2007 AMC 12B Problems/Problem 17
Problem
If is a nonzero integer and is a positive number such that , what is the median of the set ?
Solution
Note that if is positive, then, the equation will have no solutions for . This becomes more obvious by noting that at , . The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.
This puts as the smallest in the set since it must be negative.
Checking the new equation:
Near , but at ,
This implies that the solution occurs somewhere in between:
This also implies that
This makes our set (ordered)
The median is
Cheap Solution that most people probably used
Let . Then giving . Then the ordered set is and the median is so the answer is .
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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