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Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2009 Abels Math Contest (Norwegian MO) Final
2009 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
3a
1
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6 points concyclic, extending sides of a triangle at both ends
In the triangle
A
B
C
ABC
A
BC
the edge
B
C
BC
BC
has length
a
a
a
, the edge
A
C
AC
A
C
length
b
b
b
, and the edge
A
B
AB
A
B
length
c
c
c
. Extend all the edges at both ends – by the length
a
a
a
from the vertex
A
,
b
A, b
A
,
b
from
B
B
B
, and
c
c
c
from
C
C
C
. Show that the six endpoints of the extended edges all lie on a common circle. https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png
4b
1
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x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010 , when x = 1 - 2^{-2009}
Let
x
=
1
−
2
−
2009
x = 1 - 2^{-2009}
x
=
1
−
2
−
2009
. Show that
x
+
x
2
+
x
4
+
x
8
+
.
.
.
+
x
2
m
<
2010
x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010
x
+
x
2
+
x
4
+
x
8
+
...
+
x
2
m
<
2010
for all positive integers
m
m
m
.
4a
1
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( 2010/2009 )^2009 >2
Show that
(
2010
2009
)
2009
>
2
\left(\frac{2010}{2009}\right)^{2009}> 2
(
2009
2010
)
2009
>
2
.
3b
1
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existence of exactly n grid points (lattice) in a circle
Show for any positive integer
n
n
n
that there exists a circle in the plane such that there are exactly
n
n
n
grid points within the circle. (A grid point is a point having integer coordinates.)
2
1
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16 words in a language of 2 letters
There are two letters in a language. Every word consists of seven letters, and two different words always have different letters on at least three places. a. Show that such a language cannot have more than
16
16
16
words. b. Can there be
16
16
16
words in the language?
1b
1
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sum of 3 consecutive perfect cubes is difference 2 perfect cubes
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
1a
1
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infinite integers not difference between 2 perfect squares
Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.