MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania National Olympiad
2008 Romania National Olympiad
4
Triangle ABC
Triangle ABC
Source: Romania NMO 2008, 9 form, Problem 4
April 30, 2008
geometry proposed
geometry
Problem Statement
On the sides of triangle
A
B
C
ABC
A
BC
we consider points
C
1
,
C
2
∈
(
A
B
)
,
B
1
,
B
2
∈
(
A
C
)
,
A
1
,
A
2
∈
(
B
C
)
C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)
C
1
,
C
2
∈
(
A
B
)
,
B
1
,
B
2
∈
(
A
C
)
,
A
1
,
A
2
∈
(
BC
)
such that triangles
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
and
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
have a common centroid. Prove that sets
[
A
1
,
B
1
]
∩
[
A
2
B
2
]
,
[
B
1
C
1
]
∩
[
B
2
C
2
]
,
[
C
1
A
1
]
∩
[
C
2
A
2
]
[A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]
[
A
1
,
B
1
]
∩
[
A
2
B
2
]
,
[
B
1
C
1
]
∩
[
B
2
C
2
]
,
[
C
1
A
1
]
∩
[
C
2
A
2
]
are not empty.
Back to Problems
View on AoPS