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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2023 Korea National Olympiad
4
4
Part of
2023 Korea National Olympiad
Problems
(1)
geometry in P4?
Source: KMO 2023 P4
11/4/2023
Pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in circle
Ω
\Omega
Ω
. Line
A
D
AD
A
D
meets
C
E
CE
CE
at
F
F
F
, and
P
(
≠
E
,
F
)
P (\neq E, F)
P
(
=
E
,
F
)
is a point on segment
E
F
EF
EF
. The circumcircle of triangle
A
F
P
AFP
A
FP
meets
Ω
\Omega
Ω
at
Q
(
≠
A
)
Q(\neq A)
Q
(
=
A
)
and
A
C
AC
A
C
at
R
(
≠
A
)
R(\neq A)
R
(
=
A
)
. Line
A
D
AD
A
D
meets
B
Q
BQ
BQ
at
S
S
S
, and the circumcircle of triangle
D
E
S
DES
D
ES
meets line
B
Q
,
B
D
BQ, BD
BQ
,
B
D
at
T
(
≠
S
)
,
U
(
≠
D
)
T(\neq S), U(\neq D)
T
(
=
S
)
,
U
(
=
D
)
, respectively. Prove that if
F
,
P
,
T
,
S
F, P, T, S
F
,
P
,
T
,
S
are concyclic, then
P
,
T
,
R
,
U
P, T, R, U
P
,
T
,
R
,
U
are concyclic.
geometry
circumcircle