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Germany Team Selection Test
2003 Germany Team Selection Test
2
MA,MB,MC intersect the lines BC,CA,AB
MA,MB,MC intersect the lines BC,CA,AB
Source: VAIMO 2, German Pre-TST 2003
July 17, 2011
Problem Statement
Given a triangle
A
B
C
ABC
A
BC
and a point
M
M
M
such that the lines
M
A
,
M
B
,
M
C
MA,MB,MC
M
A
,
MB
,
MC
intersect the lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
in this order in points
D
,
E
D,E
D
,
E
and
F
,
F,
F
,
respectively. Prove that there are numbers
ϵ
1
,
ϵ
2
,
ϵ
3
∈
{
−
1
,
1
}
\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}
ϵ
1
,
ϵ
2
,
ϵ
3
∈
{
−
1
,
1
}
such that:
ϵ
1
⋅
M
D
A
D
+
ϵ
2
⋅
M
E
B
E
+
ϵ
3
⋅
M
F
C
F
=
1.
\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.
ϵ
1
⋅
A
D
M
D
+
ϵ
2
⋅
BE
ME
+
ϵ
3
⋅
CF
MF
=
1.
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