Difference between revisions of "2011 AMC 12B Problems/Problem 23"
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Pandabear10 (talk | contribs) 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.) |
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− | 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 shortest path from <math>(x,y)</math> to <math>B</math>, and <math>|x + 3| + |y - 2|</math> is 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> |
Revision as of 16:02, 27 December 2019
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 |
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.