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Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 2nd XMO
the 2nd XMO
Part of
XES Mathematics Olympiad
Subcontests
(1)
1
1
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TS _|_ SE if SA = SP and TB = TD
As shown in the figure,
B
Q
BQ
BQ
is a diameter of the circumcircle of
A
B
C
ABC
A
BC
, and
D
D
D
is the midpoint of arc
B
C
BC
BC
(excluding point
A
A
A
) . The bisector of the exterior angle of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects and the extension of
B
C
BC
BC
at point
E
E
E
. The ray
E
Q
EQ
EQ
intersects
⊙
(
A
B
C
)
\odot (ABC)
⊙
(
A
BC
)
at point
P
P
P
. Point
S
S
S
lies on
P
Q
PQ
PQ
so that
S
A
=
S
P
SA = SP
S
A
=
SP
. Point
T
T
T
lies on
B
C
BC
BC
such that
T
B
=
T
D
TB = TD
TB
=
T
D
. Prove that
T
S
⊥
S
E
TS \perp SE
TS
⊥
SE
. https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png