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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2015 Abels Math Contest (Norwegian MO) Final
2015 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
2b
1
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n red and 1 black marbles, game with initial fortune
Nils is playing a game with a bag originally containing
n
n
n
red and one black marble. He begins with a fortune equal to
1
1
1
. In each move he picks a real number
x
x
x
with
0
≤
x
≤
y
0 \le x \le y
0
≤
x
≤
y
, where his present fortune is
y
y
y
. Then he draws a marble from the bag. If the marble is red, his fortune increases by
x
x
x
, but if it is black, it decreases by
x
x
x
. The game is over after
n
n
n
moves when there is only a single marble left. In each move Nils chooses
x
x
x
so that he ensures a final fortune greater or equal to
Y
Y
Y
. What is the largest possible value of
Y
Y
Y
?
2a
1
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a red knights, b brown, c orange , King Arthur places around table
King Arthur is placing
a
+
b
+
c
a + b + c
a
+
b
+
c
knights around a table.
a
a
a
knights are dressed in red,
b
b
b
knights are dressed in brown, and
c
c
c
knights are dressed in orange. Arthur wishes to arrange the knights so that no knight is seated next to someone dressed in the same colour as himself. Show that this is possible if, and only if, there exists a triangle whose sides have lengths
a
+
1
2
,
b
+
1
2
a +\frac12, b +\frac12
a
+
2
1
,
b
+
2
1
, and
c
+
1
2
c +\frac12
c
+
2
1
1b
1
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functional for starters III x^2f(yf(x))= y^2f(x)f(f(x))
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
x
2
f
(
y
f
(
x
)
)
=
y
2
f
(
x
)
f
(
f
(
x
)
)
x^2f(yf(x))= y^2f(x)f(f(x))
x
2
f
(
y
f
(
x
))
=
y
2
f
(
x
)
f
(
f
(
x
))
for all real numbers
x
x
x
and
y
y
y
.
1a
1
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3x3 system, x^2 + 4y^2 = 4zx , y^2 + 4z^2 = 4xy, z^2 + 4x^2 = 4yz
Find all triples
(
x
,
y
,
z
)
∈
R
3
(x, y, z) \in R^3
(
x
,
y
,
z
)
∈
R
3
satisfying the equations
{
x
2
+
4
y
2
=
4
z
x
y
2
+
4
z
2
=
4
x
y
z
2
+
4
x
2
=
4
y
z
\begin{cases} x^2 + 4y^2 = 4zx \\ y^2 + 4z^2 = 4xy \\ z^2 + 4x^2 = 4yz \end{cases}
⎩
⎨
⎧
x
2
+
4
y
2
=
4
z
x
y
2
+
4
z
2
=
4
x
y
z
2
+
4
x
2
=
4
yz
3
1
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maximum product of distances from sides of a regular pentagon
The five sides of a regular pentagon are extended to lines
ℓ
1
,
ℓ
2
,
ℓ
3
,
ℓ
4
\ell_1, \ell_2, \ell_3, \ell_4
ℓ
1
,
ℓ
2
,
ℓ
3
,
ℓ
4
, and
ℓ
5
\ell_5
ℓ
5
. Denote by
d
i
d_i
d
i
the distance from a point
P
P
P
to
ℓ
i
\ell_i
ℓ
i
. For which point(s) in the interior of the pentagon is the product
d
1
d
2
d
3
d
4
d
5
d_1d_2d_3d_4d_5
d
1
d
2
d
3
d
4
d
5
maximal?
4
1
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3^x + 7^y a perfect squatre
a. Determine all nonnegative integers
x
x
x
and
y
y
y
so that
3
x
+
7
y
3^x + 7^y
3
x
+
7
y
is a perfect square and
y
y
y
is even. b. Determine all nonnegative integers
x
x
x
and
y
y
y
so that
3
x
+
7
y
3^x + 7^y
3
x
+
7
y
is a perfect square and
y
y
y
is odd