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Problems
Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
Math Invitational for Girls
2019 MIG
19
19
Part of
2019 MIG
Problems
(1)
2019 Individual #19
Source:
5/1/2022
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of digits of an integer
n
n
n
(For example,
S
(
17
)
=
1
+
7
=
8
S(17) = 1 + 7 = 8
S
(
17
)
=
1
+
7
=
8
). If a positive two digit integer is randomly selected, what is the probability
S
(
S
(
n
)
)
≥
8
S(S(n)) \ge 8
S
(
S
(
n
))
≥
8
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
11
45
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
13
45
<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>\dfrac19\qquad<span class='latex-bold'>(C) </span>\dfrac29\qquad<span class='latex-bold'>(D) </span>\dfrac{11}{45}\qquad<span class='latex-bold'>(E) </span>\dfrac{13}{45}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
9
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
9
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
45
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
45
13
2019 Individual