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Problems
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National and Regional Contests
Korea Contests
Korea National Olympiad
2008 Korean National Olympiad
3
3
Part of
2008 Korean National Olympiad
Problems
(1)
Tricky geo problem
Source: 2008 Korea National
6/9/2015
Points
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
lie in a counterclockwise order on a circle
O
O
O
, and
A
C
=
C
E
AC = CE
A
C
=
CE
P
=
B
D
∩
A
C
P=BD \cap AC
P
=
B
D
∩
A
C
,
Q
=
B
D
∩
C
E
Q=BD \cap CE
Q
=
B
D
∩
CE
Let
O
1
O_1
O
1
be the circle which is tangent to
A
P
‾
,
B
P
‾
\overline {AP}, \overline {BP}
A
P
,
BP
and arc
A
B
AB
A
B
(which doesn't contain
C
C
C
) Let
O
2
O_2
O
2
be the circle which is tangent
D
Q
‾
,
E
Q
‾
\overline {DQ}, \overline {EQ}
D
Q
,
EQ
and arc
D
E
DE
D
E
(which doesn't contain
C
C
C
) Let
O
1
∩
O
=
R
,
O
2
∩
O
=
S
,
R
P
∩
Q
S
=
X
O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X
O
1
∩
O
=
R
,
O
2
∩
O
=
S
,
RP
∩
QS
=
X
Prove that
X
C
XC
XC
bisects
∠
A
C
E
\angle ACE
∠
A
CE
geometry