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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2002 Abels Math Contest (Norwegian MO)
2002 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
4
1
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winning strategy for integer game, 2 players
An integer is given
N
>
1
N> 1
N
>
1
. Arne and Britt play the following game: (1) Arne says a positive integer
A
A
A
. (2) Britt says an integer
B
>
1
B> 1
B
>
1
that is either a divisor of
A
A
A
or a multiple of
A
A
A
. (
A
A
A
itself is a possibility.) (3) Arne says a new number
A
A
A
that is either
B
−
1
,
B
B - 1, B
B
−
1
,
B
or
B
+
1
B + 1
B
+
1
. The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number
N
N
N
before the
50
50
50
th has been said. Otherwise, Arne wins. a) Show that Arne has a winning strategy if
N
=
10
N = 10
N
=
10
. b) Show that Britt has a winning strategy if
N
=
24
N = 24
N
=
24
. c) For which
N
N
N
does Britt have a winning strategy?
3b
1
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can there exist a sphere tangent to all six lines , tetrahedron
Six line segments of lengths
17
,
18
,
19
,
20
,
21
17, 18, 19, 20, 21
17
,
18
,
19
,
20
,
21
and
23
23
23
form the side edges of a triangular pyramid (also called a tetrahedron). Can there exist a sphere tangent to all six lines?
3a
1
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circle having the line segment KL as diameter passes through O
A circle with center in
O
O
O
is given. Two parallel tangents tangent to the circle at points
M
M
M
and
N
N
N
. Another tangent intersects the first two tangents at points
K
K
K
and
L
L
L
. Show that the circle having the line segment
K
L
KL
K
L
as diameter passes through
O
O
O
.
2c
1
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a^3- 3ab^2 = 8, b^3 - 3a^2b = 11, a^2+b^2 =?
If
a
a
a
and
b
b
b
are real numbers such that
{
a
3
−
3
a
b
2
=
8
b
3
−
3
a
2
b
=
11
\begin{cases} a^3-3ab^2 = 8 \\ b^3-3a^2b = 11 \end{cases}
{
a
3
−
3
a
b
2
=
8
b
3
−
3
a
2
b
=
11
then what is
a
2
+
b
2
a^2+b^2
a
2
+
b
2
?
2ab
1
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x + 1 / x>= 2 if x>0 , sum x_i <= sum x_i/yi when sum x_i>=sum y_i and x_i,y_i>0
a) Let
x
x
x
be a positive real number. Show that
x
+
1
/
x
≥
2
x + 1 / x\ge 2
x
+
1/
x
≥
2
. b) Let
n
≥
2
n\ge 2
n
≥
2
be a positive integer and let
x
1
,
y
1
,
x
2
,
y
2
,
.
.
.
,
x
n
,
y
n
x _1,y_1,x_2,y_2,...,x_n,y_n
x
1
,
y
1
,
x
2
,
y
2
,
...
,
x
n
,
y
n
be positive real numbers such that
x
1
+
x
2
+
.
.
.
+
x
n
≥
x
1
y
1
+
x
2
y
2
+
.
.
.
+
x
n
y
n
x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n
x
1
+
x
2
+
...
+
x
n
≥
x
1
y
1
+
x
2
y
2
+
...
+
x
n
y
n
.Show that
x
1
+
x
2
+
.
.
.
+
x
n
≤
x
1
y
1
+
x
2
y
2
+
.
.
.
+
x
n
y
n
x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}
x
1
+
x
2
+
...
+
x
n
≤
y
1
x
1
+
y
2
x
2
+
...
+
y
n
x
n
1b
1
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(2a+b) (2b+a) =5^c diophantine
Find all integers
c
c
c
such that the equation
(
2
a
+
b
)
(
2
b
+
a
)
=
5
c
(2a+b) (2b+a) =5^c
(
2
a
+
b
)
(
2
b
+
a
)
=
5
c
has integer solutions.
1a
1
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k + 1 and 16k + 1 are perfect square
Find all integers
k
k
k
such that both
k
+
1
k + 1
k
+
1
and
16
k
+
1
16k + 1
16
k
+
1
are perfect squares.