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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1987 Dutch Mathematical Olympiad
1987 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
3
1
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2 kinds of creatures living in flatland of Pentagonia
There are two kinds of creatures living in the flatland of Pentagonia: the Spires (
S
S
S
) and the Bones (
B
B
B
). They all have the shape of an isosceles triangle: the Spiers have an apical angle of
3
6
o
36^o
3
6
o
and the bones an apical angle of
10
8
o
108^o
10
8
o
. Every year on Great Day of Division (September 11 - the day this Olympiad was held) they divide into pieces: each
S
S
S
into two smaller
S
S
S
's and a
B
B
B
; each
B
B
B
in an
S
S
S
and a
B
B
B
. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one
B
B
B
-being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.
4
1
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max line segment fitting into dodecahedron made by 2 reg. tetrahedra
On each side of a regular tetrahedron with edges of length
1
1
1
one constructs exactly such a tetrahedron. This creates a dodecahedron with
8
8
8
vertices and
18
18
18
edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.
1
1
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diophantine a^2 = 2^b +c^4
Solve into
N
N
N
:
a
2
=
2
b
+
c
4
a^2 = 2^b +c^4
a
2
=
2
b
+
c
4
2
1
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1 <2\sqrt{n} - \sum 1/ \sqrt{k} <2
For
x
>
0
x >0
x
>
0
, prove that
1
2
x
+
1
<
x
+
1
−
x
<
1
2
x
\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}
2
x
+
1
1
<
x
+
1
−
x
<
2
x
1
and for all
n
≥
2
n \ge 2
n
≥
2
prove that
1
<
2
n
−
∑
k
=
1
n
1
k
<
2
1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2
1
<
2
n
−
k
=
1
∑
n
k
1
<
2