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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1995 Flanders Math Olympiad
1995 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
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find circle radius
Points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are on a circle with radius
R
R
R
.
∣
A
C
∣
=
∣
A
B
∣
=
500
|AC|=|AB|=500
∣
A
C
∣
=
∣
A
B
∣
=
500
, while the ratio between
∣
D
C
∣
,
∣
D
A
∣
,
∣
D
B
∣
|DC|, |DA|, |DB|
∣
D
C
∣
,
∣
D
A
∣
,
∣
D
B
∣
is
1
,
5
,
7
1,5,7
1
,
5
,
7
. Find
R
R
R
.
1
1
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[solved] - Simplest problem in Flanders
Four couples play chess together. For the match, they're paired as follows: ("man Clara" indicates the husband of Clara, etc.)
B
e
a
⟺
E
d
d
y
Bea \Longleftrightarrow Eddy
B
e
a
⟺
E
dd
y
A
n
⟺
m
a
n
C
l
a
r
a
An \Longleftrightarrow man\ Clara
A
n
⟺
man
Cl
a
r
a
F
r
e
d
d
y
⟺
w
o
m
a
n
G
u
y
Freddy \Longleftrightarrow woman\ Guy
F
re
dd
y
⟺
w
o
man
G
u
y
D
e
b
b
y
⟺
m
a
n
A
n
Debby \Longleftrightarrow man\ An
De
bb
y
⟺
man
A
n
G
u
y
⟺
w
o
m
a
n
E
d
d
y
Guy \Longleftrightarrow woman\ Eddy
G
u
y
⟺
w
o
man
E
dd
y
Who is
H
u
b
e
r
t
Hubert
H
u
b
er
t
married to?
2
1
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easy (flanders '95)
How many values of
x
∈
[
1
,
3
]
x\in\left[ 1,3 \right]
x
∈
[
1
,
3
]
are there, for which
x
2
x^2
x
2
has the same decimal part as
x
x
x
?
4
1
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Flanders 5 ('95)
Given a regular
n
n
n
-gon inscribed in a circle of radius 1, where
n
>
3
n > 3
n
>
3
. Define
G
(
n
)
G(n)
G
(
n
)
as the average length of the diagonals of this
n
n
n
-gon. Prove that if
n
→
∞
,
G
(
n
)
→
4
π
n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}
n
→
∞
,
G
(
n
)
→
π
4
.