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
We declare a point 
to make up for the extra steps that the bug has to move. If the 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 
because it is symmetrical) We set 8 as the upper bound for x because the shortest distance from to added to the shortest distance from to is 
. Since the minimum value for the difference between the y-coordinates is at 
, we get 
or 
. Thus, the upper and lower bounds for are 
and 
, respectively.
Now we test each value for x satisfying and double the result because of symmetry.
For 
, the possibles values of y are such that 
for a total of 
lattice points,
for 
, the possibles values of y are such that 
for a total of 
lattice points,
for 
, the possibles values of y are such that 
for a total of 
lattice points,
for 
, the possibles values of y are such that 
for a total of 
lattice points,
for 
, the possibles values of y are such that 
for a total of 
lattice points,
Hence, there are a total of 
lattice points.
One may also obtain the result by using Pick's Theorem(how?).
(Suggestion)
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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