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Problems
Contests
International Contests
Nordic
1987 Nordic
1987 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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a/b + b/c + c/a <= a^2/b^2 + b^2/c^2 + c^2/a^2
Let
a
,
b
a, b
a
,
b
, and
c
c
c
be positive real numbers. Prove:
a
b
+
b
c
+
c
a
≤
a
2
b
2
+
b
2
c
2
+
c
2
a
2
\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}
b
a
+
c
b
+
a
c
≤
b
2
a
2
+
c
2
b
2
+
a
2
c
2
.
3
1
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f(2) = a > 2, f(mn) = f(m)f(n), increasing, minimum a
Let
f
f
f
be a strictly increasing function defined in the set of natural numbers satisfying the conditions
f
(
2
)
=
a
>
2
f(2) = a > 2
f
(
2
)
=
a
>
2
and
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn) = f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
for all natural numbers
m
m
m
and
n
n
n
. Determine the smallest possible value of
a
a
a
.
2
1
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parallelogram and 3 equal circles
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram in the plane. We draw two circles of radius
R
R
R
, one through the points
A
A
A
and
B
B
B
, the other through
B
B
B
and
C
C
C
. Let
E
E
E
be the other intersection point of the circles. We assume that
E
E
E
is not a vertex of the parallelogram. Show that the circle passing through
A
,
D
A, D
A
,
D
, and
E
E
E
also has radius
R
R
R
.
1
1
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9 journalists, different countries speak <= 3 languages
Nine journalists from different countries attend a press conference. None of these speaks more than three languages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.