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Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1986 Flanders Math Olympiad
1986 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
1
1
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limit of sum of infinite spiral inside a circle divided into twelve equal parts
A circle with radius
R
R
R
is divided into twelve equal parts. The twelve dividing points are connected with the centre of the circle, producing twelve rays. Starting from one of the dividing points a segment is drawn perpendicular to the next ray in the clockwise sense; from the foot of this perpendicular another perpendicular segment is drawn to the next ray, and the process is continued ad infinitum. What is the limit of the sum of these segments (in terms of
R
R
R
)? https://cdn.artofproblemsolving.com/attachments/2/6/83705b54ecc817b7d913468cd8467d7b8d9f8f.png
2
1
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easy ineq
Prove that for integer
n
n
n
we have:
n
!
≤
(
n
+
1
2
)
n
n! \le \left( \frac{n+1}{2} \right)^n
n
!
≤
(
2
n
+
1
)
n
(please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)
3
1
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[solved] - easy sequence
Let
{
a
k
}
k
≥
0
\{a_k\}_{k\geq 0}
{
a
k
}
k
≥
0
be a sequence given by
a
0
=
0
a_0 = 0
a
0
=
0
,
a
k
+
1
=
3
⋅
a
k
+
1
a_{k+1}=3\cdot a_k+1
a
k
+
1
=
3
⋅
a
k
+
1
for
k
∈
N
k\in \mathbb{N}
k
∈
N
. Prove that
11
∣
a
155
11 \mid a_{155}
11
∣
a
155
4
1
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Flanders 2 ('86/4)
Given a cube in which you can put two massive spheres of radius 1. What's the smallest possible value of the side - length of the cube? Prove that your answer is the best possible.