Difference between revisions of "2007 AMC 12B Problems/Problem 17"
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If <math>a</math> is a nonzero integer and <math>b</math> is a positive number such that <math>ab^2=\log_{10}b</math>, what is the median of the set <math>\{0,1,a,b,1/b\}</math>? | If <math>a</math> is a nonzero integer and <math>b</math> is a positive number such that <math>ab^2=\log_{10}b</math>, what is the median of the set <math>\{0,1,a,b,1/b\}</math>? | ||
| − | <math>\mathrm {(A)} 0 | + | <math>\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ a \qquad \mathrm{(D)}\ b \qquad \mathrm{(E)}\ 1/b</math> |
==Solution== | ==Solution== | ||
Revision as of 21:19, 18 December 2017
Problem 17
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 
See Also
| 2007 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 16 |
Followed by Problem 18 |
| 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 | |
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