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Problems
Contests
International Contests
Nordic
2019 Nordic
2019 Nordic
Part of
Nordic
Subcontests
(4)
1
1
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meaningful
A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers.
a
)
a)
a
)
Does there exist a meaningful set which consists of
2019
2019
2019
numbers?
b
)
b)
b
)
Does there exist an infinite meaningful set? Note: The geometric mean of the non-negative numbers
a
1
,
a
2
,
⋯
,
a_1, a_2,\cdots,
a
1
,
a
2
,
⋯
,
a
n
a_n
a
n
is defined as
a
1
a
2
⋯
a
n
n
.
\sqrt[n]{a_1a_2\cdots a_n} .
n
a
1
a
2
⋯
a
n
.
4
1
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an isosceles triangle
Let
n
n
n
be an integer with
n
≥
3
n\geq 3
n
≥
3
and assume that
2
n
2n
2
n
vertices of a regular
(
4
n
+
1
)
−
(4n + 1)-
(
4
n
+
1
)
−
gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.
3
1
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Quadrilateral
The quadrilateral
A
B
C
D
ABCD
A
BC
D
satisfies
∠
A
C
D
=
2
∠
C
A
B
,
∠
A
C
B
=
2
∠
C
A
D
\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD
∠
A
C
D
=
2∠
C
A
B
,
∠
A
CB
=
2∠
C
A
D
and
C
B
=
C
D
.
CB = CD.
CB
=
C
D
.
Show that
∠
C
A
B
=
∠
C
A
D
.
\angle CAB=\angle CAD.
∠
C
A
B
=
∠
C
A
D
.
2
1
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Geometric inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths of a right angled triangle with c > a, b. Show that
3
<
c
3
−
a
3
−
b
3
c
(
c
−
a
)
(
c
−
b
)
≤
2
+
2.
3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.
3
<
c
(
c
−
a
)
(
c
−
b
)
c
3
−
a
3
−
b
3
≤
2
+
2.