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National and Regional Contests
Romania Contests
Romania Team Selection Test
1997 Romania Team Selection Test
1
Romania 1997
Romania 1997
Source:
April 18, 2011
vector
ratio
geometry proposed
geometry
Problem Statement
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon, and let
P
=
A
B
∩
C
D
P= AB \cap CD
P
=
A
B
∩
C
D
,
Q
=
C
D
∩
E
F
Q = CD \cap EF
Q
=
C
D
∩
EF
,
R
=
E
F
∩
A
B
R = EF \cap AB
R
=
EF
∩
A
B
,
S
=
B
C
∩
D
E
S = BC \cap DE
S
=
BC
∩
D
E
,
T
=
D
E
∩
F
A
T = DE \cap FA
T
=
D
E
∩
F
A
,
U
=
F
A
∩
B
C
U = FA \cap BC
U
=
F
A
∩
BC
. Prove that
P
Q
C
D
=
Q
R
E
F
=
R
P
A
B
\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}
C
D
PQ
=
EF
QR
=
A
B
RP
if and only if
S
T
D
E
=
T
U
F
A
=
U
S
B
C
\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}
D
E
ST
=
F
A
T
U
=
BC
U
S
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