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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 16th XMO
2
2
Part of
the 16th XMO
Problems
(1)
Geometry inXMO need calculating
Source: 16th XMO
6/22/2024
In a triangle
A
B
C
ABC
A
BC
, let
O
O
O
be the circumcenter ,
A
O
AO
A
O
meet
B
C
BC
BC
at
K
K
K
, A circle
Ω
\Omega
Ω
with the centre
T
T
T
and the center
K
K
K
and the radius
A
K
AK
A
K
meet
A
C
AC
A
C
again at
T
T
T
,
D
D
D
is a point on the plain satisfies that
B
C
BC
BC
is the bisector of the angle
∠
A
B
D
\angle ABD
∠
A
B
D
, let the orthocenter of the triangle
A
B
C
ABC
A
BC
and
B
C
D
BCD
BC
D
be
M
M
M
and
N
N
N
. If
M
N
/
/
A
C
MN//AC
MN
//
A
C
than
D
T
DT
D
T
is tangent to
Ω
\Omega
Ω
geometry
circumcircle