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Contests
National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1998 Abels Math Contest (Norwegian MO)
1998 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(4)
1
1
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a_0 = 1 and a_i^2 > a_{i-1}a_{i+1}, prove a_i > i
Let
a
0
,
a
1
,
a
2
,
.
.
.
a_0,a_1,a_2,...
a
0
,
a
1
,
a
2
,
...
be an infinite sequence of positive integers such that
a
0
=
1
a_0 = 1
a
0
=
1
and
a
i
2
>
a
i
−
1
a
i
+
1
a_i^2 > a_{i-1}a_{i+1}
a
i
2
>
a
i
−
1
a
i
+
1
for all
i
>
0
i > 0
i
>
0
. (a) Prove that
a
i
<
a
1
i
a_i < a_1^i
a
i
<
a
1
i
for all
i
>
1
i > 1
i
>
1
. (b) Prove that
a
i
>
i
a_i > i
a
i
>
i
for all
i
i
i
.
4
1
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two collinearities wanted
Let
A
,
B
,
P
A,B,P
A
,
B
,
P
be points on a line
ℓ
\ell
ℓ
, with
P
P
P
outside the segment
A
B
AB
A
B
. Lines
a
a
a
and
b
b
b
pass through
A
A
A
and
B
B
B
and are perpendicular to
ℓ
\ell
ℓ
. A line
m
m
m
through
P
P
P
, which is neither parallel nor perpendicular to
ℓ
\ell
ℓ
, intersects
a
a
a
and
b
b
b
at
Q
Q
Q
and
R
R
R
, respectively. The perpendicular from
B
B
B
to
A
R
AR
A
R
meets
a
a
a
and
A
R
AR
A
R
at
S
S
S
and
U
U
U
, and the perpendicular from
A
A
A
to
B
Q
BQ
BQ
meets
b
b
b
and
B
Q
BQ
BQ
at
T
T
T
and
V
V
V
, respectively. (a) Prove that
P
,
S
,
T
P,S,T
P
,
S
,
T
are collinear. (b) Prove that
P
,
U
,
V
P,U,V
P
,
U
,
V
are collinear.
3
1
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n/ 1^5 +3^5 +5^5 +...+(2n-1)^5, n^2 / 1^3 +3^3 +5^3 +...+(2n-1)^3
Let
n
n
n
be a positive integer. (a) Prove that
1
5
+
3
5
+
5
5
+
.
.
.
+
(
2
n
−
1
)
5
1^5 +3^5 +5^5 +...+(2n-1)^5
1
5
+
3
5
+
5
5
+
...
+
(
2
n
−
1
)
5
is divisible by
n
n
n
. (b) Prove that
1
3
+
3
3
+
5
3
+
.
.
.
+
(
2
n
−
1
)
3
1^3 +3^3 +5^3 +...+(2n-1)^3
1
3
+
3
3
+
5
3
+
...
+
(
2
n
−
1
)
3
is divisible by
n
2
n^2
n
2
.
2
1
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T and L tetraminos in a n x n chessboard
Let be given an
n
×
n
n \times n
n
×
n
chessboard,
n
∈
N
n \in N
n
∈
N
. We wish to tile it using particular tetraminos which can be rotated. For which
n
n
n
is this possible if we use (a)
T
T
T
-tetraminos (b) both kinds of
L
L
L
-tetraminos?