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> lattice 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> lattice points here. | ||
| − | 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> | + | 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> | <math>-4 + x\le y \le 4 - x</math> | ||
Revision as of 12:31, 26 November 2017
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 desirable range because 
is the shortest path from 
to , and 
is 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.
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. 
