Difference between revisions of "2011 AMC 12B Problems/Problem 23"
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.) |
Advancedjus (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||
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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 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> | 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> | ||
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− | Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = \boxed{195}</math> lattice points. <math>\square</math> | + | Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = \boxed{(C) 195}</math> lattice points. <math>\square</math> |
− | One may also obtain the result by using | + | One may also obtain the result by using Pick's Theorem. |
== 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}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:16, 19 January 2020
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
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 |
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