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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2001 Flanders Math Olympiad
2001 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
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Weird trig
In a circle we enscribe a regular
2001
2001
2001
-gon and inside it a regular
667
667
667
-gon with shared vertices. Prove that the surface in the
2001
2001
2001
-gon but not in the
667
667
667
-gon is of the form
k
.
s
i
n
3
(
π
2001
)
.
c
o
s
3
(
π
2001
)
k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)
k
.
s
i
n
3
(
2001
π
)
.
co
s
3
(
2001
π
)
with
k
k
k
a positive integer. Find
k
k
k
.
2
1
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Tricky!
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment. Find the "?"
1
1
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divisibility
may be challenge for beginner section, but anyone is able to solve it if you really try. show that for every natural
n
>
1
n > 1
n
>
1
we have:
(
n
−
1
)
2
∣
n
n
−
1
−
1
(n-1)^2|\ n^{n-1}-1
(
n
−
1
)
2
∣
n
n
−
1
−
1
4
1
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cute problem, FMO2001
A student concentrates on solving quadratic equations in
R
\mathbb{R}
R
. He starts with a first quadratic equation
x
2
+
a
x
+
b
=
0
x^2 + ax + b = 0
x
2
+
a
x
+
b
=
0
where
a
a
a
and
b
b
b
are both different from 0. If this first equation has solutions
p
p
p
and
q
q
q
with
p
≤
q
p \leq q
p
≤
q
, he forms a second quadratic equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.