Difference between revisions of "2011 AMC 12B Problems/Problem 23"
Arcticturn (talk | contribs) (Just a suggestion (I don't know how to apply it)) |
Arcticturn (talk | contribs) (Added suggestion) |
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One may also obtain the result by using Pick's Theorem(how?). | One may also obtain the result by using Pick's Theorem(how?). | ||
− | <math>i = a - \frac{b}{2} - 1</math> | + | <math>i = a - \frac{b}{2} - 1</math> (Suggestion) |
== 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 15:37, 1 June 2022
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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