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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1989 Dutch Mathematical Olympiad
1989 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
3
1
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sum1/ \sqrt{n+\sqrt{n^2-1}}}
Calculate
∑
n
=
1
1989
1
n
+
n
2
−
1
\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}
n
=
1
∑
1989
n
+
n
2
−
1
1
5
1
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(1+\sqrt2)^{2k+1}=n(k)+a(k)
∀
k
∈
N
∃
n
(
k
)
∈
N
,
a
(
k
)
:
0
<
a
(
k
)
<
1
[
(
1
+
2
)
2
k
+
1
=
n
(
k
)
+
a
(
k
)
]
\forall k\in N \,\,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]
∀
k
∈
N
∃
n
(
k
)
∈
N
,
a
(
k
)
:
0
<
a
(
k
)
<
1
[(
1
+
2
)
2
k
+
1
=
n
(
k
)
+
a
(
k
)]
Prove:
(
n
(
k
)
+
a
(
k
)
)
a
(
k
)
=
1
(n(k) + a(k))a(k) = 1
(
n
(
k
)
+
a
(
k
))
a
(
k
)
=
1
, for all
k
∈
N
k \in N
k
∈
N
, and calculate
lim
k
→
∞
a
(
k
)
\lim_{k \to \infty }a(k)
lim
k
→
∞
a
(
k
)
2
1
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EF tangent to circle (A,EB), <EAF=45^o, ABCD is square
Given is a square
A
B
C
D
ABCD
A
BC
D
with
E
∈
B
C
E \in BC
E
∈
BC
, arbitrarily. On
C
D
CD
C
D
lies the point
F
F
F
is such that
∠
E
A
F
=
4
5
o
\angle EAF = 45^o
∠
E
A
F
=
4
5
o
. Prove that
E
F
EF
EF
is tangent to the circle with center
A
A
A
and radius
A
B
AB
A
B
.
1
1
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a_n=4a_{n-1}-a_{n-2}
For a sequence of integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
with
0
<
a
1
<
a
2
<
a
3
<
.
.
.
0<a_1<a_2<a_3<...
0
<
a
1
<
a
2
<
a
3
<
...
applies:
a
n
=
4
a
n
−
1
−
a
n
−
2
f
o
r
n
>
2
a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2
a
n
=
4
a
n
−
1
−
a
n
−
2
f
or
n
>
2
It is further given that
a
4
=
194
a_4 = 194
a
4
=
194
. Calculate
a
5
a_5
a
5
.
4
1
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BC_1 + BC_2 + ... + BC_n is fixed sum, regurar n- sides pyramid
Given is a regular
n
n
n
-sided pyramid with top
T
T
T
and base
A
1
A
2
A
3
.
.
.
A
n
A_1A_2A_3... A_n
A
1
A
2
A
3
...
A
n
. The line perpendicular to the ground plane through a point
B
B
B
of the ground plane within
A
1
A
2
A
3
.
.
.
A
n
A_1A_2A_3... A_n
A
1
A
2
A
3
...
A
n
intersects the plane
T
A
1
A
2
TA_1A_2
T
A
1
A
2
at
C
1
C_1
C
1
, the plane
T
A
2
A
3
TA_2A_3
T
A
2
A
3
at
C
2
C_2
C
2
, and so on, and finally the plane
T
A
n
A
1
TA_nA_1
T
A
n
A
1
at
C
n
C_n
C
n
. Prove that
B
C
1
+
B
C
2
+
.
.
.
+
B
C
n
BC_1 + BC_2 + ... + BC_n
B
C
1
+
B
C
2
+
...
+
B
C
n
is independent of choice of
B
B
B
's.