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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2001 Iran MO (3rd Round)
2001 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(2)
2
1
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A real sequence
Does there exist a sequence
{
b
i
}
i
=
1
∞
\{b_{i}\}_{i=1}^\infty
{
b
i
}
i
=
1
∞
of positive real numbers such that for each natural
m
m
m
:
b
m
+
b
2
m
+
b
3
m
+
⋯
=
1
m
b_{m}+b_{2m}+b_{3m}+\dots=\frac1m
b
m
+
b
2
m
+
b
3
m
+
⋯
=
m
1
1
1
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Rational Functional Equation
Find all functions
f
:
Q
⟶
Q
f: \mathbb Q\longrightarrow\mathbb Q
f
:
Q
⟶
Q
such that:
f
(
x
)
+
f
(
1
x
)
=
1
f(x)+f(\frac1x)=1
f
(
x
)
+
f
(
x
1
)
=
1
2
f
(
f
(
x
)
)
=
f
(
2
x
)
2f(f(x))=f(2x)
2
f
(
f
(
x
))
=
f
(
2
x
)