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National and Regional Contests
Korea Contests
Korea National Olympiad
2023 Korea National Olympiad
6
6
Part of
2023 Korea National Olympiad
Problems
(1)
Geometry, XYPQ concyclic
Source: KMO 2023 P6
11/4/2023
Let
Ω
\Omega
Ω
and
O
O
O
be the circumcircle and the circumcenter of an acute triangle
A
B
C
ABC
A
BC
(
A
B
‾
<
A
C
‾
)
(\overline{AB} < \overline{AC})
(
A
B
<
A
C
)
. Define
D
,
E
(
≠
A
)
D,E(\neq A)
D
,
E
(
=
A
)
be the points such that ray
A
O
AO
A
O
intersects
B
C
BC
BC
and
Ω
\Omega
Ω
. Let the line passing through
D
D
D
and perpendicular to
A
B
AB
A
B
intersects
A
C
AC
A
C
at
P
P
P
and define
Q
Q
Q
similarly. Tangents to
Ω
\Omega
Ω
on
A
,
E
A,E
A
,
E
intersects
B
C
BC
BC
at
X
,
Y
X,Y
X
,
Y
. Prove that
X
,
Y
,
P
,
Q
X,Y,P,Q
X
,
Y
,
P
,
Q
lie on a circle.
geometry
circumcircle