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International Contests
Nordic
2005 Nordic
2005 Nordic
Part of
Nordic
Subcontests
(4)
1
1
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Find all integers, NT from Nordic Math Contest 2005
Find all positive integers
k
k
k
such that the product of the digits of
k
k
k
, in decimal notation, equals
25
8
k
−
211
\frac{25}{8}k-211
8
25
k
−
211
3
1
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Boys and Girls from Nordic Math Contest 2005
There are
2005
2005
2005
young people sitting around a large circular table. Of these, at most
668
668
668
are boys. We say that a girl
G
G
G
has a strong position, if, counting from
G
G
G
in either direction, the number of girls is always strictly larger than the number of boys (
G
G
G
is herself included in the count). Prove that there is always a girl in a strong position.
4
1
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Two Circles - Nordic Math Contest
The circle
ζ
1
\zeta_{1}
ζ
1
is inside the circle
ζ
2
\zeta_{2}
ζ
2
, and the circles touch each other at
A
A
A
. A line through
A
A
A
intersects
ζ
1
\zeta_{1}
ζ
1
also at
B
B
B
, and
ζ
2
\zeta_{2}
ζ
2
also at
C
C
C
. The tangent to
ζ
1
\zeta_{1}
ζ
1
at
B
B
B
intersects
ζ
2
\zeta_{2}
ζ
2
at
D
D
D
and
E
E
E
. The tangents of
ζ
1
\zeta_{1}
ζ
1
passing thorugh
C
C
C
touch
ζ
2
\zeta_{2}
ζ
2
at
F
F
F
and
G
G
G
. Prove that
D
D
D
,
E
E
E
,
F
F
F
and
G
G
G
are concyclic.
2
1
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Nesbitt-like Inequality from Nordic Math Contest 2005
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
2
a
2
b
+
c
+
2
b
2
c
+
a
+
2
c
2
a
+
b
≥
a
+
b
+
c
\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c
b
+
c
2
a
2
+
c
+
a
2
b
2
+
a
+
b
2
c
2
≥
a
+
b
+
c
(this is, of course, a joke!) EDITED with exponent 2 over c