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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1999 Flanders Math Olympiad
1999 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
2
1
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circle chord
Let
[
m
n
]
[mn]
[
mn
]
be a diameter of the circle
C
C
C
and
[
A
B
]
[AB]
[
A
B
]
a chord with given length on this circle.
[
A
B
]
[AB]
[
A
B
]
neither coincides nor is perpendicular to
[
M
N
]
[MN]
[
MN
]
. Let
C
,
D
C,D
C
,
D
be the orthogonal projections of
A
A
A
and
B
B
B
on
[
M
N
]
[MN]
[
MN
]
and
P
P
P
the midpoint of
[
A
B
]
[AB]
[
A
B
]
. Prove that
∠
C
P
D
\angle CPD
∠
CP
D
does not depend on the chord
[
A
B
]
[AB]
[
A
B
]
.
3
1
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functional equation (flanders '99)
Determine all
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
for which
2
⋅
f
(
x
)
−
g
(
x
)
=
f
(
y
)
−
y
and
f
(
x
)
⋅
g
(
x
)
≥
x
+
1.
2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1.
2
⋅
f
(
x
)
−
g
(
x
)
=
f
(
y
)
−
y
and
f
(
x
)
⋅
g
(
x
)
≥
x
+
1.
4
1
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cheap problem - easy if you know mersenne
Let
a
,
b
,
m
,
n
a,b,m,n
a
,
b
,
m
,
n
integers greater than 1. If
a
n
−
1
a^n-1
a
n
−
1
and
b
m
+
1
b^m+1
b
m
+
1
are both primes, give as much info as possible on
a
,
b
,
m
,
n
a,b,m,n
a
,
b
,
m
,
n
.
1
1
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6-digit number
Determine all 6-digit numbers
(
a
b
c
d
e
f
)
(abcdef)
(
ab
c
d
e
f
)
so that
(
a
b
c
d
e
f
)
=
(
d
e
f
)
2
(abcdef) = (def)^2
(
ab
c
d
e
f
)
=
(
d
e
f
)
2
where
(
x
1
x
2
.
.
.
x
n
)
\left( x_1x_2...x_n \right)
(
x
1
x
2
...
x
n
)
is no multiplication but an n-digit number.