Difference between revisions of "2011 AMC 12B Problems/Problem 19"
(Created page with '==Problem== A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math…') |
(→Solution) |
||
Line 9: | Line 9: | ||
It is very easy to see that the <math>+2 </math> in the graph does not impact whether it passes through lattice. | It is very easy to see that the <math>+2 </math> in the graph does not impact whether it passes through lattice. | ||
− | We need to make sure that <math>m</math> cannot be in the form of <math>\frac{a}{b}</math> for <math>1\le b\le 100</math>, otherwise the graph <math>y= mx</math> passes through lattice point at <math>x = b</math>. We only need to worry about <math>\frac{a}{b}</math> very close to <math>\frac{1}{2}</math>, <math>\frac{m+1}{2m+1}</math>, <math>\frac{m+1}{2m}</math> will be the only case we need to worry about and we want the minimum of those, clearly for <math>1\le b\le 100</math>, the smallest is <math>\frac{50}{99} </math>, so answer is (B) | + | We need to make sure that <math>m</math> cannot be in the form of <math>\frac{a}{b}</math> for <math>1\le b\le 100</math>, otherwise the graph <math>y= mx</math> passes through lattice point at <math>x = b</math>. We only need to worry about <math>\frac{a}{b}</math> very close to <math>\frac{1}{2}</math>, <math>\frac{m+1}{2m+1}</math>, <math>\frac{m+1}{2m}</math> will be the only case we need to worry about and we want the minimum of those, clearly for <math>1\le b\le 100</math>, the smallest is <math>\frac{50}{99} </math>, so answer is <math>\boxed{\frac{50}{99}\ \(\textbf{(B)}}</math> |
− | |||
== See also == | == See also == | ||
{{AMC12 box|year=2011|num-b=18|num-a=20|ab=B}} | {{AMC12 box|year=2011|num-b=18|num-a=20|ab=B}} |
Revision as of 02:40, 9 March 2012
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
Answer: (B)
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 $\boxed{\frac{50}{99}\ \(\textbf{(B)}}$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.)
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 |