2011 AMC 12B Problems/Problem 23
Problem
A bug travels in the coordinate plane, moving only along the lines that are parallel to the -axis or -axis. Let and . Consider all possible paths of the bug from to of length at most . How many points with integer coordinates lie on at least one of these paths?
Solution
Answer: (C)
If a point satisfy the property that , then it is in the desire range because is the shortest path from to , and is the shortest path from to
If , then satisfy the property. there are lattices points here.
else let (and for it is symmetrical-7 + (x - 3)\le y \le 7 - (x - 3)-4 + x\le y \le 4 - xx = 413$lattices points,
for$ (Error compiling LaTeX. ! Missing $ inserted.)x = 511$lattices points,
etc
for$ (Error compiling LaTeX. ! Missing $ inserted.)x = 85$lattices points. <br />
Hence, there are a total of$ (Error compiling LaTeX. ! Missing $ inserted.)105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195$ lattices points.
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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