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National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2023 Mongolian Mathematical Olympiad
2
2
Part of
2023 Mongolian Mathematical Olympiad
Problems
(1)
Easy geo
Source: Mongolian Mathematical Olympiad P2
1/30/2024
In an acute triangle
A
B
C
ABC
A
BC
the points
D
,
E
D, E
D
,
E
are the feet of the altitudes through
B
,
C
B, C
B
,
C
respectively. Let
L
L
L
be the point on segment
B
D
BD
B
D
such that
A
D
=
D
L
AD=DL
A
D
=
D
L
. Similarly, let
K
K
K
be the point on segment
C
E
CE
CE
such that
A
E
=
E
K
AE=EK
A
E
=
E
K
. Let
M
M
M
be the midpoint of
K
L
KL
K
L
. The circumcircle of
A
B
C
ABC
A
BC
intersect the lines
A
L
AL
A
L
and
A
K
AK
A
K
for a second time at
T
,
S
T, S
T
,
S
respectively. Prove that the lines
B
S
,
C
T
,
A
M
BS, CT, AM
BS
,
CT
,
A
M
are concurrent.
geometry