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Problems
Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
Math Invitational for Girls
2018 MIG
24
24
Part of
2018 MIG
Problems
(1)
2018 Individual #24
Source:
1/13/2023
The sides of
△
A
B
C
\triangle ABC
△
A
BC
form an arithmetic sequence of integers. Incircle
I
I
I
is tangent to
A
B
AB
A
B
,
B
C
BC
BC
, and
C
A
CA
C
A
at
D
D
D
,
E
E
E
, and
F
F
F
, respectively. Given that
D
B
=
3
2
DB = \tfrac32
D
B
=
2
3
,
F
A
=
1
2
FA = \tfrac12
F
A
=
2
1
, find the radius of
I
I
I
.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
15
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
15
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
15
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
15
4
<span class='latex-bold'>(A) </span> \dfrac12\qquad<span class='latex-bold'>(B) </span> \dfrac{\sqrt{15}}7\qquad<span class='latex-bold'>(C) </span> \dfrac{\sqrt{15}}6\qquad<span class='latex-bold'>(D) </span> \dfrac{2\sqrt{15}}{9}\qquad<span class='latex-bold'>(E) </span> \dfrac{\sqrt{15}}{4}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
9
2
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
15
2018 Individual