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National and Regional Contests
Korea Contests
Korea National Olympiad
2020 Korea National Olympiad
6
6
Part of
2020 Korea National Olympiad
Problems
(1)
Cute Geo
Source: Korea National Olympiad 2020 P6
11/25/2020
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that quadrilateral
A
B
D
E
ABDE
A
B
D
E
is a parallelogram and quadrilateral
B
C
D
E
BCDE
BC
D
E
is inscribed in a circle. The circle with center
C
C
C
and radius
C
D
CD
C
D
intersects the line
B
D
,
D
E
BD, DE
B
D
,
D
E
at points
F
,
G
(
≠
D
)
F, G(\neq D)
F
,
G
(
=
D
)
, and points
A
,
F
,
G
A, F, G
A
,
F
,
G
is on line l. Let
H
H
H
be the intersection point of line
l
l
l
and segment
B
C
BC
BC
. Consider the set of circle
Ω
\Omega
Ω
satisfying the following condition.Circle
Ω
\Omega
Ω
passes through
A
,
H
A, H
A
,
H
and intersects the sides
A
B
,
A
E
AB, AE
A
B
,
A
E
at point other than
A
A
A
.Let
P
,
Q
(
≠
A
)
P, Q(\neq A)
P
,
Q
(
=
A
)
be the intersection point of circle
Ω
\Omega
Ω
and sides
A
B
,
A
E
AB, AE
A
B
,
A
E
. Prove that
A
P
+
A
Q
AP+AQ
A
P
+
A
Q
is constant.
geometry
Korea
pentagon