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Balkan MO Shortlist
2009 Balkan MO Shortlist
A2
A2
Part of
2009 Balkan MO Shortlist
Problems
(1)
Prove the existence of the configuration and ...
Source: Balkan MO ShortList 2009 A2
4/6/2020
Let
A
B
C
D
ABCD
A
BC
D
be a square and points
M
M
M
∈
\in
∈
B
C
BC
BC
,
N
∈
C
D
N \in CD
N
∈
C
D
,
P
P
P
∈
\in
∈
D
A
DA
D
A
, such that
∠
B
A
M
\angle BAM
∠
B
A
M
=
=
=
x
x
x
,
∠
C
M
N
\angle CMN
∠
CMN
=
=
=
2
x
2x
2
x
,
∠
D
N
P
\angle DNP
∠
D
NP
=
=
=
3
x
3x
3
x
[*] Show that, for any
x
∈
(
0
,
π
8
)
x \in (0, \tfrac{\pi}{8} )
x
∈
(
0
,
8
π
)
, such a configuration exists [*] Determine the number of angles
x
∈
(
0
,
π
8
)
x \in ( 0, \tfrac{\pi}{8} )
x
∈
(
0
,
8
π
)
for which
∠
A
P
B
=
4
x
\angle APB =4x
∠
A
PB
=
4
x