Difference between revisions of "2000 AIME II Problems/Problem 2"
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
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Revision as of 19:30, 4 July 2013
Problem
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola 
?
Solution
![\[(x-y)(x+y)=2000^2=2^8 \cdot 5^6\]](http://latex.artofproblemsolving.com/2/b/8/2b87b01628eb920d6fd80ecee4d30daf02966520.png)
Note that 
and 
have the same parities, so both must be even. We first give a factor of 
to both and . We have 
left. Since there are 
factors of , and since both 
and 
can be negative, this gives us 
lattice points.
See also
| 2000 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
