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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1989 Flanders Math Olympiad
1989 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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sort of functional equation
Let
D
D
D
be the set of positive reals different from
1
1
1
and let
n
n
n
be a positive integer. If for
f
:
D
→
R
f: D\rightarrow \mathbb{R}
f
:
D
→
R
we have
x
n
f
(
x
)
=
f
(
x
2
)
x^n f(x)=f(x^2)
x
n
f
(
x
)
=
f
(
x
2
)
, and if
f
(
x
)
=
x
n
f(x)=x^n
f
(
x
)
=
x
n
for
0
<
x
<
1
1989
0<x<\frac{1}{1989}
0
<
x
<
1989
1
and for
x
>
1989
x>1989
x
>
1989
, then prove that
f
(
x
)
=
x
n
f(x)=x^n
f
(
x
)
=
x
n
for all
x
∈
D
x \in D
x
∈
D
.
2
1
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pentagon
When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?
3
1
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Easy Trig
Show that:
α
=
±
π
12
+
k
⋅
π
2
(
k
∈
Z
)
⟺
∣
tan
α
∣
+
∣
cot
α
∣
=
4
\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2 (k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot \alpha}| = 4
α
=
±
12
π
+
k
⋅
2
π
(
k
∈
Z
)
⟺
∣
tan
α
∣
+
∣
cot
α
∣
=
4
1
1
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Easy but beatiful
Show that every subset of {1,2,...,99,100} with 55 elements contains at least 2 numbers with a difference of 9.