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China National Olympiad
2021 China National Olympiad
4
4
Part of
2021 China National Olympiad
Problems
(1)
Sum of angles are equal
Source: China Mathematical Olympiad 2021 P4
11/25/2020
In acute triangle
A
B
C
(
A
B
>
A
C
)
ABC (AB>AC)
A
BC
(
A
B
>
A
C
)
,
M
M
M
is the midpoint of minor arc
B
C
BC
BC
,
O
O
O
is the circumcenter of
(
A
B
C
)
(ABC)
(
A
BC
)
and
A
K
AK
A
K
is its diameter. The line parallel to
A
M
AM
A
M
through
O
O
O
meets segment
A
B
AB
A
B
at
D
D
D
, and
C
A
CA
C
A
extended at
E
E
E
. Lines
B
M
BM
BM
and
C
K
CK
C
K
meet at
P
P
P
, lines
B
K
BK
B
K
and
C
M
CM
CM
meet at
Q
Q
Q
. Prove that
∠
O
P
B
+
∠
O
E
B
=
∠
O
Q
C
+
∠
O
D
C
\angle OPB+\angle OEB =\angle OQC+\angle ODC
∠
OPB
+
∠
OEB
=
∠
OQC
+
∠
O
D
C
.
geometry
China MO
2021
P4