2007 AMC 12B Problem 17

Attempt Problem 17 of the 2007 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.

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17.

a

is a nonzero integer and

b

is a positive number such that

ab^2=\log_{10}b,

what is the median of the set

\{0,1,a,b,\frac{1}{b}\}?

0

1

a

b

\dfrac{1}{b}

Since

b\lt10^b

for all

b\gt0,

we have

\log_{10}b\lt b

Rule out

b\ge1

by checking whether

a=\dfrac{\log_{10}b}{b^2}

can be a nonzero integer

For

b\gt0,

we have

b\lt10^b,

hence

\log_{10}b\lt b.

b=1,

then

a=0.

b\gt1,

then

0\lt\dfrac{\log_{10}b}{b^2}\lt1,

a

could not be an integer.

Hence

0\lt b\lt1,

\log_{10}b\lt0

and

a=\dfrac{\log_{10}b}{b^2}\lt0.

Thus

a\lt0\lt b\lt1\lt\dfrac1b,

and the middle value of the sorted set is

b.

Thus, the correct answer is D.

Problem 17 in other years