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Greece Contests
Greece Junior Math Olympiad
2005 Greece Junior Math Olympiad
1
1
Part of
2005 Greece Junior Math Olympiad
Problems
(1)
Trapezoid
Source: 22nd Hellenic Jounior mo , 2005 , prb1.
11/17/2005
We are given a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
C
D
=
2
A
B
CD=2AB
C
D
=
2
A
B
and
D
B
⊥
B
C
DB \perp BC
D
B
⊥
BC
. Let
E
E
E
be the intersection of lines
D
A
DA
D
A
and
C
B
CB
CB
, and
M
M
M
be the midpoint of
D
C
DC
D
C
. (a) Prove that
A
B
M
D
ABMD
A
BM
D
is a rhombus. (b) Prove that triangle
C
D
E
CDE
C
D
E
is isosceles. (c) If
A
M
AM
A
M
and
B
D
BD
B
D
meet at
O
O
O
, and
O
E
OE
OE
and
A
B
AB
A
B
meet at
N
,
N,
N
,
prove that the line
D
N
DN
D
N
bisects segment
E
B
EB
EB
.
geometry
trapezoid