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District Olympiad
2001 District Olympiad
4
Romania District Olympiad 2001 - VII Grade
Romania District Olympiad 2001 - VII Grade
Source:
March 12, 2011
ratio
geometry proposed
geometry
Problem Statement
Consider a convex qudrilateral
A
B
C
D
ABCD
A
BC
D
and
M
∈
(
A
B
)
,
N
∈
(
C
D
)
M\in (AB),\ N\in (CD)
M
∈
(
A
B
)
,
N
∈
(
C
D
)
such that
A
M
B
M
=
D
N
C
N
=
k
\frac{AM}{BM}=\frac{DN}{CN}=k
BM
A
M
=
CN
D
N
=
k
. Prove that
B
C
∥
A
D
BC\parallel AD
BC
∥
A
D
if and only if
M
N
=
1
k
+
1
A
D
+
k
k
+
1
B
C
MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC
MN
=
k
+
1
1
A
D
+
k
+
1
k
BC
***
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