2002 AIME II Problems/Problem 8
Problem
Find the least positive integer for which the equation has no integer solutions for . (The notation means the greatest integer less than or equal to .)
Solution
Note that if , then either , or . Either way, we won't skip any natural numbers.
The smallest such that is . (The inequality simplifies to , which is easy to solve by trial, as the solution is obviously .)
We can now compute:
From the observation above (and the fact that ) we know that all integers between and will be achieved for some values of . Similarly, for we obviously have .
Hence the least positive integer for which the equation has no integer solutions for is .
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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