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Problems
Contests
International Contests
Nordic
2016 Nordic
2016 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Nordic 2016 P4
King George has decided to connect the
1680
1680
1680
islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built, but not two bridges from the same island. What is the minimal number of bridges the King has to build in order to make sure that it is still possible to travel by bridges between any two of the
1680
1680
1680
islands after the rebel movement has destroyed two bridges?
3
1
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Nordic 2016 P3
Find all
a
∈
R
a\in\mathbb R
a
∈
R
for which there exists a function
f
:
R
→
R
f\colon\mathbb R\rightarrow\mathbb R
f
:
R
→
R
, such that (i)
f
(
f
(
x
)
)
=
f
(
x
)
+
x
f(f(x))=f(x)+x
f
(
f
(
x
))
=
f
(
x
)
+
x
, for all
x
∈
R
x\in\mathbb R
x
∈
R
, (ii)
f
(
f
(
x
)
−
x
)
=
f
(
x
)
+
a
x
f(f(x)-x)=f(x)+ax
f
(
f
(
x
)
−
x
)
=
f
(
x
)
+
a
x
, for all
x
∈
R
x\in\mathbb R
x
∈
R
.
2
1
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Nordic 2016 P2
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral satysfing
A
B
=
A
D
AB=AD
A
B
=
A
D
and
A
B
+
B
C
=
C
D
AB+BC=CD
A
B
+
BC
=
C
D
. Determine
∡
C
D
A
\measuredangle CDA
∡
C
D
A
.
1
1
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Nordic 2016 P1
Determine all sequences of non-negative integers
a
1
,
…
,
a
2016
a_1, \ldots, a_{2016}
a
1
,
…
,
a
2016
all less than or equal to
2016
2016
2016
satisfying
i
+
j
∣
i
a
i
+
j
a
j
i+j\mid ia_i+ja_j
i
+
j
∣
i
a
i
+
j
a
j
for all
i
,
j
∈
{
1
,
2
,
…
,
2016
}
i, j\in \{ 1,2,\ldots, 2016\}
i
,
j
∈
{
1
,
2
,
…
,
2016
}
.