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CHMMC problems
2021 CHMMC Winter (2021-22)
7
7
Part of
2021 CHMMC Winter (2021-22)
Problems
(1)
2021-22 Winter Team #7
Source:
4/17/2022
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
6
BC = 6
BC
=
6
, and
C
A
=
7
CA = 7
C
A
=
7
. Denote
Γ
\Gamma
Γ
the incircle of
A
B
C
ABC
A
BC
, let
I
I
I
be the center of
Γ
\Gamma
Γ
. The circumcircle of
B
I
C
BIC
B
I
C
intersects
Γ
\Gamma
Γ
at
X
1
X_1
X
1
and
X
2
X_2
X
2
. The circumcircle of
C
I
A
CIA
C
I
A
intersects
Γ
\Gamma
Γ
at
Y
1
Y_1
Y
1
and
Y
2
Y_2
Y
2
. The circumcircle of
A
I
B
AIB
A
I
B
intersects
Γ
\Gamma
Γ
at
Z
1
Z_1
Z
1
and
Z
2
Z_2
Z
2
. The area of the triangle determined by
X
1
X
2
‾
\overline{X_1X_2}
X
1
X
2
,
Y
1
Y
2
‾
\overline{Y_1Y_2}
Y
1
Y
2
, and
Z
1
Z
2
‾
\overline{Z_1Z_2}
Z
1
Z
2
equals
m
p
n
\frac{m \sqrt{p}}{n}
n
m
p
for positive integers
m
,
n
m, n
m
,
n
, and
p
p
p
, where
m
m
m
and
n
n
n
are relatively prime and
p
p
p
is squarefree. Compute
m
+
n
+
p
m+n+ p
m
+
n
+
p
.
geometry