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

(Created page with '==Problem== A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> …')
 
m (→‎Solution)
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If <math>-3\le x \le 3</math>, then <math>-7\le y \le 7</math> satisfy the property. there are <math>15 \times 7 = 105</math> lattices points here.
 
If <math>-3\le x \le 3</math>, then <math>-7\le y \le 7</math> satisfy the property. there are <math>15 \times 7 = 105</math> lattices points here.
  
−
else let <math>3< x \le 8</math> (and for <math>-8 \le x < 3</math> it is symmetrical<math>, </math>-7 + (x - 3)\le y \le 7 - (x - 3)<math>,
+
else let <math>3< x \le 8</math> (and for <math>-8 \le x < 3</math> it is symmetrical, <math>-7 + (x - 3)\le y \le 7 - (x - 3)</math>,
  
−
</math>-4 + x\le y \le 4 - x<math>
+
<math>-4 + x\le y \le 4 - x</math>
  
−
So for </math>x = 4<math>, there are </math>13<math> lattices points,
+
So for <math>x = 4</math>, there are <math>13</math> lattices points,
  
−
for </math>x = 5<math>, there are </math>11<math> lattices points,
+
for <math>x = 5</math>, there are <math>11</math> lattices points,
  
 
etc
 
etc
  
−
for </math>x = 8<math>, there are </math>5<math> lattices points.
+
for <math>x = 8</math>, there are <math>5</math> lattices points.
 
<br />
 
<br />
  
−
Hence, there are a total of </math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195$ lattices points.
+
Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195</math> lattices points.
  
 
== 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}}

Revision as of 15:49, 11 March 2011

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

Answer: (C)

If a point $(x, y)$ satisfy the property that $|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20$, then it is in the desire range because $|x - 3| + |y + 2|$ is the shortest path from $(x,y)$ to $B$, and $|x + 3| + |y - 2|$ is 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 $15 \times 7 = 105$ lattices points here.

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

$-4 + x\le y \le 4 - x$

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

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

etc

for $x = 8$, there are $5$ lattices points.

Hence, there are a total of $105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195$ lattices points.

See also

2011 AMC 12B (Problems • Answer Key • Resources)
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
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