortest path" to "|x-3| + |y-2| is the *length of the* shortest path" and similar edits for clarity.)
(Solution)
Line 6: Line 6:
  
 
==Solution==
 
==Solution==
Answer: (C)
 
 
 
If a point <math>(x, y)</math> satisfies the property that <math>|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20</math>, then it is in the desirable range because <math>|x - 3| + |y + 2|</math> is the length of the shortest path from <math>(x,y)</math> to <math>B</math>, and <math>|x + 3| + |y - 2|</math> is the length of the shortest path from <math>(x,y)</math> to <math>A</math>
 
If a point <math>(x, y)</math> satisfies the property that <math>|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20</math>, then it is in the desirable range because <math>|x - 3| + |y + 2|</math> is the length of the shortest path from <math>(x,y)</math> to <math>B</math>, and <math>|x + 3| + |y - 2|</math> is the length of the shortest path from <math>(x,y)</math> to <math>A</math>
  
Line 26: Line 24:
 
<br />
 
<br />
  
Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = \boxed{195}</math> lattice points. <math>\square</math>
+
Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = \boxed{(C) 195}</math> lattice points. <math>\square</math>
  
One may also obtain the result by using pick's theorem.
+
One may also obtain the result by using Pick's Theorem.
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=22|num-a=24|ab=B}}
 
{{AMC12 box|year=2011|num-b=22|num-a=24|ab=B}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:16, 19 January 2020

Problem

A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?

$\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\  195 \qquad \textbf{(D)}\  227 \qquad \textbf{(E)}\  255$

Solution

If a point $(x, y)$ satisfies the property that $|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20$, then it is in the desirable range because $|x - 3| + |y + 2|$ is the length of the shortest path from $(x,y)$ to $B$, and $|x + 3| + |y - 2|$ is the length of the shortest path from $(x,y)$ to $A$


If $-3\le x \le 3$, then $-7\le y \le 7$ satisfy the property. there are $15 \times 7 = 105$ lattice points here.

else let $3< x \le 8$ (and for $-8 \le x < -3$ it is symmetrical) $-7 + (x - 3)\le y \le 7 - (x - 3)$,

$-10 + x\le y \le 10 - x$

So for $x = 4$, there are $13$ lattice points,

for $x = 5$, there are $11$ lattice points,

etc.

For $x = 8$, there are $5$ lattice points.

Hence, there are a total of $105 + 2 ( 13 + 11 + 9 + 7 + 5) = \boxed{(C) 195}$ lattice points. $\square$

One may also obtain the result by using Pick's Theorem.

See also

2011 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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