2011 AMC 12B Problems/Problem 19
Contents
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 1
It is very easy to see that the in the graph does not impact whether it passes through the lattice.
We need to make sure that cannot be in the form of for . Otherwise, the graph passes through the 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 (In other words we are trying to find the smallest such that .)
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 |
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All AMC 12 Problems and Solutions |
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