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Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2016 China Girls Math Olympiad
2
2
Part of
2016 China Girls Math Olympiad
Problems
(1)
Sum of length of chords equal another
Source: CGMO 2016 Q2
8/13/2016
In
△
A
B
C
,
B
C
=
a
,
C
A
=
b
,
A
B
=
c
,
\triangle ABC, BC=a, CA=b, AB=c,
△
A
BC
,
BC
=
a
,
C
A
=
b
,
A
B
=
c
,
and
Γ
\Gamma
Γ
is its circumcircle.
(
1
)
(1)
(
1
)
Determine a necessary and sufficient condition on
a
,
b
a,b
a
,
b
and
c
c
c
if there exists a unique point
P
(
P
≠
B
,
P
≠
C
)
P(P\neq B, P\neq C)
P
(
P
=
B
,
P
=
C
)
on the arc
B
C
BC
BC
of
Γ
\Gamma
Γ
not passing through point
A
A
A
such that
P
A
=
P
B
+
P
C
PA=PB+PC
P
A
=
PB
+
PC
.
(
2
)
(2)
(
2
)
Let
P
P
P
be the unique point stated in
(
1
)
(1)
(
1
)
. If
A
P
AP
A
P
bisects
B
C
BC
BC
, prove that
∠
B
A
C
<
6
0
∘
\angle BAC<60^{\circ}
∠
B
A
C
<
6
0
∘
.
geometry
circumcircle