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International Contests
Nordic
2024 Nordic
2024 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Nordic Q4
Alice and Bob are playing a game. First, Alice chooses a partition
C
\mathcal{C}
C
of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in
C
\mathcal{C}
C
. Then Bob does the following operation a finite number of times. Choose a set
S
∈
C
S \in \mathcal{C}
S
∈
C
not previously chosen, and let
D
D
D
be the set of all positive integers dividing at least one element in
S
S
S
. Then add the set
D
∖
S
D \setminus S
D
∖
S
(possibly the empty set) to
C
\mathcal{C}
C
. Bob wins if there are two equal sets in
C
\mathcal{C}
C
after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.
1
1
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Nordic Q1
Let
T
(
a
)
T(a)
T
(
a
)
be the sum of digits of
a
a
a
. For which positive integers
R
R
R
does there exist a positive integer
n
n
n
such that
T
(
n
2
)
T
(
n
)
=
R
\frac{T(n^2)}{T(n)}=R
T
(
n
)
T
(
n
2
)
=
R
?
3
1
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Nordic Q3
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
f
(
f
(
x
)
f
(
y
)
+
y
)
=
f
(
x
)
y
+
f
(
y
−
x
+
1
)
f(f(x)f(y)+y)=f(x)y+f(y-x+1)
f
(
f
(
x
)
f
(
y
)
+
y
)
=
f
(
x
)
y
+
f
(
y
−
x
+
1
)
For all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
2
1
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Nordic Q2
There exists a quadrilateral
Q
1
\mathcal{Q} _{1}
Q
1
such that the midpoints of its sides lie on a circle. Prove that there exists a cyclic quadrilateral
Q
2
\mathcal{Q} _{2}
Q
2
with the same sides as
Q
1
\mathcal{Q} _{1}
Q
1
with two of the same angles.