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 satisfies the property that , then it is in the desirable range because is the length of the shortest path from to , and is the length of the shortest path from to
If , then satisfy the property. there are lattice points here.
else let (and for it is symmetrical) ,
So for , there are lattice points,
for , there are lattice points,
etc.
For , there are lattice points.
Hence, there are a total of lattice points.
One may also obtain the result by using pick's theorem.
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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