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2020 ASDAN Math Tournament
11
11
Part of
2020 ASDAN Math Tournament
Problems
(1)
2020 Team #11
Source:
10/24/2023
△
A
B
C
\vartriangle ABC
△
A
BC
is right with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
. The internal angle bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at point
D
D
D
, while the external angle bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at point
E
E
E
. Suppose that
A
D
=
1
AD = 1
A
D
=
1
and
B
D
=
2
BD = 2
B
D
=
2
. The value of
D
E
2
DE^2
D
E
2
can be expressed as
x
+
y
z
x+y \sqrt{z}
x
+
y
z
for integers
x
x
x
,
y
y
y
, and
z
z
z
, where
z
z
z
is greater than
1
1
1
and not divisible by the square of any prime. Compute
100
x
+
10
y
+
z
100x + 10y + z
100
x
+
10
y
+
z
. Note: For a generic triangle
△
P
Q
R
\vartriangle PQR
△
PQR
, if we let
Q
′
Q'
Q
′
be the reflection of
Q
Q
Q
over
P
P
P
, then the external angle bisector of
∠
P
\angle P
∠
P
is the line that contains the internal angle bisector of
∠
Q
′
P
R
\angle Q'PR
∠
Q
′
PR
.
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