Problem:
If a is a nonzero integer and b is a positive number such that ab2=log10b, what is the median of the set {0,1,a,b,1/b} ?
Answer Choices:
A. 0
B. 1
C. a
D. b
E. b1
Solution:
Because b<10b for all b>0, it follows that log10b<b. If b≥1, then 0<(log10b)/b2<1, so a cannot be an integer. Therefore 0<b<1, so log10b<0 and a=(log10b)/b2<0. Thus a<0<b<1<1/b, and the median of the set is b.
Note that the conditions of the problem can be met with b=0.1 and a=−100.
The problems on this page are the property of the MAA's American Mathematics Competitions