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National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
11.5
11.5
Part of
2023 All-Russian Olympiad Regional Round
Problems
(1)
BY, CX, AH are concurrent
Source: ARO Regional stage 2023 11.5
2/16/2023
Given is a triangle
A
B
C
ABC
A
BC
with altitude
A
H
AH
A
H
and median
A
M
AM
A
M
. The line
O
H
OH
O
H
meets
A
M
AM
A
M
at
D
D
D
. Let
A
B
∩
C
D
=
E
,
A
C
∩
B
D
=
F
AB \cap CD=E, AC \cap BD=F
A
B
∩
C
D
=
E
,
A
C
∩
B
D
=
F
. If
E
H
EH
E
H
and
F
H
FH
F
H
meet
(
A
B
C
)
(ABC)
(
A
BC
)
at
X
,
Y
X, Y
X
,
Y
, prove that
B
Y
,
C
X
,
A
H
BY, CX, AH
B
Y
,
CX
,
A
H
are concurrent.
geometry
Russia