2011 AMC 12B Problems/Problem 23

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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$ (Error compiling LaTeX. ! Missing $ inserted.)x = 5$, there are$11$lattices points,

etc

for$ (Error compiling LaTeX. ! Missing $ inserted.)x = 8$, there are$5$lattices points. <br />

Hence, there are a total of$ (Error compiling LaTeX. ! Missing $ inserted.)105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195$ lattices points.

See also

2011 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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All AMC 12 Problems and Solutions
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