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Korea Junior Mathematics Olympiad
2003 Korea Junior Math Olympiad
3
3
Part of
2003 Korea Junior Math Olympiad
Problems
(1)
2003 KJMO P3
Source: 2003 Korea Junior Math Olympiad
6/29/2024
Consider a triangle
A
B
C
ABC
A
BC
, inscribed in
O
O
O
and
∠
A
<
∠
B
\angle A < \angle B
∠
A
<
∠
B
. Some point
P
P
P
outside the circle satisfies
∠
A
=
∠
P
B
A
=
18
0
∘
−
∠
P
C
B
\angle A=\angle PBA =180^{\circ}- \angle PCB
∠
A
=
∠
PB
A
=
18
0
∘
−
∠
PCB
Let
D
D
D
be the intersection of line
P
B
PB
PB
and
O
O
O
(different from
B
B
B
), and
Q
Q
Q
the intersection of the tangent line of
O
O
O
passing through
A
A
A
and line
C
D
CD
C
D
. Show that
C
Q
:
A
B
=
A
Q
2
:
A
D
2
CQ : AB=AQ^2:AD^2
CQ
:
A
B
=
A
Q
2
:
A
D
2
.
geometry
ratio
circle