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Contests
National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1994 Abels Math Contest (Norwegian MO)
1994 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
3a
1
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inequality with product of (x_i/{x_{i+1})^(x_i/{x_{i+1})
Let
x
1
,
x
2
,
.
.
.
,
x
1994
x_1,x_2,...,x_{1994}
x
1
,
x
2
,
...
,
x
1994
be positive real numbers. Prove that
(
x
1
x
2
)
x
1
x
2
(
x
2
x
3
)
x
2
x
3
.
.
.
(
x
1993
x
1994
)
x
1993
x
1994
≥
(
x
1
x
2
)
x
2
x
1
(
x
2
x
3
)
x
3
x
2
.
.
.
(
x
1993
x
1994
)
x
1994
x
1993
\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2}}\left(\frac{x_2}{x_3}\right)^{\frac{x_2}{x_3}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1993}}{x_{1994}}} \ge \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_1}}\left(\frac{x_2}{x_3}\right)^{\frac{x_3}{x_2}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1994}}{x_{1993}}}
(
x
2
x
1
)
x
2
x
1
(
x
3
x
2
)
x
3
x
2
...
(
x
1994
x
1993
)
x
1994
x
1993
≥
(
x
2
x
1
)
x
1
x
2
(
x
3
x
2
)
x
2
x
3
...
(
x
1994
x
1993
)
x
1993
x
1994
4b
1
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finitely many cities are connected by one-way roads
Finitely many cities are connected by one-way roads. For any two cities it is possible to come from one of them to the other (with possible transfers), but not necessarily both ways. Prove that there is a city which can be reached from any other city, and that there is a city from which any other city can be reached.
3b
1
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f(f(x)) = x+1 in Z
Prove that there is no function
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
such that
f
(
f
(
x
)
)
=
x
+
1
f(f(x)) = x+1
f
(
f
(
x
))
=
x
+
1
for all
x
x
x
.
2b
1
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x^3 +5y^3 = 9z^3 diophantine
Find all integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
3
+
5
y
3
=
9
z
3
x^3 +5y^3 = 9z^3
x
3
+
5
y
3
=
9
z
3
.
2a
1
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1/p + 1/q + 1/r = 1/n , primes wanted
Find all primes
p
,
q
,
r
p,q,r
p
,
q
,
r
and natural numbers
n
n
n
such that
1
p
+
1
q
+
1
r
=
1
n
\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}
p
1
+
q
1
+
r
1
=
n
1
.
1b
1
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PN = QN, tangent, angle bisector, diameter related
Let
C
C
C
be a point on the extension of the diameter
A
B
AB
A
B
of a circle. A line through
C
C
C
is tangent to the circle at point
N
N
N
. The bisector of
∠
A
C
N
\angle ACN
∠
A
CN
meets the lines
A
N
AN
A
N
and
B
N
BN
BN
at
P
P
P
and
Q
Q
Q
respectively. Prove that
P
N
=
Q
N
PN = QN
PN
=
QN
.
1a
1
Hide problems
cylinder inscribed in half-ball
In a half-ball of radius
3
3
3
is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is
3
\sqrt3
3
, what is the base-radius of the other one?