MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1985 AMC 12/AHSME
24
24
Part of
1985 AMC 12/AHSME
Problems
(1)
Logarithms
Source:
3/8/2009
A non-zero digit is chosen in such a way that the probability of choosing digit
d
d
d
is \log_{10}(d\plus{}1) \minus{} \log_{10} d. The probability that the digit
2
2
2
is chosen is exactly
1
2
\frac12
2
1
the probability that the digit chosen is in the set
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
{
2
,
3
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
{
3
,
4
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
{
4
,
5
,
6
,
7
,
8
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
{
5
,
6
,
7
,
8
,
9
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
{
4
,
5
,
6
,
7
,
8
,
9
}
<span class='latex-bold'>(A)</span>\ \{2,3\} \qquad <span class='latex-bold'>(B)</span>\ \{3,4\} \qquad <span class='latex-bold'>(C)</span>\ \{4,5,6,7,8\} \qquad <span class='latex-bold'>(D)</span>\ \{5,6,7,8,9\} \qquad <span class='latex-bold'>(E)</span>\ \{4,5,6,7,8,9\}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
{
2
,
3
}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
{
3
,
4
}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
{
4
,
5
,
6
,
7
,
8
}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
{
5
,
6
,
7
,
8
,
9
}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
{
4
,
5
,
6
,
7
,
8
,
9
}
logarithms
probability