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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2000 Flanders Math Olympiad
2000 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
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Trig (maybe posted before)
Solve for
x
∈
[
0
,
2
π
[
x \in [0,2\pi[
x
∈
[
0
,
2
π
[
:
sin
x
<
cos
x
<
tan
x
<
cot
x
\sin x < \cos x < \tan x < \cot x
sin
x
<
cos
x
<
tan
x
<
cot
x
3
1
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prime sequence
Let
p
n
p_n
p
n
be the
n
n
n
-th prime. (
p
1
=
2
p_1=2
p
1
=
2
) Define the sequence
(
f
j
)
(f_j)
(
f
j
)
as follows: -
f
1
=
1
,
f
2
=
2
f_1=1, f_2=2
f
1
=
1
,
f
2
=
2
-
∀
j
≥
2
\forall j\ge 2
∀
j
≥
2
: if
f
j
=
k
p
n
f_j = kp_n
f
j
=
k
p
n
for
k
<
p
n
k<p_n
k
<
p
n
then
f
j
+
1
=
(
k
+
1
)
p
n
f_{j+1}=(k+1)p_n
f
j
+
1
=
(
k
+
1
)
p
n
-
∀
j
≥
2
\forall j\ge 2
∀
j
≥
2
: if
f
j
=
p
n
2
f_j = p_n^2
f
j
=
p
n
2
then
f
j
+
1
=
p
n
+
1
f_{j+1}=p_{n+1}
f
j
+
1
=
p
n
+
1
(a) Show that all
f
i
f_i
f
i
are different (b) from which index onwards are all
f
i
f_i
f
i
at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?
1
1
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7-digit integer (flanders '00)
An integer consists of 7 different digits, and is a multiple of each of its digits. What digits are in this nubmer?
2
1
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Flanders 1
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.