Difference between revisions of "2011 AMC 12B Problems/Problem 23"
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If <math>-3\le x \le 3</math>, then <math>-7\le y \le 7</math> satisfy the property. there are <math>15 \times 7 = 105</math> lattices points here. | If <math>-3\le x \le 3</math>, then <math>-7\le y \le 7</math> satisfy the property. there are <math>15 \times 7 = 105</math> lattices points here. | ||
− | else let <math>3< x \le 8</math> (and for <math>-8 \le x < 3</math> it is symmetrical | + | else let <math>3< x \le 8</math> (and for <math>-8 \le x < 3</math> it is symmetrical, <math>-7 + (x - 3)\le y \le 7 - (x - 3)</math>, |
− | < | + | <math>-4 + x\le y \le 4 - x</math> |
− | So for < | + | So for <math>x = 4</math>, there are <math>13</math> lattices points, |
− | for < | + | for <math>x = 5</math>, there are <math>11</math> lattices points, |
etc | etc | ||
− | for < | + | for <math>x = 8</math>, there are <math>5</math> lattices points. |
<br /> | <br /> | ||
− | Hence, there are a total of < | + | Hence, there are a total of <math>105 + 2 ( 13 + 11 + 9 + 7 + 5) = 195</math> lattices points. |
== 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}} |
Revision as of 16:49, 11 March 2011
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, ,
So for , there are lattices points,
for , there are lattices points,
etc
for , there are lattices points.
Hence, there are a total of lattices points.
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 |