2011 AMC 12B Problems/Problem 19
Contents
[hide]Problem
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
Solution
It is very easy to see that the in the graph does not impact whether it passes through lattice.
We need to make sure that cannot be in the form of
for
, otherwise the graph
passes through lattice point at
. We only need to worry about
very close to
,
,
will be the only case we need to worry about and we want the minimum of those, clearly for
, the smallest is
, so answer is
Solution 2
Like in the first solution, we note that the does not affect our answer. Thus, we can ignore it and consider an equivalent problem with the line
.
A line with a slope of passes through (among other lattice points) the lattice point
. As the slope of the line increases from
, the first lattice point it hits is at
, the slope of that line being
. So the answer is
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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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